for non-differentiable convex Semi-Infinite Programming Problem. Resumen. Proponemos nuevos teoremas de alternativa para sistemas infinitos convexos que ...
TRABAJOS DE ESTADISTICA Y DE INVESTtGACION OPERAT~VA Vol, 35. Num I, 1984, pp. 32 a ,~7
AN OVERVIEW OF SEMI-INFINITE PROGRAMMING THEORY AND RELATED TOPICS THROUGH A GENERALIZATION OF THE ALTERNATIVE THEOREMS
G o b e r n a , M. A., ~ L6pez, M. A., 2 Pastor, j . t and Vercher, E. 2
Abstract We propose new alternative theorems for convex infinite systems which constitute the generalization of the corresponding to G ~ E , FARKAS,GORDANand Mowzrnq. By means of these powerful results we establish new approaches to the Theory of Infinite Linear Inequality Systems, Perfect Duality, Semi-infinite Games and Optimality Theory for non-differentiable convex Semi-Infinite Programming Problem. Resumen Proponemos nuevos teoremas de alternativa para sistemas infinitos convexos que constituyen la generalizaci6n de los correspondientes de GALE, F~XAs, Gom~,cN y MOTZXn~. Utilizando estos poderosos resultados se establecen nuevos enfoques de la teoria de sistemas infinitos de desigualdades lineales, dualidad perfecta, juegos semit Faculty of Sciences. University of Alicante. 2 Faculty of Mathematics. University of Valencia. (*) Recibido, Noviembre, 1981
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infinitos y de la teoria de la optimalidad p a r a un problema de Programaci6n Semi-tnfinita convexa no diferenciabte.
1. INTRODUCTION Let ~ r : = I ~ , t E TI be a family o f convex functions defined in IR n with finite values, T being an arbitrary index-set. Let S be the solution set o f the system ift(x) denotes the convex hull of (7; 9K(C) the convex cone generated by C; el. C its closure; int. C the interior of C; rint. C its relative interior; C b its boundary; M ( C ) the linear subspace generated by C and V(C) the linear manifold generated by C. We shall consider elements of the set IR(+r), defined as: IR~+r): = [ a : T ~ IR+lott = 0 for all t except for a finite number}
2. THE' GENERALIZED ALTERNATIVE THEOREMS The Alternative Theorems listed in the following table establish the equivalence between proposition I and the negation of proposition II. The functions appearing below are assumed to be convex, in the first two theorems, and homogeneous-convex (i.e. their epigraphs are closed convex cones) in the last three. Proposition II
Proposition I (~C-GAL~ {~(X) ~ 0 is consistent
a E cI.KlLo(~r)l ,a cI.K{ at, t E T}
~L-GogD~h'q)~ |a~x< O,t E T} O,,E (Lo(~T)) ~ ({ab, tE T}) is consistent (Under the assumption that either (Lo(~r)) or K[Lo(~r)] is closed)
O~ e (lat, t e T}) + KI{a,,sE S}]+ 'afx < O, t E T, T # 0 t a;x g O, s e S + Mllup, p E P}) ~ttpx - O,p E P is consistent (Under the asumption that the set considered in Proposition II is closed).
t
Vercher (1981) has directly proved the above finear alternative theorems, giving also a characterization o f the relative interior point o f a convex set, which allows us to obtain two more linear alternative theorems:
Proposition H
Proposition I
{afx ~ O, t e T} has a solution x ~ such that afx~ # O, for some t.
{a;x /O, or
for some
We could reasonably expect a generalization o f this last STIEMKE theorem for the homogeneous-convex case to have the following formulation: Proposition I: ~t[ft(x) ~< 0, t E TI has a solution x ~ such that ft(x ~ ~ 0 for some t>>. Proposition II: On E tint. (Lo(~r))>> It is not difficult to prove that I implies non-II. Unfortunately the converse does not hold, as can be deduced by considering the following family of functions: .h(x) = ( 0 , x ~ < 0, [tx, x > O
t E [O, 1]
Other proofs of L-GAL~ and C-GALE can be found in FAN (1968) and CHUsO-W~.x HA (1979), respectively. A slightly different version o f LFARK~ and C-Fmzg~s is contained in the same references. We would like to emphasize the meaning of the alternative L-FARg~s theorem. In fact, it gives the characterization of all the linear consequence relations o f a consistent system [ a[x ~ - o o . In some cases, the optimization problem can be reduced to a nonlinear program P:
Min./~t
TI
s.t.at = On, t E Let 0 be the value of P (if P has no feasible points, 0 = + o0)
Theorem 2.2 { a/x 1,
tl + t2 t> 0}
The mentioned rules give the following finite representation 3xl + 3x2 + x3 ~ - 2 x ~ - 2 x 2 +x3 ~< 2 2xt + 2x2 - x3 ~< - 1 - 2x2 - x3 ~< 3
4. FARKAS-MINKOWSKI SYSTEMS The F - M systems play a crucial role in the t h e o r y o f SIP, not only in the linear case (duality) but also when the constraints are not linear. W e shall show below their application in the field o f optimality conditions in SIP. In OOBERNA, LOPEZ and PASTOR (1981) we give a geometrical characterization o f the F - M systems.
Theorem 4.1 The consistent system [a[x - oo. An equivalent form of the alternative theorem:
Proposition I
Proposition II
{fi(x) ~< 0, t E T} is consistent on C
There is a X 6 IR(+r), such that P(X, x) > 0, for all x E C.
proved, for example, in of the next new result.
BERGE,
and GHOUILA-HOURI(1965), is the key
Theorem 6.1
0, for all x E C. 42
The last alternative t h e o r e m together with C-FARKAS t h e o r e m leads us to the next result: Theorem 6.2
Let us consider a convex semi-infinite game with value. It holds that: (1) Player II always has optimal strategies. (2) If L(~r) is a compact set, then player I has optimal strategies. We have f o u n d a curious interpretation of the optimal strategies of player II, if we suppose, without loss of generality, O = 0. Let us consider the concave function It(u): = inf u'x, u E IR n, and xEC
let us represent by N the h i p o g r a p h of this function (N is a closed convex cone). T h e n the only separation hyperplanes between (L(5:r)) and N have as equation:
being an optimal strategy for II. In the case that L(5:r) is compact, since every separation hyperplane contains the origin, strict separation is not possible and, therefore, NA(L(~r)) # O. If [~] E N r l ( L ( ~ r ) ) , we can write: =
k~ i
,
with
~k~=
1
and
k~>O,
tET iEIt
I ~ being finite and It ~ 0 only for a finite n u m b e r of t E T. If we define kt = ~--~,iEttk~, it can be proved (VERCHER (1981)) that kt = (k3t Er is an optimal strategy for player I. T h e o r e m 6.1 and 6.2 can also be d e m o n s t r a t e d by m e a n s of the Perfect Duality o f section 5. Cht)t~T is called a pure-strategy for player I if kt = 0, for all t # to, and kt0 = 1. We are interested in determining conditions which guarantee the existence of optimal pure-strategies for I. By using the SIolq's Minimax T h e o r e m we get the following result,
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Theorem 6.3 If T is a convex set of a linear topological space, and, for each x E C, the functions J~(x) are quasinconcaves and upper semicontinuous in t, then the game has value and player I has an optimal pure-strategy. The hypothesis of Theorem 6.3 guarantees the existence of a saddle point ([, x-) for the function.fi(x), (t, x) E T x C. It can be easily proved that k = {)~i= 1, Xt = 0 for all t ~ ?} is an optimal pure-strategy. We can establish a sort of converse statement as follows: Theorem 6.4 If player I has an optimal pure-strategy, the functionft(x), (t,x) E T • C will have a saddle point. By changing the conditions required to the functions in t, {~(x), x E C}, it is possible to obtain a result like Theorem 6.3, applying the Theorem of ISODA-NIKAIDO(1955). Theorem 6.5 If T is a compact-convex set in IRm, and, for each x E C, the functions ft(x) are continuous on T, T(x) = [ t E Tilt(x) = max.t ~ rft(x) } being a convex set, then the game has value and player I has an optimal pure strategy. I f C = [xEIRn/xi>~O, i= 1 , . . . , n , ~'/=ixi= 1} andft(x)=aIx, we obtain the Semi-Infinite Games studied by SOYSTrR (1975). V~RCrmR (1981) has given a direct approach to these games, based exclusively on the linear alternative theorems proposed in Section 1. Other special topics such as the characterization of essential pure strategies and the geometrical meaning of optimal strategies have also been studied. 8. OPTIMALITY CONDITIONS FOR NONDIFFERENTIABLE CONVEX SIP The well-known Theory of the Lagrangian Saddle Point in Nonlinear
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Programming can be extended to non-differentiable convex SemiInfinite Programming through suitable generalization of the concepts. Let us consider the general problem P: Min. [ ~o(x)/x E S } S = Ix E lRn/ft(x)
