Zeitschrift für Angewandte Mathematik und Mechanik: Volume 65, Number 8 [Reprint 2021 ed.] 9783112570647, 9783112570630

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Zeitschrift für Angewandte Mathematik und Mechanik

Applied Mathematics and Mechanics Founded by Richard von Mises in 1921

Edited at Institute of Mechanics Academy of Sciences of the G.D.R. Editor-in-Chief: G. Schmidt Editorial Board E. Becker f (Darmstadt) H. Beckert (Leipzig) L. Berg (Rostock) L. Bittner (Greifswald) L. Collatz (Hamburg) W. Fiszdon (Warsaw) H. Gajewski (Berlin) P. Germain (Paris) H. Görtier (Freiburg) J. Heinhold (Munich) H. Heinrich (Dresden) K. Hennig (Berlin) J. Hult (Gothenburg) A. Ju. Ischlinski (Moscow) R. Klötzler (Leipzig)

m

x f g y Akademie-Verlag • Berlin

ISSN 0044-2267

P. H. Muller (Dresden) H. Neuber (Bad Worishofen) K. Oswatitsch (Vienna) M. Peschel (Berlin) J. Rychlewski (Warsaw) A. Sawczukf (Warsaw) L. Schmetterer (Vienna) G. Schmidt (Berlin) J. W. Schmidt (Dresden) H.Schubert (Halle) G. G. Tschorny (Moscow) H. Unger (Bonn) F. Weidenhammer (Karlsruhe) F. Ziegler (Vienna)

Volume 65-1985 Number 8

ZAMM • Z . angew. Math. Mech., Berlin 65 (1985) 8, 329-396

EVP 1 8 , - M

Editorial Office Academy of Sciences of the G.D.R. Institute of Mechanics Prof. Dr. Günter Schmidt Dipl.-Math. Friedhild Dudel Dr. Winfried Heinrich Elke Herrmann Dr. Horst Weinert

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Terms of subscription for the journal

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Z E I T S C H R I F T FÜR ANGEWANDTE MATHEMATIK U N D MECHANIK Herausgeber und Chefredakteur: Prof. Dr. Günter Schmidt, Berlin. Verlag: Akademie-Verlag Berlin, D D R 1086 Berlin, Leipziger Straße 3 — 4 ; Fernruf: 223 6221 oder 2236229; Telex-Nr.: 114420; Bank: Staatsbank der D D R , Berlin, Kto.-Nr.: 6886-26-20712. Anschrift der Redaktion: Institut für Mechanik der Akademie der Wissenschaften der D D R , DDR-1199 Berlin, Rudower Chaussee S; Fernruf: 6743639 oder 6743643.

Veröffentlicht unter der Lizenznummer 1282 des Presseamtes beim Vorsitzenden des Ministerrates der Deutschen Demokratischen Republik. Gesamtherstellung: VEB Druckerei „Thomas Müntzer", DDR-5820 Bad Langensalza. Erscheinungsweise: Die Zeitschrift für Angewandte Mathematik und Mechanik erscheint monatlich. Die 12 Hefte eines Jahres einschließlich Tagungshefte bilden einen Band. Bezugspreis je Band 3 9 6 , — D M zuzüglich Versandspesen. Bezugspreis je Heft 33,— D M . D e r gültige Jahresbezugspreis für die D D R ist der Postzeitungsliste zu entnehmen. Bestellnummer dieses Heftes: 1009/65/8. © 1985 by Akademie-Verlag Berlin. Printed in the German Democratic Republic. A N ( E D V ) 35937

Zeitschrift für Angewandte Mathematik und Mechanik

Applied Mathematics and Mechanics Volume 65

1985

Number 8

55AMH • Z. Angow. Math. u. Mccll. 65 (1985) 8, 329-333

NOWINSKI,

J.

L.

Buckling of an Elastic Strut Made of a Nonlocal Material Es loird das Knicken eines einseitig eingespannten, schlanken, elastischen Stahes vom Standpunkt der nichtlokalen Theorie aus untersucht. Mit den Methoden der Festigkeitslehre wird die Gleichung für die Längsspannungen abgeleitet. Die so erhaltene Gleichung des Problems ist die Summe einer Anzahl von Differenzengleichungen, deren Koeffizienten sich aus den Eigenschaften des Atomgitters ergeben. Bemerkenswerterweise liefern sowohl die erste, als auch die zweite Näherung die Eulersche Knicklast. Buckling of a slender elastic strut built-in at one end and loaded at the other is examined from the standpoint of the non-local theory. Using the Strength of Materials Approach, the eguation for the nonlocal longitudinal stress is derived. The governing equation obtained represents a sum of a number of difference equations with the coefficients provided by the dynamics of atomic lattices. Remarkably, both the first and the second approximation give the value of the buckling load coinciding with that of Elder. IIpojiojiijHbiH H31HO TüHKoro y n p y r o r o BcpTHKajiwioro cTcpHuiH, ijaKpciuieHHoro ira O « H O M K O I I H C H iiarpyjKem-ioro na npyroM, paccMa'rpHBaercfl c T O H K H 3peiiHJi HCJioKajibnoii TeopiiH. Mcnojib3yn iipHÖJiHHieHHe coiipoTHBJieHMH MaTepHajiOB, BHB03HTCH ypaBiieirae HJ1H Hejioh'ajibiioro iipoHOJitnoro HanpiiwenHH. l'lojiyieHHoe ypaBiieuHe npencTaBJineT coSoli cyMMy HenoToporo qncna KoiieMHO-pasHOCTHbix ypaBHeHHil c Koa(|)imiieirraMH, nojiyqeiuiuMH 113 RHiiaMHKH aTOMiiux peujöTOK. MiiTepeciio, ;ITO i;ai; iiepBoe Tan H BTopoc iipiiöjuiiitcmiii iipuBOHiiT k BCJiimmie npuHojibiioro iwi'Höa, coBiiaHaiomcii c aiijicpoBoii.

1. Introduction A l t h o u g h problems of the nonlocal t y p e were occasionally analyzed in the past, rigorous theories of nonlocal continua based on sound physical and mathematical foundations were not established before the last three decades. ( A comprehensive historical r e v i e w m a y be found in E D E L E N ' S treatise [1].) A s c o m m o n l y known, the main feature of the nonlocal theories is their departure f r o m the traditional postulate of the zero-range, that is, of the strictly local internal interactions. T h i s postulate is replaced b y the v i e w , long ago accepted b y the atomic physics, that t h e local state at a point of a b o d y is influenced b y the actions of all of the particles of the b o d y . As a result, the stress at a point, f o r example, while remaining a function of m o t i o n and temperature at the point, simultaneously becomes a functional of these t w o factors extended o v e r the entire v o l u m e of the b o d y . F o r a small deformation this means t h a t t h e stress is (see e.g., [3]) t i j = 2/jietj + le,hkòii + / ( 2 / e ^ + l'e'kkòij) v

/

h

/

/

L

H

Jb'ig. 1. Geometry of the problem

Let the strut be referred to a Cartesian rectangular coordinate system xit ¿ = 1 , 2 , 3, and-let ty, i, j = 1, 2, 3, and «i, i = 1, 2, 3, denote the stress and displacement components, respectively. If the applied load has no transverse component, the ?act that the lateral surface of the strut is free from external tractions implies that the only non-vanishing stress component is r n . We take the latter in the form derived by Ekingen on the basis of the general theory of constitutive equations [3], which in the present case gives 5

5

^ a .

f

71

.„Sul

,

(2.1)

where V denotes the volume of the strut. Let the flexure of the strut occur in the x^-plane. B y appeal to the well-known hypothesis of the conventional theory of Strength of Materials we assume the strain component to be 6ut

x2

(2.2)

d2u 2(xt) In view of the assumed slenderness of the strut (h one may dx f reasonably expect that, as an approximation: (a) the nonlocal moduli may be made independent on the coordinate differences \x2 — x'2\ and \x3 — x's\; (b) the interactions among the particles located on the longitudinal axis of the strut and those exterior to the axis may be disregarded. (In this connection, it seems good to recall E d e l e n ' s theorem [10], Appendix, according to which the hypothesis of attenuating neighborhood implies that, for longitudinal waves in the ^-direction and media with infinite boundaries, the nonlocal moduli are independent of the coordinates x'2 and x3.) With this in mind, equation (2.1) is cast into the form L dhi'^x'i) — agii ) (2.3) - x2 dx'i + i f o — x'x) x" = - j ? f dx'*

where the radius of curvature q

where d(x1 — x\) is the Dirac delta function. From equation (2.4) in [4] we now have fi'(la?! -

ail)

+

— X\)

• o ¿j TC^n 93fi»=i a

- »'I na

(2.4)

where a is the atomic spacing and Cn's are constants determining the forces with which the given atom (at the location n = 0) is acted upon by the planes of atoms removed by na. Equation (2.4) is valid under the conditions that \xx — xil

na

(2.5)

: Buckling of an Elastic Strut

NOWINSKI, J . L .

331

for n = 1, 2, ... , N, respectively. Outside the interval (2.5) the right-hand side of the equation (2.4) becomes equal to zero. Substitution of the expression (2.4) into the equation (2.3) yields XL -f net 5 y p nX2 f (, |Si -SllW'Wzfri) ,„„. 1 (2 6) J i1 m ' Xi — na If one assumes t h a t at all cross-sections of the strut, the external load reduces to a bending couple of moment M acting in the plane xxx2, t h e n the conditions f r n dA = 0 and ftux3 dA = 0 A A are satisfied identically, and the condition

(2.7)

f rux2 dA = -M , " A with A as the area of a cross-section, implies t h a t

(2.8)

5J M=~T{x,).

(2.9)

Here J is the moment of inertia of a cross-section, 2 and x^+na . *cnn r / \xt - Xi\ \d u'2(x') , r. J I1 ~ ] ~ d x T d X l ' x1—na provided \xt — x\\ na. A lengthy calculation reduces equation (2.9) to the relation N

M = 5 J16 T M=1

C n 1 a1 na

+ na) — 2uJx,)

(2 10)

"

+ u2(x, — na)] ,

(2.11)

representing a sum of N difference expressions. Let us now imagine t h a t under the axial force P (this implies t h a t , as in the conventional theory, one has to modify the first of equations (2.7)) the strut buckles in the plane xxx2 so t h a t M — P{f — u2) .

(2.12)

This leads to the governing equation of the problem in the form jV 6 Pa. 3 £ Cn[u2(x1 + na) — 2^(3;!) + u2{xx + na)] + —u2

6 Pa?

(2.13)

I t is worth pointing out t h a t the coefficients 6 P a 3 / 5 J in the just written equation are positive and very small (of t h e order of 10 20 N) if one assumes t h a t under ordinary circumstances there is a & 3 - 10~8 cm, P — EJjL^N, L — 102 cm, and E — 107 N/cm 2 . We now proceed to the examination of two particular cases. 3. Two Particular Cases A. Suppose t h a t as a first approximation, one confines oneself to the particular case of N = 1, corresponding to the interactions of nearest-neighbor atomic planes. Equation (2.13) then simplifies to u2(x^ + a) — 2u2(x1)

Pa2 5/uJ

since < ^ = 3 ¿ta

(3.1a)

(cf., e.g., [4], eq. (1.14), or [5], p. 147). Clearly (3.1) represents a second order difference equation t h a t for a —• 0 reduces to the well-known equation EJ

d

^=P(f~u

2

).

(3.2)

Equation (3.1) m a y be solved by setting cos /3a = 1 —

OflJ

(3.3)

(again we note t h a t the (nondimensional) coefficient Pa 2 /5 J aJ is positive and of the order of 10~20 ; this guarantees t h e fulfillment of the condition cos a{} < 1), and the solution satisfying the well-known conditions at the ends of the strut (y2(0) = dy2{Q)\dx1 = 0, y2{L) = f ) is u2 = / ( l - cosj8:ea).

(3.4)

332

ZAMM • Z. Angew. Math. u. Mech. 65 (1985) 8

where

ßL = (2 n + 1)

n

n = 0, 1, 2, ....

(3.4a)

Expanding cos fia in a power series, and considering the smallness of the atomic spacing a gives easily

§ = (PjEJ)1^ .

(3.5)

This in combination with the relation (3.4a) yields for n = 0 the lowest value of the critical load equal to

n2EJ

(3.6)

Purit = 4L 2 '

It follows that in this case the results of the traditional Euler approach and of the nonlocal theory coincide. This seems encouraging since by limiting the interactions to the closest neighbors one in fact smoothes the difference between the discrete and the continuous distribution of matter. B. Suppose now that as a second approximation one admits interacting up to the second nearest-neighbor atomic planes, that is, including N = 2. Equation (2.13) then becomes

Gi u^x-l + a) — 2uz(x1) « i f ^ g r ) +

- a) + C, u2(x1 + 2a)

6 Pa3 5J f

+ «¡jfo — 2a)

(3.7)

where we set up the relations 3 Pa 3 cos pa = I — a1 5J < V We easily find that 2 /-(2 a + «1 = 4 C m ~~ 4to2 where

3 Pa* 5 0 die Molekülgröße der durch M charakterisierten Teilchen, und r = r(W') = f r W ist der entsprechende Mittelwert. Wegen d\GEjRT) = (drjrf mit dr = r{dW) gilt hier S a t z 4: Bei Gültigheit des Flory-Huggins-Ansatzes (4.1) ist die betrachtete Phase stets stabil. Die analoge Aussage trifft auch für Mischphasen aus einem Lösungsmittel A und einem Vielstoffgemisch B ähnlicher Molekülarten zu. In diesem Falle gilt *

=

X

^

+ fx

B

W.ln ' - m ,

(4.2)

wobei r = XArA + f XBWBrB. Da mit d? = rA dX^ + dX B / WBrB + XBj rBdWB und dX^-f dX* = 0 ebenfalls folgt d2(GEIRT) = (dr/r) 2 , gilt auch in diesem Falle die Stabilitätsaussage von Satz 4. Anmerkung Diese Feststellungen entsprechen der bekannten Aussage der traditionellen Thermodynamik, daß eine durch den Flory-Huggins-Ansatz beschriebene Mischung stets stabil ist. 5. Wilson-Gleichung Die Transformation der Wilson-Gleichung [8] auf die Beschreibung der Zusammensetzung durch eine Verteilungsfunktion lautet GEJRT

= -

M° F W(M) M,

M« [IN F A(M, M,

M')

W(M')

AM'] AM

(5.1)

X{M, M') - X(M, M) und X(M, M') = MM', M), V*{M) > 0. Die BT Funktionen X{M, M') und V*(M) seien stetig. Man überprüft leicht, daß mit A(M, M')

=^jexp

JLL° _

M' _

d2(GE/RT) = f [A(M\ dW)IA(M; W)f W(M) AM - 2 f [A(M; dW)/A(M; W)\ W(M) dM . M.

Dabei bedeutet A(M; W) = / A(M, M') W(M') dM'. Folglich ist jii» 6

2

W

B

T

)

= F W M M,

d W

{

M ) -

A M

'

A W )

A(M- W)

W

{

M )

2

dM > 0 .

d2(G/RT) besitzt offensichtlich genau dann den Wert Null, wenn der Ausdruck in der eckigen Klammer verschwindet, d. h. wenn AW(M) = f(M) W(M). Damit ergibt sich für f(M) die Fixpunktaufgabe M°

f A(M, M') W{M') f(M') dM' 1(M) =

:

.

(5.2)

/ A(M, M') W(M') AM'

M.

Die einzige Lösung von (5.2) lautet: f(M) = const; denn wegen der Voraussetzungen ist eine Lösung von (5.2) stetig, woraus bei f{M) ^ const die Existenz eines e > 0 und eines solchen Intervalls [Me, Me] C [M0, M°] folgen würde, daß |/(Jlf)| < max \f{M)\ - e für M i.[Me, M']. Einsetzen in (5.2) ergibt W„M>]

ME f A(M, M') W(M') dM'

\f(M)\ < max \f(M)\ lM M ] '' °

f A(M, M') W{M') dM'

M,

für alle M e [Mü, j¥°] und mithin max \ f(M)\ < max D\ f(M)\ — elt > 0. Aus dem erhaltenen Widerspruch folgt [M0, M °] [M„M ] S a t z 5: Bei Gültigkeit der Wilson-Gleichung (5.1) ist die betrachtete Phase stets stabil. B e w e i s : Da ö2(G/RT) = 0 dann und nur dann gilt, wenn dW = cW, dieses AW aber nur bei c = 0 die Eigenschaft / dW = 0 hat, folgt ÖHG/RT) > 0 für alle dW ^ 0, J" dW = 0 und folglieh auch die strenge Konvexität des Funktionais G. Damit ist [AZG]T,P > 0 für d W ^ 0.

Wenn eine Mischphase aus einem Lösungsmittel A und einem Vielstoffgemisch B sehr ähnlicher Stoffe betrachtet wird, lautet die kontinuierliche Form der Wilson-Gleichung ' GEjRT = - X . -

M>

A

In [XA + F XBWB{M')

M' F XBWB{M) M,

M,

IN [XAABA{M)

AAB{M') +

AM'}

M° / XBWB(M') M.

ABB{M,

M') DM'] DM .

(5.3)

BERGMANN, J . ; KEHLEN, H . ; RATZSCH, M. T . : I n v e s t i g a t i o n of S t a b i l i t y U s i n g C o n t i n u o u s T h e r m o d y n a m i c s

349

Hier bedeuten AAB(M)

= ^ P ' E X P V*

ABB{M,

M')

V*(M)

¿ab(M) = XBA{M) ,

XAB{M)

DZÄ A

-

XBB(M,M)

RT

- ÀBB{M, M) RT V*(M) > 0 ,

1BB{M, M') = XBB{M', M) ,

1

=

XBA(M)

VÎ exp V%{M)

ABA(M)

XBB(M, M')

exp

I n diesem Falle berechnet sich d2(GjRT) ô2(G)RT)

~1AA

RT

F* > 0 ,

nach + DXB/

•

WBAab

+ XBJ

Xa+XbJ

DWSÄAB

WbAab

1 XAAba

XbWB

+ J

XBWBAbb

so daß analog zu Satz 5 auch hier die Stabilität geschlußfolgert werden kann. Anmerkung Diese Feststellungen entsprechen ebenfalls der bekannten Aussage der traditionellen Thermodynamik, daß eine durch die Wilson-Gleichung beschriebene Mischung stets stabil ist.

6. Polymerlösungen mit einlachem Exzeß-Anteil Synthetische Polymere bestehen aus einer Vielzahl sehr ähnlicher Molekülarten unterschiedlicher Kettenlänge. Bei der Behandlung der Lösung eines solchen Polymeren B in einem Lösungsmittel A ist es üblich und zweckmäßig, nicht die oben eingeführten Größen molare freie Enthalpie G, Molenbrüche XA, XB und Verteilungsfunktion WB(M) Die Umrechnungsbeziehunzu verwenden, sondern die entsprechenden segmentmolaren Größen G, XA, XB, WB(M). gen lauten XA

= rAXAlr;

= 1 - 1

G(T,'P,XA-,XBWB)

A

XBWB(M)

;

= rB(M)

XBWB{M)JR-

=G[T,

Hierbei sind rA ]> 0 und 0 ) ,

x

0

having random coefficients (A0(w), b0(), b(co)) and u are defined by

t JJ'

where t = c0(cu)' x and p(z) denotes the second stage costs arising from the deviation z = A0(a>) x — &0(G>) between A0(co) x and 60(co), then (1) is the objective function of a stochastic linear program with recourse. E x a m p l e 1 . 2 : Optimal portfolio selection [11], [21]. Problems of this type are characterized within our framework by m — 1 and b(w) = 0. Furthermore, u = — q is then the negative of a utility function q = q(z) measuring the utility of the monetary n return z = A[w) x = £ ah((°) xk of the portfolio x = (xlt ... , xn)', where a^w), ... , an(m) denote the random returns on securities 4=1

k = 1, 2, ... , n and x% is the fraction of the investor's wealth invested in security k. E x a m p l e 1 . 3 : Error minimization and optimal design problems [2], [20]. Here A(w) x is interpreted as the output of a stochastic linear system x —> A(w) x and b(m) represents a (stochastic) target. E.g. A(ai) x = SX(oj) m a y be the final state s-p = s%>(co) of a stochastic linear decision process controlled by a sequence x = (x0, x1 xT_ j) of decisions xk, and b(co) is then the desired final state The deviation between output and target is measured by means of u(A(u) x — b(co)) . E x a m p l e 1 . 4 : Statistical prediction [1]. These problems are characterized here by A() e

,

A(w)

y — b(co) € ) x — b(co) € C^,,,), A (a>) y — b(a>) € C;,(,u). Note that (6) implies P(A(co) x - b(w) 6 Ck) = P(4(w) y - b(m) 6 Ck) ,

k = 1, 2 , . . . , r .

It is clear that (6) can hold only in very special situations. Returning again to arbitrary loss functions u, the objective function F is not constant on xy if one of the following conditions hold. L e m m a 2.3: Let x =£ y be such that A(CO) x ^ A(to) y w.p. 1. a) If u is strictly convex on the convex hull of ZXiV, where 7jx y = {A(m) x — b(m) :w € Q) |J U {A(a>) y — b(w) : a> € Q), then F is strictly convex on xy. b) Let the convex functions UA'. Rm -» R be defined by uA(z) = E(m(2 — 6(co)) | A(to) =A), strictly convex function with positive probability, then F is strictly convex on xy.

z eRm.

IfuA^m)isa

357

MARTI, K . : Computation of Descent Directions in Stochastic Optimization

c) Let A(co), b(w) be stochastically independent and define u: Rm —- R by u(z) = Eu(z — b(a>)), z £ Rm. If u is strictly convex on the convex hull of ZXiy, where Zx) x: a> e Q) (J {A.(m) y\ co e Q), then F is strictly convex on xy. Based on the above considerations, in the following we will construct vectors y lying on the i7(a;)-level Nx of F. I t is F(x) = Ju dP^(,)x-b(.)> where Pa(.)x-h.) denotes the probability distribution of A(a>) x — b(co). Basic conditions which imply the decisive equation F(y) — F(x) are contained in L e m m a 2.4:

a)//

u(A{w) x - b{a>)) =. u(A(w) y - b{w)) w.p. 1,

b) If the distribution

then

F(x) = F(y) .

(CI)

equation

= holds, then F(x) = F(y) for each loss function u. y

(C2)

It is clear that we will also obtain descent directions h of F at x if the construction yields "only" elements x of the F(x)-\n\e\ set LFix) = {y e R": F(y) F(x)}, provided — as above — that F is not constant on xy. 2.1. N e c e s s a r y (and s u f f i c i e n t ) c o n d i t i o n s for (C2)

If (C2) holds, then it is E/(^(co) x - b{w)) = Ef(A{a>) y for all functions /:

Rm

b(w))

—• R such that these /-moments exist. Especially, from (C2) follows that

E(4(co) x - b(a>)) = E(^4(co) y - b{w)) , hence and

EA(w)x = EA(io)y

(7)

cov

(8)

X - 6(0) = cov (A(.) y - &(.)) ;

furthermore, for each (r, m)-matrix II condition (C2) implies Pn{jn.)x-b(.)) = Pn(A(.)y-H.)) , especially holds PjiiOz-M ) = ^ii(-)V-H-) > where (A i t bt) is the i-th row of (A, b).

* = 1> 2, ... , m ,

L e m m a 2.5: Lei (A(a)), b(w)) have a normal distribution,

(9) Then (7). (8) are necessary and sufficient for (C2).

.While (7) is a simple linear equation, we have to consider (8) in more detail. Let (Abi)

bi(w)) of (A(m), b(w)). Denoting by Qij the covariance matrix

be the mean of the ¿-th row (Ai(w),

Qio = E(Ai(w) - Ait bt(a>) - bt)' (Aj(m) - Ah b,(w) - b})

(10)

of the rows (Aj(co), bj(a>)) and (Aj(w), 6,(to)), we find that the elements covij[A{.) x — &(.)) of the covariance matrix cov (A(.) x — &(.)) of A(.) x — b(.) are given by

cov,j (A(-)x — &(.)) = x'Qijx ,

i,j = 1, 2, ... , m ,

where x = (x', —1). Therefore (8) is equivalent to

x'Qijx = y'Qijy, Note that

i, j = 1, 2, ..., m .

(8)'

in ». Qim \ • ' : I = cov (A(.), 6(0), \Qml imi ...•Qmm/

where cov (A(.), b(.)) is the covariance matrix of the random m(n + l)-vector

vec (A(w), b(co)) = (A^w), b^co), A2(co), &2(cu), ... , Am(co), bm(co))' .

to (9).

A further necessary and sufficient condition for (C2) is contained in L e m m a 2.5: Let (A(a>), b(u>)) have independent rows (At[m), &i(ft>)), i = 1,2, ... ,TO.Then (C2) is equivalent

Note that each of theTOconditions in (9) corresponds to the case m = 1. Before we start the construction of vectors y e Nx by using one of the conditions (CI, 2) we have to mention the concept of 2.2. S t o c h a s t i c d o m i n a n c e

For arbitrary probability measures p, q on a measurable space W, the stochastic dominance relations p p ) 00 has the form where a^ai), = 0, 1, ... , are arbitrary real random variables such t h series is convergent w.p.l. In the following we give sufficient conditions for (C2) by first introducing the general method described in 5. by means of integral transformations. 4. Integral transformations

p (A{a>), b(w)) p = p(M), M = {A F{x) = / u(Mx) /i(dM) = / u(Mx) p(M) dM ,

Suppose t h a t the distribution of with respect to the Lebesgue measure of E, hence

where x = ^

has a probability density

^ j . Consider the affine (right-)transformation L: E

L(M) = MT + A ,

E defined by

where A = (G, g) is a given fixed element of E and T is a regular {n + 1, n + l)-matrix

c

H ;)•

L{A, b) = (AG + + G, Ac + yb + g) M, J f(M) dM= JdM f{L(M)) E QLjcM L, = F(x) = J u(Mx) p{M) dMdM =dM j u(L(M)x) p(L{M)) = Ju(MTx + Ax)p(L{M)) |det T\m dM , \dx-y) Let the density p have the property m p{L(M)) [det T\ = p{M) a.e. (almost everywhere). Then F(x) = F(y) for each loss junct

where C is an (n, w)-matrix, c, 8 are »-vectors and y e B, hence M'

.

Using the transformation rule for multiple integrals 9L

where / is an integrable function on write

_ t(jx where Tx = [ Theorem

c

d

and

denotes the functional determinant of the mapping

\ I, Ax — Gx — g. Because of (14) we have

4.1:

y =Cx

-

c,

(16.1)

we ca

MABTI, K.: Computation of Descent Directions in Stochastic Optimization

359

provided that d ' x = y - 1,

Ax = 0 .

(16.2), (16.3)

For the interpretation of the decisive condition (15) we consider the image L(p) of the distribution /1 under the transformation L. Again by the transformation rules of multiple integrals we see that (11) is equivalent to the symmetry- or invariance-property

MM) = M

(15)'

of the distribution ¡u of (A(ca), b(co)) with respect to the transformation L. In the next section we consider therefore arbitrary distributions ¡a being invariant with respect to affine transformations.

5. Solving (C2) resp. F(y) = F(x) for y in the case of distribution symmetries Suppose t h a t the distribution ¡jt of (A{co), 6(co)) is invariant (symmetric) with respect to a transformation A: E —> E of E = Rm-n X Rm, i.e. assume t h a t A(/a) = p , where A([i) is the image of ¡n with respect to A. If Hx: E —• Rm denotes the linear mapping HX{M):

= Mx ,

where x is a given n-vector

M = {A, b) e E , and x = I

=

I, then

= Hx(A([i))

= (Hx O A) (fi) ,

(17)

where Hx o A denotes the product of the mappings Hx and A. An easy consequence of (17) is L e m m a 5.1: Suppose that there exists an n-vector y such that Hx o A = Hv , If ¡i, is A-invariant,

i.e.

A(M) x = My .

(18)

then

Pa(.)X-1(.)

=

PA(.)V-bi)

,

hence (C2) holds. According to L e m m a 2.4 and Lemma 2.1 we know then t h a t H = y — £ is a descent direction of J 7 at a; for all loss functions u such t h a t F is not constant on xy. I n the following, the existence of a solution y of (18) is examined under the assumption t h a t always ¡jl is invariant with respect to the considered transformation A of E. a) If A = L is the affine (right-)transformation defined by (13), hence A(M) and only if (MT + A)x=My

= MT

+ A, then (18) holds if

,

hence we must have Tx = y,

Ax = 0,

(19.1), (19.2)

which was already obtained, see (16), in Theorem 4.1 for the special case t h a t ju has an invariant density p. b) The most general affine transformation A is defined by A(M)

= TM+A,

.(20)

where T: E —• E is a linear operator of E and A is a given fixed element of E. Representing T by an (m(n + 1), m(n + l))-matrix ^12 T21 \ Tm\

•••

Thl

T2.2 ...

Tin

Tm 2

...

Tmt

where Ty, i, j = 1, 2, ... , m, are given (n + 1, n + l)-matrices, then the ¿-th row Ai(M) of A(M) is defined by m MM) = X MjTy +Ait i = 1, 2, ... , m , (20)' where Mu Ai: resp., is t h e i-th row of M, A, resp. Since m

At(M) x = £ MjTijX 3=1

+

Atx,

we find now t h a t (18) holds if and only if TijX = 0 ,

i, j = 1, 2, ... , m ,

i

Titx = y ,

for all

i = 1, ... , m;

Ax = 0 . (21.1); (21.2); (21.3)

360

ZAMM • Z. Angew. Math. u. Mech. 66 (1985) 8

In the important special case /

t =

.Tn 0 ... o y 2 2 ... • •• .

\ o

o

hence Ai{M) = MiTu

0 o

|>

(22)

... ' Tm;n/

+ Ait i = 1,... , m, conditions (21) are reduced to i = 1,... ,m\

= y,

Ax = 0 .

(23.1); (23.2)

Finally, if Tu = T, i — 1, ... , m, then the affine map (13) is obtained and (23) is reduced to condition (19). Further symmetries of the distribution ¡i of (A(w), b(a>)) may be utilized if also the loss function u has some symmetry properties. Let us mention still that whenever [i is invariant with respect to an arbitrary transformation A of E, then (24)

F(x) = j u{Mx) fi{dM) = f u(A(M) x) ¡¿(dM). c) If the loss function u has the invariance property u(Sz) = u(z) ,

(25)

where 8 is an (m, ra)-matrix, then we may consider affine transformations of the special type A{M) = STM + A (or A{M) = S(TM + A)) ,

(26)

where as above T: E -* E is a linear operator of E and A Z E is fixed. Indeed, if vectors x, y are related by condition (21), then because of (24), (25) it is F(x) = fu(A{M)x)/i{dM)=

Ju(S(TM)x+Ax)[i{dM)=

J u{SMy) ¡x{dM) = J u{My) /¿{dM) = F(y) ,

hence y € Nx. A special class of symmetric loss functions u is defined by

S = Sn =

; • en(m)'

where n is a permutation of the indices ¿ = 1 , 2 , . . . , m and ej is defined by eji = 0

j, and eji = 1 for i = j; (25) means then that

w(Zjj(1), ••• . MM)) ='«(Zl> ••• > «m) » i.e. u is invariant with respect to the permutation n of zlt ... , zm.

d) We need also the special class of affine maps defined by A(A,b) =(T1A

+G,T2b

+g),

(27)

(.A,b)=:MtE, mn

where A = (G, g) is a fixed element of E and 1\: R ¿-th row {'l\A)i of 'L\A is given by m ( = E Aicv '

rn n

-» R

m

, ï\\ R

1

IV' are linear operators. Therefore, the (2V.1)

3= 1

where Cy, i, j = 1, ... , m, are given (n, »)-matrices, and r

l\b = 8b ,

(27.2)

where S is a given (to, m)-matrix. dl) Knowing only that u is a convex loss function, then we have to work with condition (18) which requires in the present case that {TXA) x -Sb

+ Ax=

A(A, b) x s {A,b) y = Ay - b .

This identity is fulfilled if and only if and.

CijX = 0 ,

i=£j;

Cax = y,

i = 1, ... ,m\

Ax = 0

S = I ( = identity matrix) .

(28.1); (28.2); (28.3) (28.4)

d2) If $ # / , then, corresponding to (26), we consider the affine transformation A(A, b) = (A^A), A^b)) having the special form A^A) m

where R (28.3), then

= S^A

n

+ G

mn

' ->• R is

A(A, b)x=

(or A&A)

= S^A

+ G)) ,

Ajfi) = Sb + g ,

again a linear operator defined as in (27.1). If now two vectors x, y are related by (28.1) to

A^A) » - Mh)

= S{T±A) x - Sb + Ax = S(A, b) y .

Hence, from (24), i.e. under the assumption that ¡x is /1-invariant, follows that F(x) = Ju(A(M)

(29.1), (29.2)

x\/i(dM) = / u{8My) /¿{dM) = / u(My) fi{dM) = F{y)

and therefore y e Nx, provided that u has the invariance property (25), i.e. u(Sz) s u(z).

MABTI, K . : Computation of Descent Directions in Stochastic Optimization

5.1. T h e e x t e n d e d o b j e c t i v e

361

function

Condition (21) may contain very strong or even unsolvable constraints for y. We consider therefore the following extended objective function / AM) »(i) - &iH

V4m(a>)

x(m)

-

bm(m),

where z(l), ... , x(m) £ R and (^i(cy), &((«)) is the i-th row of [A(co), b(to)). This represents the "ideal" situation that each output zt = A^co) x — bi(w) may be controlled individually by some x = x(i). I t is easy to see that using FQ, then (21) may be weakened to n

Tyx = 0 ; which imply that Fix)

Tax = y{i)

= F0(x,

for some

y(i) € Rn , ,

x, ... , x) = F0(y( 1), y(2),...

i, j = l, ... ,m;

Ax = 0 ,

(30.1); (30.2); (30.3)

y(m))

for an arbitrary loss function u. Consequently, K

=

(vW

-

> 2/(2)

x

-

y(m)

x

-

x

)

is.then a descent direction of Fa at (x, x, ... , x), provided that F0 is not constant on the line segment joining (x, ... , x) and (i/(l), ... , y(m)). Unfortunately, contrary to the projection of the gradient of F0, the projection 1 m l m h = m

0

=

z (yd) /II i = 1

- * ) = "I

L y(») ¿^1

-

«

of h0 onto Rn is in general not a descent direction of F0(x, ... , x) = F(x) at x. Nevertheless, the line through x defined by h = ITh0 is a promising search line at x and the optimization problem min ^„(»(1), ... , x(m))

s.t.

x(i) € D ,

i = 1, ... , m ,

(31)

may be used to approximate the original problem (2). If F*, F*, resp., denote the infimum of (2), (31), resp., then Ft ^ F*. Upper bounds for F* — F* may be obtained if we know a Lipschitz-constant Lu of the convex loss function u with respect to the union of the supports of ^4(.) x — 6(.), x £ D. 6. Methods for finding distribution symmetries In the following we are looking for necessary and sufficient conditions for the invariance A{ft) = [i of the distribution of {A{m), b(co)) with respect to the affine transformation A(M) = TM + A, M e E , of E = Rm n x Rm defined in (20). If A[fi) = fi, then

H

£f{A(A(u),

b{a>)))

= E/(4(co), 6(c))

for all functions f : E — * R such that these /-moments exist; especially we have that b(co))) =

A(E(A(w),

6(c)) ,

cov (A(A(.), b(.))) = cov

(A(.),

&(.)) .

(32.1), (32.2)

Note that (32) is also sufficient for A(ju) = fi if (j, is a normal distribution on E. 6.1. I n v a r i a n t c h a r a c t e r i s t i c

functions

I t is well-known that a probability distribution is uniquely determined by its characteristic function. Hence ¡j, is characterized by ji(M)

where

i

:=

E e x p (i t r M(A(a>),

: = |/ —1. Furthermore, tr

MN'

&(«))')

=

,

M

m £ j=i

MjN'j,

e E

,

where

Mu

Nt

denotes the j-th row of the

(m,

n , +

l)-matrices

M, N € E, is the inner product of the Euclidean space E. Therefore A{jx) == fi where

A(fj)

if and only if

A([i) = jx ,

is the characteristic function of A(fi)

(M)

=

e x p (i t r MA')

where vec T*M

= x' I

[

;

\

is affine, we get

T21

(33)

••

T22

) and r ' = | •

\M'J .

A

,

Z7!! r

Since

A{/i).

¡i{T*M)

•

T im

..

,.

T, T,

•

| is the matrix representing the adjoint operator

T,

T* of T. Obviously, the invariance of ¡x with respect to the affine transformation A is characterized by the identity e x p (i t r MA')

j*(T*M)

=

p.(M)

,

which is a condition for the matrices r, A defining A. 24

Z. angew Math. u. Mech., Ed. 65, H. 8

(34)

362

ZAMM • Z. Angew. Math. u. Mech. 65 (1985) 8 Let us assume now that (A(w),

b(oj)) is an affine transformation

(¿(to), 6(o))) = 5 + r(A0(w),

60(o>))

of a further random matrix (-40(co), b0(a>)), where 3 = ( 0 , 6) is a fixed location matrix and F: E -* E is a linear scale operator. I f U: E -* E is defined by (35.1)

Z ( M ) = r M + S , then '

¡i - ¿ f a ) > where ,«„ is the distribution of (^40(co), b0(a>j). p,(M)

- exp (i t r MS')

fi0(r*M)

.

According to (34) the distribution p, defined by (35) is symmetric with respect to A(M) exp (»tr M(A

+ TE)')

(35-2)

Consequently,

fi0{(Tr)*

= exp (i tr ME')

M)

fia(F*M)

= TM

+ A if and only if

,

(36)

which yields L e m m a 6.1: Let ¡i be defined

by (35). If

A(E)

= E.,

fi0((TD*

then fi is invariant

with

respect to A(M)

If 3 = E(A{oj),

=

far*M)

= TM

+

M)

,

(37.1), (37.2)

A.

6(co)), then (37.1) is identical with (32.1). We ask now whether (37) is also necessary for A(/x) = ¡1.

L e m m a 6 . 1 . a : Suppose

that E(^40(oj), b0(a>)) exists.

If fia is real, then A(/x)

= n implies

(37).

Proof: If fi0 is real, then /¿0 is invariant with respect to L(M) — —M, hence E(^0(co), b0(w)) = 0. If A(p) — fi, then (32.1) yields 3 = E(A(a>), 6(b))) = A(E(A(co), b{m))) = A(3), hence (37.1). The rest follows from (36). While (37.1) is a simple linear constraint for T and A, condition (37.2) for T only is more involved. Since distribution symmetries may be described by means of (36), using (33) we observe that (37.2) is equivalent to the condition (TD (fio) = r(fi0) (37.2)' for T, where (TF) (¡u0) and I\/ji0), resp., are the images of fi0 with respect to TT and r , resp. I f we know that the distribution fi0 of (^40(a)), b0(a>)) is invariant with respect to the affine map A0: E —• E, defined by A0(M)

= T0(M)

+

Ao,

tHen (37.2)' holds if and only if 01

= r(A0(p0))

(TD

(fi0)

r(fi0)

= (TD

(A0(fi0))

= ( r o A0) 0«,,) = {(TD

Equation (38.1) holds certainly if TF T r = rT

0

,

rA0

=

° A0)

(Tr)

(38.2)

.

— F o A0, which is equivalent to

o;

if T " 1 exists, then A0 = 0 and T is defined by T = Equation (38.2) is implied by F =

(38.1)

FT0F-\

o A0

which holds if and only if r if r -

1

= TrT

0

,

exists, then also

= 0;

TTAa T^1

must exist, hence A0 = 0 and T =

L e m m a 6.2: Let ¡ia be invariant

with respect to Aa(M)

= T0M

FT^T-1. +

a) If FA0 = 0, then (37.2) holds for any T satisfying TF = J T 0 . b) If T is a solution of r = TFT0 and TFA0 = 0, then T satisfies 6.2. I n v a r i a n t

A0. (37.2).

densities

Suppose that ¡i has a density p = p(M) with respect to the Lebesgue measure on E. I f A: E —• E is an arbitrary differentiable 1—1-transformation of E, then also the image A(p) of p, has a density pA given by PA(M)

=

1

v{A-\M))

,

(39)

where QA/dM is the functional determinant of A and A'1 denotes the inverse of A. I f A is affine, i.e. if A(M) = TM + A, where the linear operator T is represented by a regular matrix r, then (iAjciM = det t and A_1(M) = T~l(M — A). For the A-invariance of fi we have the criterion A(p)

= p,

if and only if

Pa(M)

= p(M)

a.e.

= =

MARTI, K. : Computation of Descent Directions in Stochastic Optimization

363

(almost everywhere), hence, because of (39), density p of /j, must have the invariance property p(M)

• p(A(Mj)

=

a.e.

6.3. I n v a r i a n t c o n d i t i o n a l

(40)

distributions

Denote by v(. \ b) the conditional distribution of A(a>) given b(a>) = b and let v(A \ b) be its ^ -acteristic function. If /? = P4(.) is the distribution of b(a>), then the characteristic function ji of // = &(.)) may , -^presented by ju(A, b) = / exp (ib'v) v{A | v)

,

{A,b) = M 6 E .

(41)

I n order to describe invariances of v(. | b) we consider the special class of affine transformations A(A, b) = ( A M ) , A2(b)) defined in (27), where A±(A) = 1\A + G and A2{b) = Sb + g. If A^vi- | b)) denotes the image of v{. \ b) with respect to Alt then we get L e m m a 6.3: Assume that A M - 1 b )) = "(• I

v-P- 1 '

i42-1)' (42-2)

^ . ( P * ) ) = P«.) •

Then A(fi) = ¡u. Proof: Fourier transformation of (42.1) yields w.p.l

exp (i tr AG') v(T*A \ b) = v(A \ A2(b)) for all A e M™* . Bccause of T(A, b) = (TXA, TJ>), A = (G, g) and T*(A, b) = (T*A, T*b) from (33), (41) follows

A(ii) (A, b) = exp (i tr AG' + ib'g) /1(T*A, T*b) = exp (i tr AG') exp (ib'g) f exp {i(S'b)' v) v(T*A \ v) P(dv) = = / exp [ib'A2(v)) exp (i tr AG') i(T*A \ v) P(dv) = / exp (ib'A2(v)) v(A \ A^(v)) $(dv) = = / e x p (ib'v) v(A | v) A2(P) (di>) = ¿(A, b) ,

hence A(/i) = /i.

Special cases: a) If A(co), b(w) arc stochastically independent, then v(. | b) = P^c(.) and (42.1) requires that the distribution P,i(.) of A((o) is 4i-invariant. — b) If A2(b) = b, then (42.1) means that w.p.l the conditional distributions v(. \ b) of A((o) given b(tu) = b are /^-invariant. If fi has a density p, then (42) may be formulated also by means of densities. Let q(A \ b) denote the density of v(. | 6) and let p0 be the density of /? = P&(.), hence p{A, b) = q(A | 6) • p„(b). According to (40) the invariances (42) may be described by ? ( ^ | 6 ) = |detr(l)|?(/l 1 ( J 4)|/l 2 (6)) where t(1) is the matrix representing

a.e.,

Pts(b)

= |det S\ p0(A2(b))

a.e.

(42.1)', (42.2)'

E x a m p l e 1: Let q(A | b) = /t(^4j — 11(1) b, ... , A'm — U(m) J), where //(1), ... , //(to) are (re, »i)-matrices and h is a nonnegative function on R m •» such that

h(SW[,... , SW'm) = h(W[

W'm) ,

Wit

Rn,

for some n X n matrix S with det S = ¿ 1 . Defining A(A) = AS' + A, At = g', i = I, ... , m, and A2(b) = Sb + g, where S is a matrix such that SH(i) = H(i ) S and H(i) g = g, i = 1, ... , m, we obtain

I dot T(l)| q{A1(A) | A2(b)) = q{Ai(A) \ A,(b)) = h(SA { + g- li( 1) (-S6 + g), ... , SA'm + g= h(S(A[ - 11(1) b), ... , S(A;n - U(m) b)) = q(A \ b) ,

H(m) (Sb + g)} =

hence (42.2)' holds. E x a m p l e 2: i f i ( . | b) is real valued, then v(M \ b) — v(M \ b) — v(—M | b),i.e.v(. \ b) is invariant with respect to A(M) = —M.

6.4. R o w

symmetries

A) We suppose first t h a t ¡i is a mixture of (4(a>), b(co))-distributions 1 5S i ^ m, i.e. let ¡i be defined by m t* = J-® MitMde),

having independent rows (Ai(io), bi((o)),

(43)

i = 1

n+1

where /¿¡Q for each i = 1, 2, ... ,TOis a probability measure on (the Borel a-algebra of) li

depending on a parameter m q & It which has a probability distribution X on the measurable parameter space (R, cF) and (x) ¡j,is is the product of »=i the measures ¡iie, ... , /im„. If each ¡x-iQ has a density pi„, then this class may be represented by p(M) = j II pUMJ ¡=i

A(dg) ,

MzE

,

(44)

where p is then the density of ¡x. Symmetries of this class resulting from row changes and invariances of individual rows m a y be covered by affine transformations A of t h e type (26). For the present case T and S are selected such t h a t t h e rows At(M) of A{M) are given by Ai{M) = Mn{i)Tnii)n(i) 25 Z. angew. Math. u. Mech., Bd. 65, H. 8

+ An(i),

i = 1, ... , m ,

(45)

364

ZAMM • Z. Angew. Math. u. Mech. 65 (1985) 8

where n is a permutation of the row indices i = 1, 2, ... , m and T1V ... , Tmm are {n + 1, n -f l)-matrices. From (43) and (45) follows by means of (33) that m

^

= exp (i

A ( f i ) ( M )

m

-¡M*»)

Z j=i

J/7

A(de)

^ ( M ^ T ' j j )

=

3-1

m

= J 77 exp (iM„-wAi)

n-'(j)T'jj)

i

A(de)

=

m

m

= J77exp

(iMfA^)

Q{M^n{j)^

3=1

6.4:

where

rj:

R

R

Proof:

- — '

'

m

transformation

of

(47) R.

If

= A, f/ien

r]{X)

is

invariant

with

respect

to

A.

(47) into (46) yields

^

m

m

m

A(fi) ( M ) = f n h n i . e ) { M i ) ^ e ) = f n k j= l 3= 1

( M i ) r)(X)

e

(de)

=

/ n k e W 3= 1

We)

=

.

hence A(fi) = fi. Note. a) If the distributions //jo have a density pje, then because of (39) condition (47) may be given in the form pje(w')

=

|det Tjj\ Pn-'(j)

v(e)(w'Tti

+

¿j)

a-e-

.

1

^

j

b) For a given parameter g (47) means that the transformation with a possibly modified parameter rj(g). If L: E E is defined by L sented by m L(®

3=1

c)

Special

ft'j)

=

m ® /in-'O)

If

cases:

vie)»

3= 1 rj(g)

=

g,

¿-a-s-

then

SS

(47)'

of a row distribution j ( M ) =

M j T j j

A j, I

yields the distribution of another row j ) = b can be given in the form m v(.\b)

= f ® v

j e

(48)

( . \ b ) X ( d Q ) ,

3=1

where, for each j = 1, ... ,TO,q 6 R and b € Rm, VjQ(. | b) is a probability distribution on Rm. If each Vje(. | 6) has a density qje{. \ b), then this class may be described by q ( A \ b ) = j n q j=l

( A l \

k

-

) W e ) >

b

(49)

where q(A \ b) is the density of v(. \ b). Symmetries of the present class of distributions can be described by means of affine transformations of the type (27). Given a permutation n of the row indices i = 1, ... ,TO,definition (48) of v(. I b) suggests to choose S and T1 in (29) such that A2(b) = S„b -f- g, where S„ is the permutation matrix defined in section 5c and the rows Au(A) of A^A) are given corresponding to (45) by Au{A)

where

C

n

=

A

are

, . . . , Cmm

n i i )

(n,

C

n i i ) n ( i )

-h

G„

{ i )

,

m | b))

{A)

(50)

w)-matrices. Using a similar argument

^

A^vi.

1,..., TO ,

=

i

=

exp

(i

£

as in (46), we obtain

m S

A j G ^ ) )

3-1

1u)C'jj | 6) A(dQ) —

n

1

m

=

f

n

L„u)(v„u)

Q(.

I 6))

[ A , ) A(de)

,

(51)

3=1

where

L f . Rn

R

is defined by

Lf(a')

=

a'Ci}

+

G

j t

j

=

1, ... , TO.

Refering to section 6.3 we obtain L e m m a 6.5:

I *>))

Li{vje(• where

rj: R

- »

R

Assume

is

that

=

a measurable

X-a.s. *)(•

I

transformation

+

9)

(52)

, of

R.

I f

rj(A)

=

A and

A2(?b^))

=

P t ( ), then

f i is

A-invariant.

MAKTI, K . : Computation of Descent Directions in Stochastic Optimization

365

P r o o f : Under the above assumptions (51) yields

m m A^. i b)) (A) = / nvMe) (A i A2 (b)) A(de) = s rr'M A i ^

3=1

j=1

vm (,((>) (a'Cj}

+