Parametric Integer Optimization [Reprint 2021 ed.] 9783112529027, 9783112529010

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MATHEMATICAL

Parametric Integer Optimization

B.Bank/R. Mandel

Band 59 Akademie-Verlag Berlin

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B. Bank / R. Mandel

Parametric Integer Optimization

Mathematical Research

• Mathematische Forschung

Wissenschaftliche Beiträge herausgegeben von der Akademie der Wissenschaften der DDR Karl-Weierstraß-Institut für Mathematik

Band 39 Parametric Integer Optimization by B. Bank and R. Mandel

Parametric Integer Optimization

by Bernd Bank and Reinhard Mandel

Akademie-Verlag Berlin 1988

Autoren: Doz. Dr. sc. nat. Bernd Bank Dr. sc. nat. Reinhard Mandel Humboldt-Universität zu Berlin Sektion Mathematik

Die Titel dieser Schriftenreihe werden vom Originalmanuskript Autoren

der

reproduziert.

ISBN 3-05-500398-5 ISSN 0138-3019 Erschienen im Akademie-Verlag Berlin, DDR-1086 Berlin, Leipziger Str. 3-4 (c)

Akademie-Verlag Berlin 1988

Lizenznummer: 202*100/425/87 Printed in the German Democratic Republic Gesamtherstellung: VEB Kongreß- und Werbedruck, 9273 Oberlungwitz Lektor: Dr. Reinhard Höppner LSV 1085 Bestellnummeri 763 763 1 (2182/39) 02000

Preface. The subject matter of this volume Is concerned with optimization problems in which some of the data involved are seen as parametric quantities and some or all of the variables are required to be integer. The characterization of the parameter-dependent behaviour of the feasible region, the extreme value, and the set of optimal Doints elaborated here is of great importance both *n theory and practice. This monograph is Intended to be a first comprehensive contribution to the theory of parametric nonlinear and linear integer programming and, it is directed to mathematicians as well as to researchers, practical workers and students who are familiar with the basic knowledge of mathematical programming. The book mainly contains research results obtained by the authors during the last five years and reflects the Important contributions to the theory of parametric integer programming due to C.E. Blair and R.G. Jeroslow and that of R. Hansel to the quadratic Integer case. Our work was essentially influenced by fruitful discussions with several colleagues, including, above all, Professor F. NoSiCka, of the division Mathematische Optimierung at the Sektion Mathematik of the HumboldtUniversitSt zu Berlin, and stimulated by the collaboration with a research group at the Lomonossow University Moscow headed by E.G. Belousov. In particular, the authors would like to express their deep gratitude to D. Klatte and B. Kummer for their Insightful discussions and helpful comments, and to 0. Guddat, who steadily encouraged our interest in this subject and writing this monograph. We have become greatly Indebted to 0. Kerger, Ch. Reimann and M. Willenberg for the assistance in preparing the final version of the manuscript, to S. Schmidt for her careful typing, and we would like to thank R. Hoppner and G. Relher of the Akademie-Verlag for their patience and support. Finally, we welcome all comments, criticisms and suggestions, which should be dlrecteu to the authors: Humboldt-Universitat Berlin Sektion Mathematik PSF 1297 Berlin, DOR 1086 Berlin, August 1986

Bernd Bank Reinhard Mandel

5

Contents Preface 1. Introduction 2. Multifunctions and Constraint Sets 2.1. Definitions 2.2. Parameter Depending Constraint Sets

Page 9 19 19 20

3. Stability of Some Continuous Optimization Problems 3.1. Definitions 3.2. Stability Properties

28 28 29

4. Quasiconvex Polynomial Optimization Problems 4.1. Properties of Quasiconvex Polynomial Functions 4.2. Multifunctions and Constraint Sets Defined by Quasiconvex Polynomial Functions

40 40 46

4.3. Stability Properties for a Fixed Objective Function

61

n

5. Integer Points in Certain Subsets of the Space R 5.1. Subsets Described as the Sum of a Compact Set and a Convex Cone 5.2. Subsets Described by Quasiconvex Polynomial Functions 5.3. Polyhedral Subsets

64 64

6.

83

Stability Properties of Nonlinear Integer Optimization Problems 6.1. Introduction 6.2. Integer Problems with Quadratic Objective Function 6.3. Integer Problems with a Convex Polynomial Objective Function 7.

72 76

83 84 97

The Existence of Optimal Points for Integer Optimization Problems 7.1. Concave Objective Functions 7.2. Polynomial Objective Functions 7.3. Some Special Mixed-Integer Optimization Problems

104

8.

Quantitative Stability of (Mixed-) Integer Linear Optimization ¡Problems

115

9.

On Relations between Parametric Optimization, Solution Concepts and Subadditive Duality for Integer Optimization

123

104 106 ill

9.1. Penalty Functions for (Mixed-) Integer Linear Optimization 9.2. Partitioning Procedures

123

9.3. Subadditive Duality

128

Bibliography

133

124

7

1. Introduction -Stability theory and the parametrization of optimization problems are of great importance both in theory and applications. Optimization models of real-life problems, like any mathematical model of such problems, only touch, in a certain approximative sense, the surface of the relations in the reality. For instance, if the functional dependences in the optimization problem are sufficiently well seized, then the applicability of the model essentially depends on the accuracy of the Initial data and, a possible methodology to treat this problem is obtained by parametrization of the data in the optimization models. On the other hand, a parametrization of the date in an optimization problem reflects a typically mathematical frameworkj successfully applied in almost all branches of mathematics and their applications. During the last decade a remarkable success has been made on stability analysis and parametrization of the models in nonlinear programming. For a comprehensive survey we refer the reader to the monographs: BANK/GUDDAT/KLATTE/KUMMER/ TAMMER (1982) (for shortness in the following: BANK et al. (1982)), BROSOWSKI (1982), DINKELBACH (1969), DONTCHEV (1983), FIACCO (1983), GAL (1973,1979), N0ZI R is the objective function and the constraint set M(A), for every ^fcA/ is a subset of R n . (As usually P(A) is called a mixed-integer, a pure integer or "continuous" optimization problem if 0 < s < n , 8 = n or s = 0 , respectively.) The behaviour of the problem P(i\) is essentially characterized by the properties of the constraint set-mapping n M : A — s » 2 R given by M ( A ) C R n , the value function

and the optimal set-function p" defined by ^ ( A ) = { x £ M ( A )

9

For the problem P(^) it is desirable that these mappings possess certain continuity or Lipschitzian properties (of global or local kind). The semi-continuity analysis of the above mappings, which will play a central role in our considerations, permits answers to the questions: 1. Does the accuracy of the solutions and extremal values increase with the degree of approximation of the problem data X ? ,2. Can a solution of P(X°) obtained for a fixed 'Xo be considered an approximative solution for the problems PCX) occurring under small perturbation of the data^.? We shall use the notation "stability" of P(A), which is not of a uniform meaning, in the literature to optimization in order to announce that certain semicontinuity properties of the multifunctions above are ensured. First papers related to parametric (mixed-) integer linear optimization problems were published in the sixties by 60M0RY (1965), FRANK (1967) and NOLTEMEIER (1970). The further development of this direction was mainly influenced by the contributions of MEYER (1974,1975,1976), BANK (1977), GEOFFRION/NAUSS (1977) and BLAIR/OEROSLOW (1977,1979,1982). While the first three papers were essentially oriented to solution methods for one- and more-parametric linear problems with parametrized linear objective functions and right-hand sides of the linear constraints, MEYER was the first who introduced the multifunctional considerations into the investigation of parametric integer problems in order to analyse the existence of solutions and the stability behaviour of such models. A comprehensive global and local analysis of the constraint setmapping, the value function and the optimal set-function for (mixed-) integer linear problems with variable objective functions and right-hand sides of the constraints is due to BANK. BLAIR/OEROSLOW obtained an important quantitative characterization of (mixed-) integer linear problems with variable right-hand sides in terms of a kind of Lipschitzian properties for the value function and the optimal set-function. In 1982 BLAIR/OEROSLOW could extend their results and show that the value function of pure integer linear problems with variable right-hand sides can be identified with thr so-called GOMORY-functions and, thus they obtained an algorithmic insight into the character of these value functions. All these results mentioned before can be considered to be a comprehensive theoretical background for (mixed-) integer linear problems (with a fixed constraint matrix; in the case that the constraint matrix is parametrized, no deep results are known up to now). Numerical procedures for the analysis of parameter-dependent (mixed-) integer linear optimization problems are considered from different points of view in several papers, e.g. FRANK (1967), NOLTEMEIER (1970), MARSTEN/MORIN (1977), SHAPIRO (1977), HOLM/KLEIN (1978), SEELANDER (1980, 1980a). Similar as in the case of"continuous" optimization problems the para-

lo

metric framework related with (mixed-) integer, problems shows a great importance for the development! of a duality theory and solution methods. Remarkable progress in the (mixed-) integer linear case can be realized in the subadditive duality approach, cutting-plane theory, for penalty functions and partitioning procedures, see e.g. GOMORY (1965,1969), CHVATAL (1973), BURDET/OOHNSON (1974), JEROSLOW ($.978,1979), SCHRIOVER (1979), BACHEM/SCHRADER (1978), BLAIR/3ER0SL0W (1981,1982), BENDERS (1962), BANK/MANDEL/TAMMER (1979), TIND/WOLSEY (1981). For postop^imality and sensitivity analyses in (mixed-) integer linear optimization we refer the reader to the articles due to GEOFFRION/NAUSS (1977) and WOLSEY (1981). Various papers deal with parametric combinatorial problems (0-1 problems), which will not be treated in this book; for references see the BIBLIOGRAPHIES Ii , II., III. In contrast to the situation just analysed rather little is known about nonlinear problems in parametric (mixed-) integer programming. The stability analyses explored in the articles by RADKE (1975) and ALLENOE (1980) are obtained under compactness requirements imposed on the constraint sets of the nonlinear problems. The basis of our considerations of parametric (mixed-) integer nonlinear optimization problems is given by the contributions of HANSEL (1980), BANK/HANSEL (1984), MANOEL (1985) and BANK/BELOUSOV/MANDEL/CEREMNYCH/SHIRONIN (1986) (cited in the following as BANK/BELOUSOV et al. (1986)), where no compactness for the constraint sets is assumed. The main part of our considerations of parametric (mixed-) integer nonlinear programming is related with the stability analysis of the general class of problems P(p,b) inf {f (x) + p T x / x tM(b),x 1 ,...,x

i n t e g e r p £.R , b £ R ,

n

where the fixed function f : R — » R is quadratic or convex polynomial (without a linear part in both cases) and the constraint sets M(b) are of the form M(b) = C(b) + V C R n , b c R m ,

(1)

and given by a compact-valued upper semicontinuous multifunction C and a fixed convex polyhedral cone V. Further explorations are carried out for problems P(p,b) with constraint sets of the implicit form M(b) = j x t R n / pj(x) ^ b j ( j=l,...,m], b e R m ,

(2)

where Pj(x), j°l,...,m, are quasiconvex polynomial functions on R n . These latter constraint sets belong, under a certain assumption ((CPC), see below), to the class defined by (1). The following three aspects of a stability theory for P(p,b) are treated in the book: - Existence and stability of feasible points, - Semicontinuity properties of the value functions of P(p,b), their 11

level-sets and optimal sets, - Existence of optimal points. Parametric (mixed-) integer linear programming is a special case of the general problems just explained, for which additional, more sbphisticated results (quasi-Lipschitzian properties) can be derived. Further, for this class of problems several applications of stability results to duality and solution concepts are demonstrated. Now we elucidate the.arrangement and the contents of the single chapters. Chapter 2. After defining the different semicontinuities used for the multifunctions in our context we present some known basic results for parameter-dependent constraint sets in R n without integer requirements which are fairly general and, in particular, closely related to (mixed-) integer nonlinear programming cases. Some of the theorems are valid in more general spaces (see BANK et al. (1982)), our limitation to R n is natural and allows certain simplifications. The considerations of this chapter will be completed by the investigation of constraint sets M(b) described by quasiconvex polynomial inequalities (Chapter 4.) and linear constraints (Chapter 4. and 8.) where also Lipschitzian properties of the corresponding multifunctions are treated. Chapter 3. Here, the aim is to give a survey on the known qualitative stability results for quite general nonlinear optimization problems in R n in a form associated with (mixed-) integer problems (see BANK et al. (1982)). This presentation is carried on by the results in Chapter 4. for (quasi) convex polynomial optimization problems as well as for the case 8 = 0 (i.e. no integer variables) in the (mixed-) integer oroblems P(p,b) fron Chapter 6. and P(b) from Chapter 8. Chapter 4. Firstly, we derive basic properties of quasiconvex polynomials and constraint sets M(b) of the form (2) (where we partially a similar way proposed by BELOUSOV (1977) for convex polynomial

follow

func-

tions). Taking into account results and techniques from Chapter 2. these properties enable us to examine the constraint sets M(b) in view of the continuity of the corresponding multifunction M, the representability of M(b) as a sum of a compact set C(b) and the recession cone V and the identification of such inequalities Pj(x) i bj which only are essential for the existence of (mixed-) integer points in M(b). An important result with respect to the stability study for (mixed-) integer problems P(p,b) in Chapter 6. consists in the fact that the constraint sets M(b) from (2) belong to the class defined by (1) if at least one set M(b°) is representable in the form (1) (condition

(CPC)).

Moreover, in (1) a compact-valued multifunction C which is continuous "On the effective domain of M if the condition (CPC) is satisfied can be chosen. The proof of this result, which makes use of an idea of KUMMER

12

(1977) for the included special case' of linear constraints P j ( x ) ^ bj, permits a linear programming proof of a version of HOFFMAN'S (1952) well-i known Lipschitzian result for the mapping M. Further, in this linear case we obtain that there is even a Lipschitzian multifunction C allowing the representation

(1), which is a result of importance for o.ir

considerations in Chapter 8. In order to be able to characterize (mixed-) integer points (Chapter 5 . ) in sets M(b) of the form (2) we investigate the so-called stable mapping M

st

aS30c

^ated

to M which respects only the inequalities P j ( x ) - bj

essential for the existence of such points. Summarizing stability results for quasi-convex polynomial

optimization

problems we complete this chapter. Chapter 5. is concerned with (mixed-) integer constraint sets G(b) ={x€.R n / x C M ( b ) , x 1 ( . . . , x 8 integer], b t R®,

(3)

First we discuss the "uniform distribution" of (mixed-) integer points in subsets of the space R n which have a representation as the sum of a bounded set and a convex cone. We extend some results of BELOUSOV

(1977)

from the pure integer case to the mixed-integer one, where we restrict ourselves to such questions necessary for our further

considerations;

a comprehensive study of the distribution of integer points in convex sets in R n can be found in SHIRONIN

(1980). For the case that M ( b ) in

(3) has the form (1) we can characterize the structure of the sets G(b) and their convex hulls and show that the corresponding

multifunctions

are upper semicontinuous on their common domain if the included polyhedral cone V from (1) has a system of mixed-integer generators oxtion

(con-

(MIG)).

The second part of this chapter deals with sets G(b) where M ( b ) is given by quasi-convex polynomial inequalities (i.e. M ( b ) from (2)). In particular, we present conditions for the existence of (mixed-) integer points in sets M(b). Under the assumptions (CPC) and (MIG) we can expose subsets of the effective domain of the multifunction G on which this mapping shows continuity. A result due to TARASOV/KHACHIYAN

(1980) giving

an estimate of the norm of at least one point belonging to G(0) (where M(b) from (2) is described by convex polynomial functions with integer coefficients) completes this part of Chapter 5., which can be considered a first more comprehensive study of (mixed-) integer quasiconvex polynomial inequality

systems.

The third section is related with the particular linear dlophantine case intensively studied during the seventies, where we give a quasi-Lipschitzian analysis of the multifunction G and related

compact-valued

mappings. The chapter terminates with a characterization of the vertices of the convex hull of G(b) (where M ( b ) has the standard form of linear programming with integer data), which is similar to that for a feasible

13

basic point in linear programming. .Chapter 6. contains the stability analysis of the parametric

(mixed-)

integer problems P(p,b) (as introduced above). We characterize the conr tinulty behaviour of the value functions cP, the optimal Tp and the

set-functions

£ - o p t i m a l set-functions'ty . Most continuity results for

both kinds of the fixed part f(x) in the objective function are derived for the general class of constraint sets given by (1). The more special case pf constraint sets M(b) defined in (2) where the condition (CPC) is fulfilled has only to be considered in order to guarantee the upper 8emicontinuity of the

¿ - o p t i m a l set-function

on certain subéets of

its domain. Already the following simple example of a parametric linear integer problem shows that neither the value function is upper semicontinuous nor the optimal set-function is upper semi-continuous if the constraint set-mapping is not continuous. P(b) lnfj-x / 0 ¿ x ' b, x integer], For | b t J w » t h bfc -

beR.

1 - ^ — * b Q = 1 one has

and, further, tf(b t ) = {o} and

tf(bfc) = 0 and 0:

rt A°)r>o i a J- I v ^ i ^ V

(iv) H^ljgwer^senücor^^ V £ >0

X

3 6 = ¿U

A

if

) > 0:

r ( a ° ) c u £ r ( A ) v a c Uj (v)

continuous at

X

(vi) H-continuous at

o

if it is both upper and lower semicontinuous ata*] 3.° if it is upper and H-lower semicontinuous at A 0 .

For convenience we will use the following abbreviations throughout the text: u.s.c.

and

JjJtcEi

for upper semicontinuous and lower semicontinuous, respectively. Furthermore, we say that a multifunction is continuous (or u.s.c., or l.s.c., or H-l.s.c., or H-continuous) if it shows that respective property at every point A . The properties of a multifunction to be J2££S¿Í5Í2¡¡. w i l 1 b e introduced later. For further (semi-) continuity-terminologies, connections and elementary properties we refer to BANK et al. (1982).

19

If two multifunction»

=> 2

Pjs A j

Riv n n < 1

—

b«

2

are given,

then we define two composed multifunctions as follows: )n < r i n r 2 > ( X ) Df r i ^(.K), (r x + r 2 ) ( a ) g f r y x . ) + r 2 ( \ ) ;

their effective domains are given by A

Df {'•CA1nA2/ri(a)nr2(A)

A

Bf

A

2

/q(A>

t 0}

• r2ca) /

m.

0],

respectively. 2.2. Parameter Depending Constraint Sets We shall start by stating some basic results on constraint sets of the following form: M( A )

Bf

[x £ R V f j i x , X ) 6 a, i - 1

m} ,X £ A ,

(1)

where we impose convexity or quasiconvexity requirements on the realvalued functions f. : R n x A

?*R, i = 1, ... ,m,for every fixed A. . By M

we denote the corresponding multifunction, which is called the jconstraint, set-mappinq and defined on A = dom M

[XC

Bf

rVm( * ) /

0} .

(2)

To show the u.s.c.-property of constraints sets to have the form (1) we apply the following lemma: Lemma 1. Let

X°eA

, S c R n be a compact convex set and x ° e M ( A ° ) n S .

Further,

let the functions f^, i = l,...,m, be quasiconvex on R n for each 7\. € A and lower semicontinuous on R n x [2.°} . Moreover, the functions fj(x°,.), i = l,...,m, are supposed to be upper semicontinuous at Finally, for an — x

t

£ > 0 and two sequences

e M ( a

t

) ,

x'^cl U£ S

and {x'} c R

n

a,0. let

V t . Then the intersection of the

line segment slCx 0 ^') for an arbitrary t and the boundary bd U £ S is a t r t) * single point z , and the sequence ¿z J has an accumulation point z which belongs to the intersection M ( b d Proof. i

U^S.

Obviously, by the assumptions of this lemma the intersection t

s (x°,x )n bd U^S contains, for every fixed t, a single point only and there exist numbers z

t

t

= «t x

allowing the representation

+ ( l - « t ) x ° e bd U £ S

Vt.

The compactness of bd U^S without loss of generality 3 z * C bd UjS : zfc

s»z*.

By the quasiconvexity of ^ and x fc e

20

M ( t h a t

(3) *

implies that (4)

n

on R , for every X c A , it follows from (3)

* max { 0 , f.(x°

x1')} .

The continuity conditions of f. and the assumptions on the sequences and (x t i then according to (4) give f ^ z * , X ° ) ^ linv f ^ z ' ,

*

i max {o, lim f i (x°, C\ t )| < i max [0, f ^ x 0 ,

°)J

=0

from which the proposition follows.

#

Theorem 1. Let

A°e A

(i)

the functions f. are quasiconvex for all A.€./\ and all i = 1 , . . . , I R ,

be and nold that

(ii) the functions f^ are lower semicontinuous on R n x

for all

i = I,...,m, (iii) M ( i s

a bounded set.

Then tne multifunction M defined by (1) is u.s.c. at Proof.

'X0.

If we can show that

30: M ( X ) c C i

U1M(X°)

V ^ t U ^

0

!

(5)

holds, then one may verify the validity of Theorem 1. in the following way. Assume that M is not u.s.c. at a sequence

A

, "X ^

M(xfc)\ UfcM(A°) / 0

then there exist an

£ > 0 and

with Vt.

(6)

Let x*", for all t, be elements of the difference sets f6). By (iii) and (5) the sequence {x*"} has an accumulation point x* belonging to M( 3.0), because of the lovyer semicontinuity property of the functions fj the A0

closedness of M at

is guaranteed.

It follows that U j M ( X ° ) contains an infinite number of the points x^ which contradicts (6). Now, assume (5) is not valid. Then there exist e

1^-°} and

a

for each t, a point

point x £. R n satisfying

x t e M( 7lfc) and x l 4 W ¡jf cl UjMC

Vt.

If we now set S = M( A 0 ) , then the hypotheses of Lemma 1. are fulfilled Thus, there exist a point z e R n and numbers 0, U£(X)OM Hence,

( X°) •

it f o l l o w s

fc

2

x C M

that

0

( -X ) for a l m o s t

Therefore,

all

the m u l t i f u n c t i o n

( 1 3 ) aijd (14) into account,

t. Ij

(M

I2 M

0

) is l.s.c. at

the same h o l d s

Now w e i n t r o d u c e m u l t i f u n c t i o n s M

:A

for M .

> 2

defined

M ( A ) R f C ( \ ) + V, A C A C R ™ , n where C : A ^2R is an u . s . c . c o m p a c t v a l u e d having

non-empty

The f o l l o w i n g by

images

theorem

and

shows

V is a p o l y h e d r a l

that the c l a s s

(15) is c l o s e d with r e s p e c t

multifunctions the form

Theorem

by

(M1+M2),

The same

w h i c h one easily

5. : A

1

multifunctions Further,

2

by (15)

multifunction

of m u l t i f u n c t i o n s

defined

of finitely

is true for

many

such

the c o m p o s i t i o n

of

confirms.

Rn

r Let C ^

taking

cone.

to the c o m p o s i t i o n

intersection.

and,

#

and C,,: A

defined

on

2

A^AgCR

¡>»2 1

"

be two c o m p a c t - v a l u e d and V ^ V g C R

0

polyhedral

u.s.c. cones.

let

25

V

Df

V

inV2' A =

)

Df

2 A(Ci(A)+Vi;,

[ur" / M U )

Then for the multifunction M :A

/ 0]. *2

R

define^] by M("X) there exists a

compact-valued u.s.c. multifunction M(X) = C(X) + V

» 2"

such that

Vit A

(16)

holds. Further, M itself is anu.s.c. multifunction. Proof. Let H be the subspace of maximal dimension contained in the polyhedral cone V. By H we denote the orthogonal complement of H. Further, let S 1 ,...,S q and T 1 ,...,T P be the closed faces of VjA h"1 and V ^ H 1 , respectively. We show that the multifunction C defined by C(\) = Df

{ J . . { ( C - C A J + S ^ n (c„(a) + Ti)} 1 2 J •(i ( j):S 1 nT 1 «lo[ 10}J

has the required properties. First, we show (16). Obviously, M(/l C(a. )+V. For an arbitrary X & A let x € M("X) be an arbitrary point. We select now closed faces S 1 and i J. J. T J of VjA H and V 2 n H , respectively, which are of smallest dimension fulfilling x e [(c1(A)+si)+v]n[(c2(x)+Tj)+v], Then there are points c X e C j O . ) , v*e V, 1 = 1,2, u 1 e S 1 and u 2 e T ^ such that x = c 1 + u 1 + v 1 for 1 = 1,2 hold. Because of the minimality of the dimensions of S 1 and T^ we have 1 i 2 i u £ ri S and u £ r i T J . In order to prove that x c . C ( A ) + V, it suffices to show that S1r\ t j = {oj. Assuming 3 u / 0 : u e S x O T^ for £ > 0 and £ sufficiently small we have u 1 - d u t S 1 and u 2 - £ u e T i . Since is the only linear subspace contained in the intersection , 1. . i. * (Vj^o H )r\ ( V 2 n H ), there exists an £ 2 as considered in Chapter 2. Obviously, for a general study of P(x)

it is not necessary

to distinguish between components of \ appearing only in the objective function and such appearing only in the constraint set M ( ^ ) ,

respec-

tively . Since in the following we mainly consider the behaviour of P(^.) in terms of the extreme value and the optimal set as functions on A plicity, we restrict ourselves to the case that A c dom M.

28

; for sim-

The value function

Cf : A

is defined by

c f ( A ) = inf {f (x, "X ) / x £ M ( ^ ) l . J Df n Further, the multifunction Y : A >2 given by the optimal sets Y U )

= { x £ M U ) / f ( x , A ) = Cp(*)l, J Df is called the optimal set-function. Its domain d o m V " i s the solvability set of the parametric optimization problem P ( A ) . As usual in parametric optimization, stability properties of PÍA.) are characterized by semicontinuity properties of the mappings

and "Y associated to P(3.).

In the literature a unified notation of stability of optimization lems does not exist, here we use the notation

prob-

"stability"as an

abbreviation for certain continuity properties of the mappings above. 3.2. Stability

Properties

Theorem 1. Let the following assumptions hold: TCRn

(i) (ii)

and

r

Q

>• 2

R

c R m are open convex sets, is a multifunction which is l.s.c. at

CI

and

fulfils the condition \

r u

1

)

+

\

r u ^ c - r ^ *

1

V A1,

*

a ,

(iii) f is independent of A and convex on the set T, (iv)

M is defined by M(X) =

Df

(v)

rcA ) O

T

V A C

Q ,

M(A°)/0,

Then the value function cp is continuous at

A°-

Proof. First, we show that the multifunction M is l.s.c. at open set with

X o . Let W be an

i 0 and x a point belonging to this

intersection.

Then by definition of M the following holds: Vé. > 0 3 y £ £ M( ? ? ) : ||x-y£|| c £ and, if £. is taken sufficiently small, the point y^ Furthermore there is a neighbourhood W'c W of y t set. The lower semicontinuity of P at

belongs to W.

because T is an open

A ° ensures the existence of a

¿ > 0 such that

r(\)n w

/ 0

Vac

U6 {A°}.

Hence, the multifunction M is l.s.c. at Let

¿ > 0 and x ° £ M ( A . ° ) such that f (x°)

the 1 .s.c.-property of M at

A0. tf ( A. °) + £ .

that for each sequence

there exists a sequence

{x^ C R

n

Then we know by ÍA^ C Q

with

such that

29

x t C M ( X t ) for almost all t and xt

—>x°.

Using the continuity of f over T one obtains íiü 2 by

is u.s.c.

N ( X ) = { x £ R n / f(x, A ) - C p ( A ) í O} . 1 Df ' Obviously, then we have ^ ( X ) = M( \ )n N ( A )

V X e A.

Furthermore, IV = el U,(M( A°)o N( X 0 ) ) . 1 Df If one now can show that 3 0

VUU^A

0

!,

(8)

then the upper semicontinuity of Tj. is obtained as follows.

31

The multifunction N is closed at

A " because for every sequence

with A*" and for every sequence { x H c R n having the f" f t * properties x € N( A ) and x — = > x one obtains (A^^A

f(x*, A ° ) -x .

implies that

^

^ f ( x \ x ° ) ^ h( A 0 ) holds, contradicting the above conclusion of (12) that h cannot be lower semicontinuous at This completes the proof of Theorem 3.

#

Remark 1. The preceding Theorem 3. allows the following two consequences for the so-called

¿ - o p t i m a l sets. If the hypotheses of Theorem 3. are ful-

filled, then (i)

the 6-optimal set mapping Y,( A ) =

H

Df

£ x £ M ( X ) / f(x,

; — R (ii) the mapping ^ : A x R + — > 2

: A

>2

< ( A )

given by + df , £ >0,

is l.s.c. at

defined by

33

= t x e M ( x ) / f (x, x ) i f U ) Df is u.s.c. at the point ( A

+

ef

,0).

The proofs are carried out in BANK et al. (1982). Theorem 4. Let the constraint set M ( o f

P ( X ) be given in the form (2.1). Fur-

ther, let the assumption (3) hold and let (4), (5) and (6) be required for both the objective function f and the functions f., i=l,...,m,which appear in the constraints. Then the value function cp is lower semicontinuous at Proof. As in the proof of Theorem 3. we define G ( X ) = M( \ ) A {x e R n / f(x, A ) ¿ ' f C V 0 ) } = Df = { x £ R n / g.(x, X ) i 0, i = l

f(x, X ) - R

E = {Xe R Df

is non-empty and fixed and

is again denoted by cp and the optimal

. Further we use the m

(14)

is an affine linear function for each fixed x e M .

value function of P ( A ) function by y

XCRm,

,

The

set-

definition

/ cp( \ ) > - ool . J

The following results hold under fairly weak assumptions on the constraint set M (i.e. at most closedness). Thus, they may directly be applied to mixed-integer problems with fixed constraint

sets.

Theorem 6. (i)

E is a convex set and the value function cp is concave on E and, consequently, continuous on ri E t

(ii)

If f is upper semicontinuous on M x E, then cp is upper

semi-

continuous on E. (iii) If M is a closed set and f is continuous on M x E, then

is

closed on E. Proof. (i) is clear by the assumption that f(x,.) is an affine linear

function

for every x€.M and by the well-known continuity property of a concave function. (ii) Let with

\ ° € . E be arbitrarily chosen and let —

F

o

r

an arbitrary

be a sequence

£ > 0 and a point x ° € M such that

37

f(x°, x one has m s Cp( A (iii) Let

Iii f(xfc,

£

A°£ E,

\t

+ £ and an arbitrary sequence £ f(x°,

fxfcj C M with

cp(^°)

.

t

{^. }cE and { x ^ C M be arbitrarily chosen such that

> A 0 and x f c e Y (

Then one has x°£ M and f(x°,

with x 1 — = > x ° . = TTlii f(xfc, X t ) = lTiii (

from which it follows that x°e Y ( X°).

£ 0 3t(6): N ( A t , a t ) C

U £ N ( X°,a o )

Vt>t(i).

Proof. Let {x'j C M be an arbitrary sequence with x t e N(A. t ,a t ). If this sequence is convergent (without loss of generality *) to an x°e M, then one has oc 0 = l i m * t * lim (g(x k )+ A t T x k ) ^ g( x °) + ^ o T x ° , from which it follows that x ° e N( X a Q ) . Let us suppose without loss of generality * that "xtll

and — i — >x*. llxfcU We may suppose that C U ^ ( A ° ) for a U

(

> CO

int

E

3ci Now, for the sequence

38

0 such that still

=

Df

Xfc

-

II x^l

one obtains that cfiA 1 ) ^ g i x ^ + ^ V holds,

from

= g(xfc)+ A t T x t - ciilxfcU i A ° - =

This implies, in particular, that n n,a a u t L ll u l + lj u j = ' must hold. (4) represents a linear subspace U which contains L and which has an integer basis because all coefficients appearing in (4) are integer numbers. This basis can be extended to an integer basis of the space R n . Now, the original quasiconvex t'orm q m ( x i p(0). According to the definitions of the points u° and u* one has, by Lemma 2, that

44

u(oc) = «.u 1 + ( l - « ) ( t u ° ) 6 Q , Df

°

V2R

such

are compact and convex.

Proof. Let H be the subspace of maximal dimension contained in the recession cone V. Then the closed convex sets M(;0, ; i € A

, can be

written in the form MOV) = ( M ( X ) A H i ) + H. If vrv H 1 = {o|[ , then the non-empty sets M G O A H

1

are compact and con-

vex. By Remark 5., the multifunction C having the latter sets as images is the desired one. Thus, we have to investigate the case VflH struct a continuous function t : A — ^ R a

C : A

r"

— » 2

/

0ur

a

*m

to

con-

such that the multifunction

defined by

C(A) = U i L M C O Of 1

/ a T x Ì t(A)\nH a , i

(14)

a = T Z dì Df j=l satisfies (13). Then, the choice of the vector a guarantees the compact-

50

ness of C(A.) and the continuity of C follows from the continuity of t and Remark 5. In order to construct the function t we define •T f . C O = sup{d J x / x t M U K , j = 1.....V,. . °f -T \ M*(\) = (xt R n / d J x i f .(SO, j = l,...,rl, *€.A . J Df J Theorem 2. and the fact that the conditions (12) are satisfied for /V? imply that the functions }/ i(Pi) * 01/

, j = 1

I = 1 Df . t. A\) = min { for j 1 1 and i £I(pj).

'

m,

V k eC*(A.U

The index set I is non-empty because of the definition of R(Pi> and L Tt V o H i -jOj. We will use the functions tjj(A) to construct the function t(A) in question and, first, we show that each t.j(A) is continuous on A. We assume that tjj(A) is not upper semicontinuous, then we have ~i{xl\c A

3 £>o

: X1 — > A ° e A

and tjj (A 1 ) > ^

]

U°) + 6 , 1 -

1,2

1

Hsnce, there are points k £ C * ( A ° ) with P j i k ^ C t ^ a 0 ) + f ) u i ) > A j, 1 = 1,2 Underlying without loss of generality* that k 1

> k * e C * ( A ° ) it

follows that Pj(k* + ( t ^ C A 0 ) + | ) u l ) ^ A°. Since the definitions of I(Pj) and i e l ( P j ) cause that p^(x) decreases on { x e R n / x = k* + pu 1 , p ^

, we have p j (k*+t. ^ (f)i?- )>

contradicts the definition of are upper semicontinuous on

which

Therefore, the functions tjj(A)

A.

Tn order to show that t. ^ ( A) is lower semicontinuous on A we assume that 3 {A1} C A 3 s > 0

— > A°e A

Assume without loss of generality

and ti .(A 1 ) < t i .(A°)

, 1 = 1,2

*

tjjCA 1 ) — o l < tjjft 0 ) - £ . * o * Let k e C (A ) be arbitrarily fixed. By the l.s.c property of C there is a sequence ^k 1 } : k ' t C (A^) and k* p j (k J + t j j (A1 )u 1 )

3>k. From

one obtains p ^ k + tu 1 )

and, therefore, it

hoi ds PjCk + tu 1 ) ±

V k fcC*( A 0 ) .

This contradicts the definition of t ^ C A 0 ) because of t ^ tjjCA 0 ) - £ • We define the function t1 0 0

52

= max ft., J (A) / j e l , Df

i e K pJ . H

which i s continuous function t(A)

on A

and w h i c h a l l o w s

= t*(x) Df

+ t-00

t(x)

It

t o show now t h a t

remains

property

is

c Z I a^u1, i =1

Obviously,

the

desired

U A

continuous. the sets

d e f i n e d by ( 1 4 )

have

the

(13).

The i n c l u s i o n opposite fixed.

us t o d e s c r i b e

t:

C(x)

inclusion

+ VtM(l), A t A is

valid,

Then t h e p o i n t

too.

< is

obvious.

Let X e A

x can be r e p r e s e n t e d

x = k + ¿ 1 .T M(X) = < x c R n / d J x ± A.., j = 1 ,ml > .T + (obviously, it holds that 0 M(x) = V ={ue.R n / d ] u i 0, j = l,...,m},' Rn then the compact-valued multifunction C :A >2 according to Theorem 3. can be chosen such that there is a constant number j»>0 independent of X and it holds

54

Ix|

é f lAli

VxecU)

VA€ A

Proof. For the orthogonal complement H

used in the proof of Theorem 3. we may

underly the representation H 1 = { x £ R n / c*x = 0, i = 1, ... ,r - nj , where Ic 1 ! M , i

= 1

r. The multifunction C may be chosen according

to (15) and (16) and we obtain C(A) = [ x £ R n / d j x i X., j = 1,

m;

( I I d j ) T x i min ( w k - v k ) T X ; j=l lík¿q c 1 x = 0, i = 1 Now, let \ e A

(17)

r } , ~k€. A .

be arbitrarily fixed and the point x a vertex of the poly-

hedron C(X) (17). Then there are a linearly independent system of vectors

contained in the system

c

/h1

"^"h]

Df

„1

„r -)

j=l

and numbers

'

0 such that M(ß) = C(ß) + V, ß e dorn M, |l(u,v,y,z)ll - *vi II ß || V ( u , v , y , z ) e C ( ß ) . *

If xe.M(A ) is a fixed point, then the point (u,v,y,z) = (x, \-Dx,x, ;\*-Dx) belongs to M(x->-*). Hence, there are points (u, v, y, z) c. C ( X - A ) and ( u , v , y , z ) £ V such that (x, A- Dx, x, A* - Dx) = (u,v,y,z) + (u,v,y,z) and ||(u,$,9,z)ll

i^U-All.

We consider now the point (x*,v*) = (y, z) + (ü,v) Df for which we have * * A A _«* *" Dx +v = Dy + z + Du + v

56

and, by the definition of V, Dx

+ v

= D y + z + D y + z =

D(y+y) + (z+z) =

= Dx + X - Dx = X * * * and, since v ^ 0, we have x t M ( A ). Now we estimate Itx—x*|l = |(u-u) ± II Oil and by f =

+

= Uu-y || ±

II y II S Z^IIX-X*«,

the proposition follows.

Corollary 4. The multifunction C : A

>2R

#

defined by (17) is Lipschitzian on A .

Proof. -i

Let A , A

*

be

t A

fixed and let

C(X, 711 C O ) =Df CO.) , x e c C X ) , then, by Corollary 3., there are a constant number ¿ > 0 independent of ^ and a point x e c ( A ,

)) such that

" (ia.) "

Hence

•

«x-x*| = / *

> - oof.

(22)

For simplicity we consider the mapping M ^ on the set A although, in general, not all parameters

j appear in (2l).

For a fixed \ the set M g t ( A ) is called the s^able^set to

and the

corresponding multifunction M . is called the stable mappinq to M. We r St /yvw^^-vvC^«^*' note that according to Theorem 2. the index set'o dependent of

t

(22) is actually in-

\ t A .

In order to characterize the stable stets M s t (A.),i.tA

(or, more pre-

cisely, to determine the index set ^ s t ) we.apply the following elimination procedure for systems of quasiconvex polynomial constraints which describe the sets M(;\). The elimination procedure, Steg^O.

Set 0 Q

:= {l,...,m}.

Stejw^. 'If for a j p 6 - ^ ^ Uj t 0 + { x £ R n /

there is a direction

P j ( x ) ^ Xy

j^p.l}

with u. £ R(pj ), then set Jr Jr

and go to Step (r+1).; *

otherwise, set 0

:=

and stop.

Note, that the considered direction u. if the set {x t R n / Pj(x) ^ X y ^x / x = x° +ocu- ,o(i o ! J'rn unbounded from below

in Step r. exists if and only contains a half-line

on which the polynomial function p. (x) is Jr

Lemma 9. Let M(X), X £• A , be given by (5). The index set 3

generated by the

above elimination procedure coincides with the index set J

t

of the

stable sets Mgj.(X). Proof. Because of the stopping criteria the set 0

*

contains only indices j

59

such that the polynomial functions Pj(x) are constant on the recession cone 0 + M

of the sets

3

M *(X) = [x R n / p.(x) é A.., . 3 Df > ' Hence the recession cone 0 M is a linear subspace and the closed 0 convex set M may be represented as the sum of a compact set and 0 the linear subspace 0 + M Therefore, all polynomial functions p,(x) + J j c 0 are bounded from below on M ^(A) and, with M ( x ) c M it * ~i 1 follows that 0 C O st.. In order to show the opposite inclusion we assume the following situation after the execution of Step r.: °r-lcast'

but

V

i 0

s f

Then we have

and

c = inf(p i (x) / x £ M ( A ) î > - oo J J Of r inffpj (x) / Pj(x) i -X., j e O r _ 1 ] = - oo.

According to the choice of u. ,...,u. V-l obtain

and Theorem 2. we

successively

|x £ R n / p j (x) £ A., j €.0 r _ 2 , p (x) ^ c - l j i 0, J Jr [ x t R n / pj(x) i A . ,

j £Oq,

Pj

(x) i c-l} 4 0,

which contradicts the definition of c. *

Therefore, 0 ^ ^st f o U ^ s -

#

Corollary 4. (i)

The recession cone V3 . = •'st Df of the stable sets M g t ( A ) , A t A

, is a linear subspace and the

multifunction M . is continuous on A st (ii) The domain dom coincides w i t h A =

. dom M and it is a closed set.

Proof. By Lemma 9. and its proof, Remark 5. and Corollary 1. The proposition set and a stable

#

(i) of Corollary 4. justifies the notations of a stable mapping.

The following lemma is strongly related to the consideration of integer points in constraint sets of the form

(5).

Lemma 10. Let

Ae.A

and M t(a.) be the stable set to M(X) (5). Then, for every

there exists a point x £ M & t ( A ) such that 60

£ >0,

p (x)

y/xe.u t {**}

V j ¿Ost

(23)

holds. Proof. We apply the above elimination procedure to the system describing M(A). h

V

By u

,...,u

we denote the direction considered in the single steps.

Let x° be an arbitrarily chosen point of M g j.(A).Since the quasiconvex polynomial function Pj (x) .decreases over each half-line of the form S = { x C R n / x = y + r t u 3 r , t ^ 0} , y C R n , we can choose a number t r > 0 , sufficiently large, such that p, (x) 'r

Jr

V x c c l U, (x°+t u'r I j

Furthermore, the point x° + t r u Pj(x°+truS

=

P j (x°)

r

belongs to

since

V j cast.

Now we choose a number t f , _ 1 > 0 sufficiently large such that V . ' Jr-1; Pi (x)U, V x e c l U£{x°+tpu r+t u r"M J J r-1 r-1 :>lds. From Pj(x°+tru

]r

+tr_1u

3r

-1)

= P J ( x ° + t r u , r ) i.aj

Vj£Os

and p 5 (x) i supfpj (x) / x £cl U £ {x°tt u ) r j j i 'r

'r

V xecl

.

>r

u £ { x ° + t r V r + t r _ 1 u' l r " 1 j

one concludes x°+tru3r+tr_1u]r"1e M

s t

U ),

PiJ (x) i AJ , P i (x) i 3 Vxccl r Jr V-l J r _i

U, f x ° + t r u ] r + t r 1 u V " l i . ti r r-i j

Proceeding in this way we obtain the point * n Jr J1 , X = x°+t p u + ... + tjU cMst(A) fulfilling Pi

Jr

(x) ^

Jr

,-..,Pi (x) i X : V x e c l JX Ji

U,{x*l. t

#

4.3. Stability Properties for a Fixed Objective Function We are looking for the stability of the following parametric optimization problem: P(X)

inf{p(x) / x e M C O j

,Ae A

where M(X) is given by (5) and p: R

= dom M, n

— > R is a quasiconvex polynomial

function.

61

By our considerations above we k n o w that: 1.) M is a l.s.c. mapping on A ( T h e o r e m 1.), 2.) M is H-continuous on A if the sets M(x) satisfy (CPC) (Corollary 1.) 3.) If the condition 3*°£A

:cf(A°)>-

oo

(24)

is satisfied, then the value function on A ; otherwise,

given by (8) is continuous

tf(7\) = - oo for all TleA (Theorem 2.).

From Theorem 7.4. one easily deduces for P(a.) that: 4.) The optimal sets

2.&A

are non-empty if (24) is satisfied.

As a consequence of Corollary 1. we obtain Theorem 5. Let M(A-) be given by (5) and p be a quasiconvex polynomial

function.

Further, let (24¡) hold. If the set M = [ x t R n / p(x)ép(0), p.(x) - p. (0), i = l,...,mi 1 1 Df satisfies the condition (CPC), then there is a compact-valued mapping *

Rn i»2 which is H-continuous on A

:A

Tf C O

+ 0+M

=

V PI £ A .

Proof. One can write, for each -If U )

= M(A)n{x€R

such that

n

A. e A ;

/ p(x) - i^) I (epi f n L ¿ )

62

(Here, Mj^ | M 2 means "Mj is parallel to M 2 ",i.e., sup d ( x , M 2 ) < oo). x c M^ Due to SHIRONIN (1980) we have: (i)

The primitivity of a convex polynomial function is invariant with respect to a linear

(ii)

transformation.

The sura of finitely many primitive convex polynomial yields a primitive polynomial

functions

function.

(iii) Convex forms are primitive. (iv)

A convex polynomial function is primitive if it is non-linear in at most two variables.

(v)

A convex polynomial function f on R

o is primitive if its degree

does not exceed 14. As a consequence of his considerations on the H-continuity of M(;0

(5)

involving only primitive convex polynomial function SHIRONIN obtained the following theorem: Theorem 6. If all polynomial functions appearing in P(X) are convex and M(¡0 is defined by primitive polynomial functions/ then the optimal

set-function

is H-continuous on /[ if (24) is satisfied. This theorem remains valid if the objective function p is only a quasiconvex polynomial

function.

63

5. Integer Points in Certain Subsets of the Space R n This chapter is devoted to properties of the constraint sets G(b) of (parametric) mixed-integer optimization problems. In particular, such properties of sets of (mixed-) integer points G(b) will be considered which are of importance for the analysis of the problems studied in the later chapters. Mixed-integer points in sets in R n which are representable as the vector sum of a bounded set and a convex cone can be characterized

in view

of their "distribution"; this is carried out in Section 5.1. For sets M(b) of the special form (2.15) we can show that the mappings G and G

conv

defined by '

G(b) = { x & M ( b ) / x 1 ( . . . , x s integer J and G c o n v ( b ) = conv G(b) are u.s.c. if the included polyhedral cone V has a system of

(mixed-)

integer generators (i.e.Vsatisfies the (MIG) assumption; see below). In Section 5.2. we consider the sets G(b) of the (mixed-) integer points in sets M(b) described by quasiconvex polynomial inequalities

(4.5).

If, in particular, M(b) belongs to the class of sets defined by (this is the case in Theorem 4.3.), then, under the

(2.15)

(MIG)-assumption,

we shall expose subsets of dom G on which the mapping G is even continuous . For linearly described sets M(b) one can derive more extended

properties

of G(b) and the related mappings. This is pointed out in Section - 5.3. ; here we shall also show a "quasi-Lipschitzian"

behaviour of the con-

straint set-mapping G and characterize the vertices of the convex hull conv G(b).

5.1. Subsets Described as the Sum of a Compact Set and a Convex Cone

If M is a subset of R n and s, 0 ^ s ^ n, a natural number, then we denote the M

s

= { x £ M / x1(...,x Of

belonging to M by integeri.

We (analogously to BELOUSOV (1977)) say that the points of M formly distributed in the set M C R

are un^Lj;

if

Ms * 0 and sup{d(x,M g ) / x E M ^ < oo. The latter condition means that the ball around every point of M having

54

this supremum as radius contains at least one point of M From Diophantine Approximation we cite the following lemma, which is related to KRONECKER's famous Approximation-Theorem:

Lemma 1. If k € R n is a non-zero vector, then, for all £ > 0 , the integer points of the set S £ = { x t R n / x = tk + y, IIyll 0 , the integer points of the

set L

={xfcR n / x = z + y, z e L , llylt 0 we find an £ > 0 such that the set L* = { x e R n / x = x 1 + y 1 + ... + x q + y q , Of IIyJll * i = 1

x'es,, q}

e is a subset . It is sufficient to show that the integer points of * of L, * the set L are uniformly distributed in L . However, this is clear since

by Lemma 1. the integer points of every set S^ £

= lxfeR n / x = x 1 - ^ , x ' e s . , Ily'll^*}

are uniformly distributed.

#

Lemma 2. If V C R n

is a convex cone, then, for all £>0, the integer points in the

set

V, = [ x t R n / x = z+y, zeV,llyll 0 , then, from

Corollary 1., it follows that the integer points of the set L £ ( l ) are uniformly distributed in L^. Let 6 be such that

6Z

0 , the integer points of \ = ^ y t R d / y = z + u, z eIN»V , l|u|lc£} are uniformly distributed in V c . fc

/V,

Since C is a bounded set and O e i n t M, we may conclude that the integer points of M are also uniformly distributed in M, i.e. ¿=

sup {d(x,(M) d ) / x e.M } ^ oo.

Now let x e M

be an arbitrary point.

Then there are a y e M and an integer point y €. (M)^ such that d . * . x = > a 1 y i and lly-y II H o i * = > a y.£.M we have i=l d |lx-x*ll = 11 STa^Cy.-y*)II ¿6 JZ IIa1 II 1 1 i=1 and, therefore i

=

1

For the point x

*

sup|d(x,M g ) / x e M ^ Z

oo.

#

The main part of our considerations will be related to sets represented by the sum of a compact set- and a polyhedral cone. If the cone V in Theorem 1. is such a polyhedral cone, then the assumptions may be replaced by requirements on the generators of this polyhedral cone. We use such requirements in the following form:

68

Let V C R n

be a polyhedral cone. IVe say that V satisfies the assumption

+

* and G c o n v ( b ) = K c o n y ( b ) • V

where either W = V

70

s

= f u e V / u.,...,u Df

integer ? •>

or

r

* = { u e R n / u = 2 Z OsV1,!!. natural, 1 i=l 1 1 r If v , ...,v (ii)

i=l,...,r? '

are (mixed-) integer generators of V;

The multifunctions G and G C Q n v are u.s.c. on B;

(iii) The domain B is closed if A

is closed.

Proof. Let us define the K(b) = (x €.R Df

n

multifunctions K and K c o n v by / x = y + ¿ 1 * . v l , 0 i ot. i 1, i-l

i=l,...,r;

y e C ( b ) , x1,...,x8

7

^ '

integer}

and K C.U„I InVJ b ) =0

f

conv K(b),

(6) 1

respectively, which are given on B. The appearing vectors v , i=l,...,r, are (mixed-) integer generators of the cone V. Using now the u.s.c. property of C and the fact that Z 9 x R n ~ 9 is a closed set (here, Z denotes all integer numbers) one obtains the claimed upper semicontinuity of K, of which the same property of K

is an immediate consequence. The

desired representation G(b) = K(b) + W is in both cases obvious. For an arbitrary bff B one has conv G(b) = conv(K(b;+W) = = conv K(b) + conv W = =

K

conv *

V


b o = 1.

71

However, we shall show in the following section that for a constraint set mapping G, where the included M is given by the set of solutions of a system of quasiconvex polynomial inequalities with perturbed right-hand sides, it is possible (of course under certain assumptions) to characterize subsets of B on which the restriction of G is continuous.

5.2. Subsets Described by Quasiconvex Polynomial

Functions

This section is devoted to sets of (mixed-) integer points given by G(b) = ( x £ M ( b ) / x 1 ( . . . , x 1 s Df

integer] >

(7)

where M(b) is given by (4.5), i.e. M(b) = { x t R n / p • (x) < b i ( j=l ,m] n ^ and p. : R > R , j = 1,...,m,are quasiconvex polynomial

functions.

By A and B we denote the domains of the corresponding mappings M and G, respectively. First, we derive conditions under which a quasiconvex polynomial nf inequalities has a (mixed-) integer

system

solution.

Lemma 5. If 0 is an index set fulfilling 0 ^ C O c{l,...,i|, where J g t is given by (4.22), then the set G(b) is non-empty if the (mixed-) integer

points

in the set M,(b) = {x £ R n / p.(x) * b., j

(8)

are uniformly distributed in M^(b). Proof. Let ¿ > s u p { d ( x , ( M - ) ( b ) ) s ) / xeM-j(b)}. *

Then, by Lemma 4.10., there is a point x e M g t ( b ) such that Pj(x) ' b. V j i O s t

V x eu^(x*)

holds. Because of the choice of c$ there exists an x £ ( M j ( b ) ) g 11 x - x l K c i and we obtain x e G ( b ) .

satisfying

#

If one requires the (MIG) assumption for the recession cone V-. of the st stable sets M g t ( b ) , then the (mixed-) integer points of M g £ ( b ) are uniformly distributed in M g t ( b ) whenever the latter set contains a (mixed-) integer

point.

Theorem 3. Let the recession cone V-. of the stable set M .(b) fulfil the assumption st st 72

(MIG). Then H)

G(b) ^ 0 < j = > ( M s t ( b ) ) s i 0;

(ii)

the domain B of the multifunction G is closed.

Proof, (i) By Corollary 4.4. (i) the stable sets M g t ( b ) fulfil the condition (CPC), and now the (MIG.) assumption for V^

yields the st

implication (Mst(b))s / 0

= = »

G(b) / 0,

if one takes Lemma 4. and Lemma 5. into account. The opposite implication is obvious. (ii) By Corollary 4.4.(ii) the domain of the stable mapping M g t

defined

by (4.21) is closed. Using now Theorem 2. one obtains that the set [ b £ R r a / ( M g t ( b ) ) g / 0} is closed and, applying (i), the proposition (ii!) follows.

#

The following example shows that the (MIG; assumption for V-. is st essential for the validity of Theorem 5. Example 2. The stable sets M g t ( b ) of the sets M(b) = { x t R 2 / Y ? x 1 - x 2 £ b, - V ? x 1 + x 2 ^ 0, -Xj i -if are given by M g t ( b ) = [ x £ R 2 /{z x1-x2 ± b, - V ? x1+x2 i o|. The recession cone V n of M .(b) is the linear subspace st st V-,

= { u £ R 2 /V? u1-u2 = Oj

and, it has no integer generator. If s = 2, then G ( b Q ) = 0 for b Q = 0. But ( M s t ^ b p ^ 2 bt = i

=

l(o)]ancl'

for

the

se(

luence

> 0 = b Q one has G ( b t ) / 0.

By Remark 4.6. we know from Theorem 2. that the multifunction G (7) is u.s.c. on its domain if one imposes the assumptions (CPC) on the mapping M involved in (7) and (MIG) on the recession cone of M(b). However, in order to be able to guarantee the continuity of this mapping G one has to restrict it to certain subsets B(b°) of the domain B. Let b°6. B be a fixed point. We define B(b°) = ( b £ B / P v v G(b) = P y v G(b°)\ x x l'",'xs l'""x8 ' Df

(8)

denotes the projection into the subspace of R n correXj,...,x s sponding to the integer variables

where P

One immediately shows that the following holds for all fixed b ° C B : B(b°) f 0;

(9)

73

b1CB(b°)

=»B(b

1

)

-- B(b°);

(10)

1

b e B S B ( b ° ) » ^ B C b ^ n B(b°) = 0.

(11)

Theorem 4. Let (CPC) be satisfied for M in (7) and let (MIG) be fulfilled for the recession cone V. Then (i)

G is u.s.c. on B;

(ii) G is continuous on B(b°) for an arbitrarily fixed b ° C B. Proof, (i) is clear by the remark preceeding this theorem. (ii) Because of (i) we have only to show the l.s.c. property of G on B(b°). Let b * e B(b°), ( b ' j c B t b 0 ) Obviously,

and bfc

*b*.

it suffices to verify that for each arbitrarily

fixed

) there exists a * sequence { x H . C R* n with# x t € G ( b * ' ) / for almost +• all t, such that x ^ x . Hence, let x e G(b ) be fixed. We consider

x £G(b

the multifunction G(xj,...,x ) defined on B(b°) by G(x*

x*; b) = {x e M(b) / x. = x* J Df '

j =l

si.

(12)

We have G ( x * , . . . , x * ; b t ) i 0 V t because of the definition of B(b°) and -l £ # * * it holds that x £ G ( X j , . . . , x g ; b ). By Theorem 2.5. and Theorem 4.3. we conclude that the mapping G(xj,...,x ) is continuous on B(b°). Hence, 0

there is a sequence ( x ^ C R t

t

x eG(x*,...,x*;b ) V t t

such that

and xt t

Since G ( ) < j , . . . , x * ; b ) C G ( b ) 1

s>-x*.

the desired sequence is obtained by { x t } .

# The following example shows that the validity of the proposition (ii) of Theorem 4. essentially depends on the underlying description of the involved multifunction M.

2

For the compact-valued continuous multifunction M: [ -

— * g i v e n

by M(b) = { X 6 . R 2 / X 2 ^ 1 , 2 x 1 - x 2 i l , - 2 X 1 - X 2 i

1, ( l - b ) x 1 + b x 2 = - b }

and the constraint set mapping G by G(b) = { x t R 2 / x £ M ( b ) , x x

integer]

we have B = A

= [-

, B(0) = B,

but, obviously, G is not l.s.c. at b = O t B ( O ) .

74

Now w e cite a r e s u l t p u b l i s h e d by TAfeASOV/KHACHIYAN

(1980), w h i c h gives

an e s t i m a t e of the norm of at least one p o i n t b e l o n g i n g to G ( 0 ) / 0 in the case that all p o l y n o m i a l s P j ( x ) a p p e a r i n g in (7) are convex

with

integer c o e f f i c i e n t s . T h e r e f o r e , let us denote by dj the degree of P j ( x ) and d = max d ^ further, if d, * d. ^ ... ^ d. , then D = d. ,...,d. J 1 1 1 Df j 1 2 m Df l l *p where p = minim,n\ , and finally, let h b e the m a x i m u m of the moduli of Df the c o e f f i c i e n t s

in all p o l y n o m i a l s a p p e a r i n g in

(7).

Let all the p o l y n o m i a l s pj in (7) be convex w i t h integer c o e f f i c i e n t s , T h e o r e m 5. and let d, D.and h be d e f i n e d as above. If d > 2 and G ( 0 ) is n o n - e m p t y , then there is an x€ G ( 0 ) flxll ^ a?, w h e r e 1 4 log ta = £ D (d^n)

fulfilling

I

(13) log(hdn).

T h e o r e m 5. was u s e d by T A R A S O V / K H A C H I Y A N

in order to discuss the

r i t h m i c c o m p l e x i t y for convex polynomial

optimization

algo-

(for a comprehensivje

survey the reader is r e f e r r e d to the paper of K H A C H I Y A N

(1982)).

In the c o n t e x t of the c o n s i d e r a t i o n s p r e s e n t e d here, the

essential

d i f f i c u l t y of deriving an e s t i m a t e of the form (13) c o n s i s t s in o b t a i n i n g an e s t i m a t e r = r ( d , D , h , n ) such Mst(0)nVj

c^xeR st

n

that

/ llxll i r }

holds for the s t a b l e set M g t ( 0 ) of G ( 0 ) / 0, This b e c o m e s clearer by the facts

that

(Mst(0))s / 0 < = ^ . G ( O )

/ 0

(Theorem 3 . ( i ) ) and that the r e c e s s i o n cones 0 + { x £ R n / p j ( x ) £ 0, j e o j , J c. jl,...,mI, can be d e s c r i b e d applying the t e c h n i q u e s of C h a p t e r *

Using the r a d i u s r one can easily obtain a r a d i u s r that 3 x*C (Mst(0))s n [ x £ R n S t a r t i n g w i t h such an x

#

= r (d,D,h,n)

1

k

* x

i=l

i t V e 1

such

/ llxll 4 r*|.

and f o l l o w i n g the way d e s c r i b e d in the proof of

L e m m a 4.10. one can c o n s t r u c t natural n u m b e r s t 1 ( . . . , t . teger v e c t o r s u ,...,u

4.

K

I

(where k ^ m - | O g t l ) s u c h that

G(0).

K

and (mixed-)

in-

(14)

In a more p r e c i s e analysis one can verify that the n u m b e r s tj and v e c t o r s u1

in (14) can be c h o s e n in such a way ||x*

+

¿ 1 tVll i=l 1

^ r*

+

¿ 1 t.llu1!! ' i=l 1

holds, w h e r e p has the d e s i r e d

that f

form.

75

5.3. Pplyhedral Subsets. In this section we consider (mixed-) integer points in subsets of the form (4.5) where the polynomial functions are linear, i.e. Pj(x) = d J x , j=l,...,m. Obviously, then the recession cone V of the sets M(b) / 0 is given by V = { u e R n / d' u i 0,

j=l,...,m}.

By Corollary 4.2. the multifunction G given by G(b) = {xe. R n / d J x i b i ( Df >

j=l,...,m; x 1 ( . . . , x integert i s j

(15)

belongs to the class considered in Theorem 2. where the compact sets K(b) from (5) are specified by C(b) in the form (4.17), Theorem 6. Let G be given by (15) and, let V satisfy the (MIG)-condition. Further, let K be given by (5) and (4.17), ana let K C Q n v and G c o n v have a corresponding meaning as in Theorem 2. Then (i)

each set K(b) is a union of a finite number of compact convex polyhedra; the sets ^ c o n v ( b )

(ii)

(iii) the polyhedron

G

an(

1

G

b

tl C onv^ ^ has a

conv( )

are

convex

polyhedra;

vertex if and only if {o^ is the

only subspace contained in V; (iv)

there exist positive numbers llxll

bit + T

and T (independent of b) such that

VxtK(b).

Proof. The statements (i), (ii) and (iii) are simple consequences of Theorem 2. and the well-known representation theorems for convex polyhedra. In order to prove (iv) we use the constant number f>0

from Corollary 4.2.

1

corresponding to the multifunction C (4.17). If v , . . . , v r are (mixed-) integer generators of V and if we take t = ^T^ llv1!! , then we obtain the Df i=l desired estimate for the sets K(b) (5). # Rn We call a multifunction i :A >2 quasi-Lipschitzian on a subset ^

qC _

A

if there are constant numbers

"Xe A ) such that

d(r(3Q,r(V)) - f 11^-

'A'II

+T

p>0

and

VX.X'eA,,.

Theorem 7. Let G(b) (15) be given in the form G(b) =[x€.R n / Dx i b, x 1 ( . . . , x g integer J with a rational (m,n)-matrix D. Then (i)

G is quasi-Lipschitzian on B;

(ii)

there exists a positive number ci such that

76

T>0

(not depending on

6 * sup{d(x(G(b)) / x e M ( b ) l

VbCB; Rn

(iii) tnere is a c o m p a c t - v a l u e d m u l t i f u n c t i o n K : B

^2

having

the

properties: 1° G ( b ) = K ( b ) + V , b £ B, 2° K is q u a s i - L i p s c h i t z i a n on B. P r o o f . In order to p r o v e (i) one can follow a similar way as in the p r o o f of C o r o l l a r y 4.3. Let b,b £ B and x e G ( b ) be a r b i t r a r i l y We

chosen.

define G(B) =

{ ( u , v , y , z ) e R 2 ( n + m ) / D u + v - D y - D z = 6,

Df

v ( z ^ 0, u 1 , . . . # u 8 , y 1 ( . . . , y g

integer]

and V-

= { ( u , v , y , z ) £ R 2 ( n + m ) / D u + v - D y - z = 0, Df v,z ^ 0, u 1 ( . . . , u s , y 1 ( . . . , y s

1 integer^.

Then, the T h e o r e m s 2. and 6. imply that there e x i s t s a m u l t i f u n c t i o n K and c o n s t a n t n u m b e r s ^ > 0

and p > 0 s u c h

chat

G((3) = K ( B ) + V 2 s , l i £ d o m G , and ll(u, v,y,z)ll - l^lllit| + p

V ( u , v , y , z ) £ K(6).

Now, by the * same* a r g u m e n t s as in the proof of C o r o l l a r y 4.3., one a p o i n t x £ G(b ) such that

finds

lix-x*l ^ 2-^11 b-b*ll + 2p holds and, setting

= 20^ and f = 2p, (i) is s h o w n .

One obtains proposition

and

M = {(*) £ R Df D

n + m

(ii) by

/ Dx - b *

considering

o| '

G = { ( * ) £ R n + m / Dx-b i 0, x 1 , •..,x 1 8 Df D

integer t.

A

A

The p o i n t s of G are u n i f o r m l y d i s t r i b u t e d in M (Lemma 4 . ( i ) ) and we assume

f

.A

A -J

oi •> s u p ^ d ( x , G ) /

xeMf.

If x t M ( b ) , then ( * ) £ M and there is a p o i n t ( £ o ) £ G w i t h II (¡jo) - (£)'ll ' « . Since x ° £ G ( b ° ) ,

by (i) there is a point x

£ G ( b ) s u c h that

'

llx°-x*H ¿ p l l b ° - b II + T , where

and t are c o n s t a n t n u m b e r s i n d e p e n d e n t of b and b°. Now, we can

estimate llx—x*ll

¿llx-x°ll + II x°-x*ll

(iii) Let |v

r

+

i| b°-b II

p a

+1T= Df

ci .

, . . . , v ^ be a s y s t e m of (mixed-) integer g e n e r a t o r s

of

77

r and let 1 > max fcf, J Z l| v111?, where i is a J i=l constant number with the property (ii). Using C(b) defined by (4.17) V = { x £ R n / Dx i

we define K(b) = { x £ R n / x €G(b), x = y+v, y £ C ( b ) , llvll ¿ t j . Df Obviously, K(D) _ is compact and satisfies 1°. In order to show 2° let * b, b e B and x £ K ( b ) be arbitrarily fixed. Then there are a point y e C ( b ) and a vector v with llvll i t

such that x = y+v. Since C, by Corollary 4.4. *

+

is Lipschitzian on B, there are a point y e C(b ) and a constant number *

f > 0 (not depending on b and b ) such that lly-yll

¿fl|b-b*li.

*

*

*

By C(b ) C M(b ) and (ii) there is a point x c G(b ) with II x*-y*ll< cf . Now we ^have

m

^

||x-x II = lly+v-y +y -x

N

^ II y-y*ll + llvll + l| y*-x*ll i and

ifII b-b*ll setting =

+Z+6 the desired proposition follows.

#

For the constraint sets G(b) from (15) we can discuss the question of a parametric description. By Theorem 2. the constraint set G(b), for a fixed b e B , has the representation G(b) = K(b) + W, W = [u£Rn

r / u = ¿2. O.v 1 , B. natural, i = l,...,r|. 1 i=l 1

(16)

and, by Theorem 6., the set K(b) is a finite union of compact convex polyhedra (in the pure integer case K(b) consists of a finite number of integer points). If one now underlies that the polyhedra appearing in K(b) are given by the convex combinations of their vertex points, then (16) may be considered a parametric representation of the constraint set G(b). The validity of (16) is also guaranteed for a fixed b e B if one uses, in the construction of K(b) (due to the proof of Theorem 2.) and 1 r W, such a (mixed-) integer generator system v ,...,v , for which the greatest common divisor of the integer components is equal to one and if the convex hull of the vertex points of the polyhedron M(b)r> H

is

taken instead of C(b) defined in (4.17), where H is the subspace of maximal dimension contained in the cone V. Sometimes it is desirable to have a parametric representation of G(b) includ-ing a minimal number of unbounded parameters Bj only. This minimal number coincides with the dimension of the polyhedral recession cone V, which one sees by applying a well-known result of CARATHEODORY.

78

We consider the finite number of variations k 1 = (k*,...,k*),...,k t =

of the indices l,...,r, where

d = dim V such that the corresponding vector systems t M

,v

kï ]

are

t linearly independent and such that V = l j v ( k J ) holds, j=1

where

d ki V(k J ) = ( u t R n / u = J Z BjV 6. à o, i = 1, d}. 1 1 Of i=l ' Thereby one can choose the variations in such a way that the cones V(k^) have only common boundary points. If one denotes by K , (b) the respective compact sets related to K(b), kJ then one has t d k? G(b) = \J {K .(b) + { u t R n / u = H B.v 1 B. natural, i = l , . . . , d W . J 1 1 j=l k i=l ' Next, we will mention some properties of the domain B of G. Thereby we use the special description G(b) = (x € R n / x = (z,y), Az + Qy ^ b, z € R 8 , integer J

(17)

where D = (A,Q) is an (m,n)-matrix. In Theorem 2.(iii) it was pointed out that the domain B of the multifunction G given by (3) is closed if the domain A of the involved mapping M is closed and if the cone V satisfies (MIG). This result is also true for the multifunction G defined by (7) if the coefficients of the quasiconvex polynomial functions are rational numbers. However, if G is given by (17), then one gets more information about the structure of the domain B and, moreover, one may derive properties of the sets B(b°) defined in (8), on which G is continuous. These facts are listed in the following two remarks. The proofs can be found in BANK/BELOUSOV et al. (1986). We shall use the following (known) definitions. Let S c r " . Then S is called a monoid if O S S , and u , v e S implies that u + v e S . The set S is said to be star-shaped if there exists a point v e S such that the line segment s|(u,v) between u and v belongs to S for every u e S , u / v. Remark 1. Obviously, the domain B is a monoid. If the matrix (A,Q) is rational, then the domain B is closed and connected and may be represented as the union of a countable family of convex polyhedral cones which all are translations of the following fixed m-dimensional cone: W = {g€.R,n / g i Qy for some y e R n - s V . Of Remark 2. If the matrix

(A,CJ) in (17) is rational, then the set B(b°) (8), b ° £ B,

arbitrarily fixed, is star-shaped and neither open nor closed in general 79

There exist a finite set T of s-dimensional integer vectors z and a point d°€ R m such that B(b°) = (d°+W)\

U

(Az+W).

zeT The monoid B possesses a complete decomposition into a countable number of sets B(b*) (8)

b*e B.

We illustrate Remark 2. by the following figure and an example.

Example 3. M(b) = { x £ R 2 / 2 x 1 + x 2 i b 1 ,

x ^ x ^ b ^ ,

T

b° = (2,2) . (i)

G(b) = [x 6. M(b) / x x , x 2 integer j, B(b°) = {b€ R 2 / 2£b i < 3 , i=l,2 J.

(ii) G(b) = { x € M ( b ) / Xj integer j, B(b°) = R 2 . Remark 3. Using the parametric descriptions of the sets G(b) sketched in connection with the representation (16) one can also derive parametric representations for the domain B and the sets B(b°). Obviously, such a representation for the sets B(b°) will be of very complicated, not tractable form. The next theorem will give a characterization of the vertex points of the convex hull of constraint G(b) = { x t R

n

sets having the form

/ Ax = b ( x > 0, x 1 ( . . . , x

integer]

(18)

where A is an integer (m,n)-matrix. This latter assumption and the form of G(b) ensure that the convex hull is a convex polyhedron having a vertex point if it is non-empty. Theorem 8. Let G(b) be given by (18) and, let A have the rank m. Let A denote the maximum of the moduli of all m-order minors of A and J their greatest common divisor. If x° is a vertex point of conv G(b), then there exists an index set I = { i-, , i t C { l , . . . , n j such that the matrix i, i (A ,...,A ) is regular and A holds where A

80

Y i 4-1 ,oc=l,...,m, represents the i^ -th column of A.

(19)

Proof. First, one has that the vertex point x

belongs to G(b). Obviously, the

case m = n is trivial. Therefore, we suppose that m ^ n . We consider the system of all index sets 3 1

m

(A ,...,A

•••

xîR

w 1

,

Obviously, we have . ^ 0, x - ^ 0 m+1 m+1 id, for

je{l,...,m+lj,

Xj and x. are integers if x° is an integer number. Further, x^ ^ 0, i=l,...,m, since X °1

-

ti T

+

!i T

X n°, + 1

'

X ) = {* c G ( b ) / f(x) + p T x i f(p,b) + 6 I Df where £ £ 0. In section 6.2. we consider the fixed function f: R n —

(2)

R

in (1) to

be a quadratic form and, in section 6.3. to be a convex polynomial function. Most continuity results are derived for the general class of constraint sets which are allowed in (1). But, in order to show the upper semicontinuity of the

¿-optimal set-function ^ffon certain sub-

sets of its domain we underly the special constraint sets G(b) from (5.7) where the (CPC) condition is satisfied. In our considerations the level-set-mapping of the problem P(p,b) plays an important role. This mapping is defined by the level sets N(p, b, a) = j x t R n / x t G ( b ) , f(x)+p T x ^ a j J Df

(3)

of the problem P(p,b)^ These sets define a multifunction, mapping, N:

i>2 R

with

£ n = {(p,b,a)c. ? * R / N(p, b, a) / 0 } Df where £ = i ( p , b ) e R n x B / cp( p b) > - ooi , Df ' i.e.

is the set of all pairs (p,b) for which the constraint sets

83

G ( b ) i 0 and the o b j e c t i v e function

is b o u n d e d from b e l o w . This set is

c a l l e d the domain of f i n i t e n e s s of cp . A s w e shall see, the b e h a v i o u r of the o b j e c t i v e function on the p o l y h e d r a l cone V (appearing in the r e p r e s e n t a t i o n of the

fixed

constraint

G ( b ) ) is of great i m p o r t a n c e in our d e v e l o p m e n t s . T h e r e f o r e , we use the following

definitions:

E(b) = [p£Rn Df

/ c p ( p , b ) > - oo 1, , b £ B ,

p° and we choose an £>0. Then there exists a point x° e V such that x° T 0x° • p t T x ° ^ v ( p ° ) + £ j T for almost all t. Since fyip*) - x° Dx° + pfc x°, me have

r(k) .

Finally, if V satisfies (IG), then v can be considered to be an integer vector. Proof. We can suppose that the convex polyhedral cone V is described in the form

.T V = { v e R n / d 1 v ^ 0, i =

89

and, that the elements of x

fc

t

= z

may be represented by

1

+ ocj.u

t

(12)

1

where u £ V Ju !! = 1 ,

t

t

z tC(b ),at

>00.

Without loss of generality* we can assume that u '

> u ° where u ° c V

and l|u°ll = 1. Because of the upper semicontinuity of C and since C(b°) / 0 is a compact set, the sequence {z^j must be bounded. The inequality xt

Dxfc + pfc xt

= zt

Dzfc + pfc zt

yields u° Du° = 0

+ at(2Dzt+pt)Tut

+ < X 2 ufc Dufc i afc

if one takes into account that ufc Dufc ï 0.

Using the index-set I = (i«-ii, ... , q V

d 1 u° = 0} we obtain

Df

u°e V, |u°ll = 1, d i T u° = 0

V i ei. iT

Since V satisfies (MIG), and because of d v e V, there must be a vector v e R n integers v e V, Ilsuch v II à that 1, u° v ^ I

, d

1

v ^ 0, for all i £ I and all

whose components vj,...,v

v = 0

are

Vi tl.

If V satisfies (IG), then v cart be considered to be an all

integer

vector. Setting now y*" = x^-v one has Df

B y 1 2 = llxfc II2 + iv|2 - 2xt v Hlx1!!2 + llvll2 - 2zfc v-2ot t u t v and, taking into account t h a t « . —>cO , v T u ° ^ i and u*" — > u ° * t t we obtain without loss of generality the assertion

hold,

that ||y || 0 0 the discussion of * II* both cases above implies that without loss of generality x -v £ G ( b ) and, for every natural k, there is an index r(k) such that xfc - k v e G ( b t ) V t > r(k) . The proof of (i) and (iii) will be completed by estimating as follows: (x t -kv) T D(x t -kv) + p f c T (x t -kv) = = x t T Dx f c + A * tT

i x z Dx

t

tT

- k(2Dxt+pt)Tv ^ t

+ pc xL

£

Vk i 0,

at

where we took into account that (pt,b^)e. O , for infinitely many indices r, and, using Lemma 5. (iii), we could conclude, for an arbitrary natural number k, that (x r -kv) T D(x r -kv) + p r (x r -kv) i ± x r D x r + p r x r -k oc i a p - k « and x r - k v £ G ( b r )

if r > r ( k ) . Then it would follow, for k sufficiently that lim Cp(p r ,b r ) = - oo, which would be impossible

large and a p

and the fact that cp is lower semicontinuous on £

because of ( p ° , b ° ) t £

Therefore (18) holds, which together with p^ r

T

o

implies that

T

Ov = - p u v.

lim 2x

(19)

Now A let us define I T I G(b,6) = { x C G ( b ) / 2x Dv = BV , Df B = ^ ( b , B ) t B x R / G(b,(1) t 0 } , Df V

=

Df

(ui V

/

2

UTDV

= oi

.

By Theorem 2.5., tjjere exists a compact-valued u.s.c. multifunction C : (B x R)

* 2R

such that

G(b,B) = [ x £ R n / x €C(b,6) + V, x 1 ( . . . , x 2 A

integer} A

£

holds, for all (b,B)t B. For the multifunction G we declare the sets £ A

and

A.

and the mapping N in an analogous way as in the case of the

multifunction G. Because of the rationality of D and as v is an all A T . integer vector, the set { u £ V / u Du = OS satisfies, by Lemma 6.2., the condition (IG). If we set now 15 it follows from (19) that

= - p° v and B nf

l i m t p r , ( b r , B r ) , a r ) = (p°,(b°,B 0 ),a 0 ).

94

= 2 v T D x r , then nf

Furthermore, (pr,(br,Br),ar)e-g r

r

r

and

x e N ( p , ( b , R r ) , a r ) N U 6 N(p°,

(b°,B o ),a o ).

This says that we have constructed a further counterexample for which the statement of this theorem is false. However, by (16) it holds dim conv G(b°,6 o ) < d i m conv G(b°), which contradicts the choice of the original counterexample.

#

Corollary 1. If the assumptions of Theorem 4. are fulfilled, then N(p°,b°,a o ) / 0. Proof. We choose

such that

t

llx H = min { | x | / x e N ( p t , b t , a t ) } for t = 1,2,... . By Theorem 4., the sequence ( x ^

has an accumulation

point x° because otherwise one would obtain a contradiction to the choice of x®". Then one immediately sees that x° e N ( p ° , b ° , a Q ) . In Theorem 5. we show that the ¿-optimal set-function ^ ( 2 ) the property to be u.s.c. on a certain subset of

jg

#

possesses

x R + if besides

the hypotheses of Theorem 4. one additionally requires that the value *

function £

is continuous on the subset

jc

of its domain of finiteness

. On account of Theorem 3., it is in fact the requirement that cf is

u.s.c. on

£ .

Theorem 5. Let D be a rational matrix and let V satisfy (IG). If is a subset _ "i of t on which CP is continuous, then the £-optimal set-function is * t u.s.c. on

£

x R+.

Proof. Let (p^b®", 6- t ) £ E * x R + , t = 1,2,..., be a sequence which converges to (p°,b°, £ _ ) . Then, for the vectors (p t ,b.,a.) with t t a,. = CP(p ,b ) + £.. we have c Df fp t .b t ,a t ) — * ( p ° , b ° , a o ) and (p®", b^, a^. )£•

^, for t = 0,1,2,... .

Since V ( p t , b t , e t ) = Nip'.b'^j.), t = 0,1,2,,.., one obtains the proposition by Theorem 4.

#

95

Now we consider the quadratic parametric (mixed-) integer

optimization

problem P(p,b) inf£x T Dx + p T x / x £ G ( b ) j

, bfcB,

where the constraint sets G(b) ar.e given in the form G(b) = { x t R n / Pj(x) ^ b., 1=1, ...,m; x 1 ( . . . , x s integer]

(20)

with quasiconvex polynomial functions p^(x), i = l,...,m. This k i n d of constraint sets we have investigated in section 5.2.

For

the case that the involved sets M(b) ={xe.R n / p.(x) i bj, i = 1

m}

satisfy the condition (CPC) and their common recession cone V = 0 + M ( b ) possesses the (MIG) property we could identify subsets B(b°) (5.8) of the domain of the multifunction G on which G is continuous. Employing this sets B(b°)

we can expose subsets

c

£

on which the continuity

of the value function cj> , assumed in Theorem 5., is ensured. Theorem 6. Let the constraint sets G(b) be given by (20) and let (CPC) be satisfied for M. Further, let b ° c B be a fixed parameter point and

£

given

by t*

= {(p,b)t£ /

btB(b°)l J

Df

where B(b°) is defined in (5.8). Then (i)

The value function

is continuous on

V = 0 + M ( b ) satisfies

(MIG).

(ii) The £ -optimal set-function V

£

if the recession cone

is u.s.c. on

E

x R + if the matrix

D is rational and the recession cone V satisfies

(IG).

Proof. We have only to show (i) and here, because of Theorem 3., it suffices to verify that tf> is upper semicontinuous on 0

. Let (p°,b°) £ €

be

and x ° c G(b°) such that

x ° T D x ° • p ° T x ° * Cf(p°,b°) • £ . W e assume that ( p t , b t ) — > ( p ° , b ° ) ,

By Theorem 5.4. (ii), the constraint-

set mapping G is l.s.c. at b° and therefore, there are points xtCG(bt),

for t sufficiently large, such that

ITm cpip^b*) i l T m ( x t T D x t + pfc xfc) = = X ° T D X ° • p ° T x 0 < cp(p 0 ,b°)

96

+

£ ,

— H e n c e ,

one has

from which it follows that cp is upper semicontinuous at (p°,b°).

#

6.3. Integer Problems with a Convex Polynomial Objective Function In this section we treat parametric (mixed-) integer problems P(p,b) (1) where the objective function is a convex polynomial function with a variable linear form, i.e. the function f(x) in (1) is a convex polynomial function without a linear term. The way we proceed is similar tio that in section 6.2. The polyhedral cone V = K(f)r\ V Df

(21)

where K(fj = [x£.R n / sup(.f(tx) / U Df

O^coo]

will play the same role as the set { x € V / x^Dx =

in the previous

section. By section 4.1., we know: 1. K(f) = 0 + {xe.R n / f(x) ^ Oj = R(f) is a linear subspace (this holds since f has no linear term) and therefore, f (x+kv) = f (x) V x € R n V k t R V v e K ( f ) .

(22)

2. The linear subspace K(f) may be represented as the solution set of a system of homogeneous linear equations having only integer coefficients if all the coefficients of the polynomial function f are rational numbers. The following lemma yields a tool which may be used to characterize the domain of finiteness E(b) and the value function cpif the parameter b e B of the constraints is fixed. We still need the definition cPn(p,b) = inf {f (x) + p T x / x e c ( b ) + V l . J ° Df Lemma 6. If f(x) is a convex polynomial function without a linear term and oo,

that

97

„t

|u°|- 1, u ° £ V, fc

and x e A = [ x c R n / f( x ) + p T x * olfor all t. Of A well-known fact from convex analysis (cf. ROCKAFELLAR (1970)) says that u°c 0 + A . On account of Lemma 4.5. and Lemma 4.4. one concludes u ° t K ( f ) and p T u ° ^ 0. Because of (23) this gives u ° e V, llu°ll»l,

p V

= 0.

(24)

Now we consider the problem inf {f (x) + p T x / x t C ( b ) + A

qT

f

where V = ju€.V / u

(25)

J

u = O J . Taking

cpQ(p,b) = - oo into account we

obtain fPSm (24) and (25) that inf [f (x) + p T x / x e c ( b ) + v } = - oo o

t

A

A

must hold. Since u f V , it follows that dim V ^ d i m V. Thus, the counterexample (25) yields a contradiction to the choice of the original counterexamples # Corollary 2. When b e A , then we have f 0 (p,b) = - oo 4F=»cf>v(p) = - oo where

cj>y(p) is defined by (5).

Proof. One concludes this proposition from (22) and Lemma 6.

#

Corollary 3. When f(x) is a convex polynomial function without a linear term, b e B , and the cone V satisfies the condition (MIG), then the equivalence Cf 0 (p,b) = - oo ^=^cp(p,b) = - oo holds. Proof. Obviously, it suffices to show the implication '

. Let

V

x tG(b) be arbitrarily chosen. Since V satisfies (MIG) and because of ~ T Lemma 6. there is a generator v of V with p v < 0 , and v 1 ( ...,v are integer numbers. For each natural number k we have x + k v t G ( b ) and f(x+kv) + p T (x+kv) = f(x) + p T x + kp^v, from which follows.

0o.

we can assume that u

IIu U = 1 and u ° C V. S i n c e the m u l t i f u n c t i o n C is upper

t

>u

o

where

semicontinuous

and the n o n - e m p t y set C ( b ° ) is c o m p a c t , the s e q u e n c e {z*"} must be

bounded.

The set A

^ X ' * n

+

l>

£ R n +1

/

f

(x)

+

x

n +l ^

is c l o s e d and convex and we have

99

xfce. N ( p t , b t , a t ) —1

(x t ,p t ; T x t -a t ) E A,

/ t ,p t^x t- a )\ (x t

_ / O ,p O^ u 0\). 5>(u

fc

T As in the proof ot Lemma 6. we can conclude that (u°,p° u°)e.0 + A, from which f(ku°) + (k-u°) T p° i 0

Vk i 0

follows and therefore, f (ku°) = O V k

i 0,

i.e. u° e R(f).

If we set X =. I i e { l , . . . , q ) / d 1 u° = DF .T u°£ V, lu° II = 1, d 1 u° = 0, i £ 1 .

, then

Since V satisfies (MIG) and since we have d 1 u ^ 0, for all i e l

and

all lit V, there is a vector v e R n , vj,...,v , possessing the properties vtv,

|vIt ^ 1, v V

* 1/2, d 1 v = 0

Viei.

t

= x - v . Then, because of ( p t , b t ) c £ Of obtain the estimate

Now we set

, t = 1,2,..., we •

f(x fc -kv) + pfc (xfc-kv) = f(xfc) + pfc xfc - kp k v ^ £ fix*) + pfc xfc i afc V k - 0. Using this estimation one completes the proof in the same way as it was done in the case of Lemma 5.

#

The following two theorems demonstrate that both the lower semicontinuity of the value function

cp and the upper semi-continuity of the

level-set-mapping N are ensured if the cone V satisfies the condition (MIG). Theorem 8. If f(x) is a convex polynomial function without a linear term and V satisfies the condition (MIG), then the value function

oo. Because of (26), this is x t £ N(p t ,b t ,cp(p°,b°

Therefore, we can apply Lemma 7., i.e. there exists a subsequence { x r ^ and there is a vector v £ V , xr - v £ G ( b r ) ,

v / 0 and v^,...,v g integer, such that

|lxr-v||-llxr|l.

(27)

Since one concludes the inequality p

r

r

r

v ^ 0 from ( p , b ) c £

, we have

the estimation f(x r -v) + p r (x r -v) = f(x r ) + p r x r - p r v « i f(x r ) + p

r

V

£ f ( p r , b r ) • f.

Because of (27), this inequality -gives us a contradiction to the choice of xfc, which proves the theorem.

#

Theorem 9. Let f(x) be a convex polynomial function without a linear term and let V satisfy the condition (MIG). Further, let (p t ,b t ,a t )6. 1 t

(p*',b ,a t )

N,

t = 1,2

;

> ( p ° , b ° , a o ) with

(p°,b°)e£.

Then the level-sets possess the property V £ > 0 3t(£): N ( p t , b t , a t ) C U £ N ( p 0 , b ° , a o ) V t > t ( £ ) . Proof. We assume that the theorem is false. Then, without loss of generality A. = c

*

we have

N ( p t , b t , a . ) \ UeN(p°,b°,a ) / 0, t = 1,2

Df

for some £ > 0 . Now, we select points x ' c A t ,

t = 1,2,...,of minimal

norm. This sequence {x^j cannot possess an accumulation point because it would have to belong to the set N(p°,b°,a o ) implying that x t Ui(p°,b°,a ) would have to hold for infinitely many t. Thus, > t< * / [x I >oo. By Lemma 7., there is a vector v e V with v p 0 and v1(...,v integer and there is a subsequence {y r J of { y ^ defined by c t y = x -v which possesses the properties (i), (ii) and (iii) of Lemma Df 7.

On account of the choice of the points x r it follows that

y r c. U f c N(p°,b°,a 0 ). p n n r* Let z' e N(p , b , a ) be chosen such that lly - z |l a Q that liincf(p r ,b r ) = - oo, which contradicts and the fact that cf is lower semicontinuous on

(p°,b°)e S

. #

Corollary 4. If the hypotheses of Theorem 9. are satisfied, then N(p°,b°,a o ) / 0. Proof. Analogously to Corollary 1.

#

Corollary 4. directly allows to formulate the following existence theorem for the class of problems P(p,b) considered here. Theorem 10. When f(x) is a convex polynomial function without a lineor term and the A, cone V satisfies the condition (MIG), then the equivalence P(p,b) has an optimal solution (p,b) £ S

is true. The next theorem is related to the upper semicontinuity of the f-optimalset function ' V . Theorem 11. Let f(x) be a convex polynomial function without a linear term and let V satisfy

(MIG).

*

If the value function cj> is continuous on set function y i s

u.s.c. on

£

£ c £

, then the

£-optimal-

x R+.

Proof. The proof runs analogously like the one of Theorem 5.; one has only to replace Theorem 4. by Theorem 9.

#

If the constraint sets G(b) of the problem P(p,b) are given by (20), then, proceeding as in the previous section, we can expose such subsets

J; c £

on whicn the value function coincides with the set of all pairs

(p,b) for which P(p,b) has an optimal solution. By Theorem 7. the set "¿can be divided into disjoint "stability-sets" of the form - vPxB(b°). On every such stability set the value function is continuous and the optimal-set function Y i s

u.s.c. Moreover, the

£-optimal-set function

is u.s.c. on every set - V^x B(b°)x R + .

103

7 . The E x i s t e n c e of Optimal Points for Integer O p t i m i z a t i o n

Problems

T h i s c h a p t e r deals w i t h the q u e s t i o n under w h i c h c o n d i t i o n s

(mixed-)

integer o p t i m i z a t i o n p r o b l e m s possess an optimal Let the f o l l o w i n g general

solution.

(mixed-) integer p r o b l e m be given:

inf (f(x) / x € M s j n

where M C R ,

Mg

= { x e M / Xj,...,*

It is clear that the

(NC)

M

s

(1) integer} and f: M

— ^ R .

conditions

* 0,

inf {f (x)l x e M g ^ > - oo are n e c e s s a r y for the e x i s t e n c e of an optimal

solution.

Our aim is to expose s u c h c l a s s e s of p r o b l e m s

(1) for w h i c h at m o s t

weak additional

a s s u m p t i o n s have to be imposed b e s i d e s the

necessary conditions mal s o l u t i o n of

trivial

(NC) in order to ensure the e x i s t e n c e of an o p t i -

(1).

O f t e n it may turn out that c h e c k i n g w h e t h e r the n e c e s s a r y are f u l f i l l e d is a tough task, therefore,

conditions

it will be useful

to have a

m o r e t r a c t a b l e e q u i v a l e n t v e r s i o n of the n e c e s s a r y c o n d i t i o n s

(NC).

In p a r t i c u l a r , we consider cases for w h i c h the s e c o n d c o n d i t i o n in (NC) may be r e p l a c e d by inf^f(x) / x 7.1. Concave Objective

oo if M g / 0.

Functions

The f o l l o w i n g simple p r o p o s i t i o n for p r o b l e m s q u a s i c o n c a v e ) o b j e c t i v e function t h e o r e m s for integer

(1) with a c o n c a v e

is b a s i c for d e r i v i n g

(or

existence

problems.

L e m m a 1. Let M g be n o n - e m p t y and f a q u a s i - c o n c a v e f u n c t i o n on conv M g .

Then

p r o b l e m (1) has an optimal s o l u t i o n if and only if the p r o b l e m infff(x) / x e conv M g ^

(2)

has an optimal s o l u t i o n . M o r e o v e r , each optimal s o l u t i o n of (1) solves problem

(2),

too.

P r o o f . If x c c o n v M , then, by a w e l l - k n o w n t h e o r e m of C A R A T H E O D O R Y , 1 r there are f i n i t e l y many points x ,...,x € M with s r . r x = > .x , w h e r e ot • - 0 and X Z • = 1. S i n c e f(x) is q u a s i c o n c a v e 1 i=l ijl 1 on conv M g , we have f(x) ± min If(x ),...,f(x and, using this fact, it is easy to verify the p r o p o s i t i o n s . # L e m m a 1. e n a b l e s us to e x t e n d k n o w n e x i s t e n c e r e s u l t s for

optimization

p r o b l e m s w i t h o u t integer r e q u i r e m e n t s to (mixed-) integer cases;

of

course, one has to impose a p p r o p r i a t e a s s u m p t i o n s on the convex hull of the f e a s i b l e set VI .

104

T h e o r e m 1. The n e c e s s a r y c o n d i t i o n s (NC) are s u f f i c i e n t for the e x i s t e n c e of an optimal p o i n t of (1) if one of the f o l l o w i n g two a s s u m p t i o n s

is f u l -

filled: (a) The convex hull conv M g

is a convex p o l y h e d r o n and, f is c o n c a v e

on

it. (b) The c o n v e x hull conv M g may be r e p r e s e n t e d as the sum of a c o m p a c t convex set and a c l o s e d convex cone and, f is a c o n t i n u o u s

concave

f u n c t i o n on conv M . s Proof. One o b t a i n s the p r o p o s i t i o n s as c o n s e q u e n c e s from Lemma 1. and k n o w n e x i s t e n c e results for the c o r r e s p o n d i n g (2) (see e.g. B E L O U S O V Remark

(1977)).

"continuous"

problems

#

1.

The i m p o r t a n c e of the r e q u i r e d c o n c a v i t y of the o b j e c t i v e f u n c t i o n f becomes clear by c o n s i d e r i n g the q u a s i c o n c a v e f u n c t i o n f(x) = e x . Obviously,

the p r o b l e m inf { e x / x integer^ has no solution,

of the fact that e x

is not c o n c a v e on the set R of real

because

numbers.

Lemma 2. Let M C R

n

be r e p r e s e n t e d by M = C + V w h e r e C is a c o m p a c t set and V a

c l o s e d c o n v e x cone. F u r t h e r m o r e ,

let the (mixed-) integer p o i n t s of M g

be u n i f o r m l y d i s t r i b u t e d in M . If now f(x) is a c o n t i n u o u s function on conv C + V, then the inf {f (x) / x £ M g } > - oo


concave

equivalence inf {f(x) / u M } > - oo

holds. Proof. We have only to show that the

implication

{f(x)

inf^f(x) / x e M}= - oo = i n f

/ x e M g } = - oo

(3*)

is valid. If the l e f t - h a n d side of (31) is true, then, b e c a u s e of the fact that f is a c o n t i n u o u s c o n c a v e f u n c t i o n on conv C + V, we k n o w from convex analysis that there is a vector u £ V for

which

inf |f (x) / x = x + tu, t i 0]= - oo

(41)

holds for any fixed p o i n t x 6 conv C + V. Let us consider such a vector u and an x =

ctM.

S i n c e the p o i n t s of M g SIj^. s u p ^ d ( x , M g ) /

are u n i f o r m l y d i s t r i b u t e d in M, we have xeM^oo.

For the s e q u e n c e ^ x ^ c m a sequence [ y ^ c M g

that

such

given by x ' = c t tu, t = 1,2,..., t h e r e

exists

that

I x 1 - y ^ l c - oo. Then problem (3) has an optimal solution if the objective function f(z,y) has one of the properties (a) or (b): (a) f(z,.) is a concave function on the convex polyhedron G(z) for every z £ B , and, for every integer d > 0 , t h e r e is an integer oc(d) > 0 such that

0 lcp(b) - cp(b*)l

i g-il b-b*l| + 6

integer J ; and

d » 0 such that

Vb,b*eB.

(If As = 0, i.e. the "continuous" problem P°(b) is considered, then ty j. and K are Lipschitzian and, o = 0 in (ii).) Proof. (i) We start showing the first part of (i), which will be used to demonstrate proposition (ii) and, both enables us to verify the remaining part of (i). By assumption, we have ^ ( b ) / 0, for all Following theu) e R G(0) = {(y,

b£B.

2n technique

proof of Corollary 4.3. we define / Dyused - Duin= the Ii, y,u i 0,

Df

y x , . . . , y s , u 1 ( . . . , u s integer ^

and V 9 y 1 , - - - , y s , u 1 ( . . . , u g integer j .

Then, by Theorem 5.2., there is a compact-valued multifunction K and, by Theorem 5.6. (iv), there are constant numbers ^ > 0 and T > 0

such

that G(B) = K(B) + V 2 s , 0 €.dom G, H (y, u) II - yUBIl +TT, V ( y , u ) e K(B). * * Let b, b e B and x e * y ( b ) , x ' £

(b ) be arbitrarily fixed. Then

(x,x' ) C G(b-b*) and therefore, one finds points ( y , u ) e K(b-b*), (y,u) e v 2 g such that (x,x*) = (y,u) + (y,u), ll(y,u)H i §llb-b*l|

+T.

We show that y"t If (Oy). Obviously, y t G ( D y ) . Assume that there is a point z £ G ( D y )

satisfying c T z

c T y . Then we have

cT(y+z)^ cT(y+y) = cTx. However, this contradicts x e ' f ( b ) because of

116

y + z ^ 0 and y .

+ Zj,

i = 1 , . . . , s, i n t e g e r

and D(y+z) i.e.

= Dy + Dz = Dy + Dy = Dx = b ,

y +

zeG(b).

One a n a l o g o u s l y Since

*

that

u£l|'(Du).

Dy = Du, we h a v e

cTy" = and,

verifies

cTS

for

the

point _

x

= u + y Df we o b t a i n T * T c x = c u

T + c 7

T = c u

Because of x*e. G ( b * )

T T . c f f = c x .

+

( w h i c h one e a s i l y

sees)

this

implies

x * £ ( b * ) .

Estimating llx-x*H = II y + y l l - u - y II = l( y - u II ^ 2 l l ( y , u ) M £ i

2(j>llb-b*l|

and s e t t i n g

+T)

2 R

way a s u s e d i n t h e p r o o f

a quasi-Lipschitzian such that N(b,a)

of

Theorem

compact-valued

= i 0 be the corresponding constant

numbers such that d(K(b,a), K(b*,a*)) ^oC||(b,a) - (b*,a*)ll+ B V(b,a), ( b * , a * ) e dom N. If one chooses a =cp(b) and defines K(b) = K(b, CP(b)), Df then one has Of(b) = K(b) + V s

VbeB

and, taking (ii) into account, one gets d(K(b),ft(b*)) -i*(Kb, cf(b)) - (b*, cf(b*))l/ +

8t

iot(|lb-b*|+ I (b*) I ) t 6 i i

b-b*l| + ^llb-b*rt+ 0 such that iq>(b) -Cf(b')l

ifllb-b'll 2

Further, l|b-b'l| ^ llb-b'll

V b , b ' e. B.

for all b,b'e B.

For an arbitrarily fixed u €. R m we put r (u) = 1 +P+llull, Df Now, let z ^ 0 integer and Az 4 b, and let r ^ r(u). Then L(z,u,r) = c T z + u T (b-Az) + r llAz-bll2* - tp(Az) + u T (b-Az) + r IIAz-bll * i cf>(b) - I f(b) - cf (Az)l + u T (b-Az) + r II Az-bl/ ^ ^ cp(b) - ^ llb-Az II - Itu II llb-Az tl + r llAz-bll > >cf(b), which the propositions of the theorem are simple consequences of.

ft

Theorem 2. shows that L(z,u,r) yields a penalty function (for the pure integer problem P(b)) which, obviously, possesses "nicer" properties than the one considered in Theorem 1. We remark that lower bounds for the "sufficiently large" numbers r occurring in both the Theorems 1. and 2. can be derived using estimates of the constants taken from the Theorems 8.1. and 8.2.. 9.2. Partitioning

Procedures

Here we establish the general BENDERS') partitioning framework for integer programming in terms of parametric optimization. There are many concrete realizations, in particular, for special linear problems, which we will not treat here and refer the reader to the BIBLIOGRAPHIES on integer programming.

124

Let the general mixed-integer problem be given by inf{f(z,y) / (z,y)€ M, z integer} n

where M C R , f

:

For every X £ R S

m — * r, z £ R

8

and y e R

(1) n_s

.

we associate with M the set

M(\) = {y C R n _ s / (X,y)e m | Df ' and the extremal value Cf>(X) =

(2)

inf{f(X,y) / y e M U ) J .

(3)

Furthermore, we set

A = {Xe R s / M(X) / 0}.

(4)

Df

3y (3) a value function

Cp: A

>R

is defined. Using the definitions

above the pure integer problem inf {cp(a) / A e. A , X integer}

(5)

is assigned to the mixed integer problem (1), and there is a similar relation between (1) and (5) as it was pointed out in Remark 7.1.

The Partitioning Principle The pair (z°,y°) is an optimal point of the problem (3) if and only if z° is an optimal point of the pure integer problem (5), and y° is optimal for the problem inf {f(z°,y) / y e M ( z ° ) j .

(6)

The partitioning principle permits to design an algorithmic approach to solve problem (1) by following the idea: First one solves the pureinteger problem (5), which requires to find the set A and the value function cp in advance. If z° is an optimal point of problem (5), then one still has to solve the "continuous" optimization problem (6). Obviously, the realization and efficiency of such an approach depend on the properties of the original problem (1) and the auxiliary problems (5) and (6). Since, for practically relevant cases, it is impossible to determine the function cp and the set A (not even in the linear case), the only thing one can expect is to follow the above idea in an approximative manner, in particular in such a way that only an approximative determination of ^ and A is necessary. In order to be successful in such a determination it is desirable to have "nice" properties of the function cp and the set A

, i.e., for example, that cj> is a convex or concave

function and A is a convex set.

125

For instance, for a problem (1) with a convex objective function and a convex polyhedron M as the constraint set it is easy to ¿how that the set A is a convex polyhedron and, cf is a convex function (if the infimum of f is finite on M). We discuss a possible partitioning procedure for the simplest case where the objective function f is linear, too. Let us consider inf{c T z + d T y / ( z , y ) e M , z e R 8

integerj

where M = { ( z , y ) e R n / Az + Qy ^ b, (z,y) i ol . 1 Of In order to characterize the set A from (4) and the value function y> from (3) we define the convex polyhedron IV = fu e.Rm / Q T u i d, u i Ol. 1 Of Let U = { . . . , u ^ j

(8)

denote the set of all vertices of W and

H = {h*,...,h^l a generating system of the recession cone 0+lV. Then, from linear parametric programming, one knows (see e.g. NOZlCKA et al. (1974)): 1°

A = [ H R ®

/ (b-A A ) V

^ 0, h 1 «. H}

(if H = {o}, then A = R®); 2°

If there is a A ° t A (A) = c \

such that cp(A°) is finite, then

+ max {(b-AA) T u k / u k £ uj , A t A-

If W = 0, then (5) has no solution. Therefore we suppose that VV 4 0. Let U ' c U, U' / 0 and H ' c H be arbitrarily chosen subsets. We define A(H') = [ U R Df A(0)

8 +

/ (b-AA)V

^ 0,

h^eH'l, '

= R®, Df

cf>.,.(A) = c T A + max Df

i(b-A A ) T u k / u k t U'} .

1

Because of 1° and 2° we have A C A(H'), cfu'CO

6