Theorems of the Alternative and Optimality Conditions for Convexlike ...

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 101. No. 2. pp. 243-257, MAY 1999

Theorems of the Alternative and Optimality Conditions for Convexlike and General Convexlike Programming1,2 T. ILLES 3 AND G. K.ASSAY4

Communicated by D. G. Luenberger

Abstract. A general alternative theorem for Convexlike functions is given. This permits the establishment of optimality conditions for convexlike programming problems in which both inequality and equality constraints are considered. It is shown that the main results of the paper contain, in particular, those of Craven, Giannessi, Jeyakumar, Hayashi, and Komiya, Simons, Zalinescu, and a recent result of Tamminen. Key Words. Convexlike functions, alternative theorems of Farkas type, duality theory, constrained optimization problems.

1. Introduction Theorems of the alternative of the Farkas and Gordan types for generalized convex functions play an important role in many applications, especially in optimization theory regarding optimality conditions for nonconvex programming problems and duality theory of these problems. In the last 15 years, many results concerning so-called Convexlike functions have been published. We refer, for instance, to Hayashi and Komiya (Ref. 1), Simons (Ref. 2), Craven, Gwinner, and Jeyakumar (Ref. 3), Borwein and 1

This research was partially supported by the Hungarian National Research Foundation, Grants OTKA T-014302 and T-019492. 2 The authors are indebted to a referee for proposing the definition of (t, Oy2; K)-convexlike functions and pointing out that the main results of this paper have straightforward generalization to the bigger class of generalized convex functions. Section 6 is based on this remark. 3 Associate Professor, Department of Operations Research, Eotvos Lorand University, Budapest, Hungary. 4 Associate Professor, Department of Analysis and Optimization, BabeS-Bolyai University, Cluj, Romania.

243 0022-3239/99/0500-0243$16.00/0 C 1999 Plenum Publishing Corporation

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Jeyakumar (Ref. 4), Craven and Jeyakumar (Ref. 5), Gwinner and Jeyakumar (Ref. 6), Tardella (Ref. 7), Zalinescu (Ref. 8), and Tamminen (Ref. 9). A systematization of convexity concepts for sets and functions and some applications to multiobjective optimization problems can be found in Breckner and Kassay (Ref. 10). The aim of this paper is to establish a general alternative theorem for convexlike functions and to deduce, from this, optimality conditions for convexlike programming problems. Our results permit studying problems such that, in addition to inequality constraints, there are equality constraints as well. Except for Refs. 6 and 9, the results mentioned above are concerned only with inequality constraint problems. In the paper of Gwinner and Jeyakumar (Ref. 6), equality constraints appear for a special class of functions; here, the regularity condition is different from ours. The main results of the recently published paper of Tamminen (Ref. 9) are particular cases of our theorems. Results concerning convex problems (with so-called cone convex functions) have been obtained, e.g., by Giannessi (Ref. 11). The paper is organized as follows. In Section 2, we introduce some notations and recall the definitions and elementary properties that we need. The main result (Theorem 3.1) is stated in Section 3. Example 3.1 shows that the regularity condition in Theorem 3.1 cannot be omitted; in fact, it is essential only in the case of infinite-dimensional spaces. In Section 4, we investigate the convexlike programming problem, and using Theorem 3.1, give necessary optimality conditions for it. In Section 5, it is shown how to obtain, from our theorems, results due to Craven and Jeyakumar (Ref. 5), Hayashi and Komiya (Ref. 1), Craven (Ref. 12), Zalinescu (Ref. 8), Simons (Ref. 2), and Tamminen (Ref. 9) concerning convexlike programming. Finally, in Section 6, we extend our results (Theorems 3.1 and 4.1) to a new class of generalized convex functions, called (T,O y2 ;K)-convexlike functions. These are common generalizations of all previously mentioned results and related results of Craven and Jeyakumar (Ref. 5), Jeyakumar (Ref. 13), and Jeyakumar (Ref. 14). We also give an example for convexlike programming problems. Part of these results were first announced in Ref. 15. 2. Preliminaries

First, we introduce some notations and give the definitions we need. Let X be a nonempty set, let Y be a topological vector space over the reals, and let K be a cone of Y. We denote by Y* the dual space of Y, by OY the origin of Y, and by K' the dual cone of K, i.e., the set

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A subset M of Y is said to be nearly convex if there exists ae(0, 1) such that, for each y1, y2eM, we have We recall the following result (see for instance Aleman, Ref. 16). Lemma 2.1. If Me Y is nearly convex, then: (i)

int M is a convex set, where int M is the set of all interior points of M (it may be empty); (ii) the set B={/3e[0, 1] V y 1 , y 2 e M : fiy1 + (1 -p)y2eM} is dense in [0, 1]. Let F : X - > Y . We say that F is K-convexlike if there exists ae(0, 1) such that, for each x1, x2eX, there exists x 3 eX with In the special case when Y= Rz, with Z=O and K= Rz (the cone of all nonnegative real-valued functions on Z), let H: X x Z -> R be defined by Then, relation (1) can be written as and we obtain the convexity introduced by Konig (Ref. 17, a = 1/2). In the following, we consider two topological vector spaces Y1 and Y2 over the reals. Let kic Yi, i= 1, 2, be two convex cones with vertices OY1 and OY2 such that int K1 =O and let f: X -> Y1, g: X -> Y2 . Put We first prove the following auxiliary result. Lemma 2.2. The set F(X) + K has nonempty interior if, and only if, the interior of the set M=F(X) + ((int K1) x K2) is nonempty. Proof. The "if" part is obvious, since M Y is K-convexlike and the set F(X) + K has nonempty interior. Then, the following assertions hold: (i)

If there is no xeX such that

then there exist y * e K 1 and y*eK2, with ( y * , y * ) = ( O Y 1 , OY2) such that (ii) If there exist y * e K 1 \ { O Y 1 } and y2eK2 such that (4) holds, then there is no xeX such that (2) and (3) hold. Proof.

Assertion (ii) is obvious; hence, we have only to prove (i). Let

Since F is K-convexlike, then M is nearly convex. By Lemmas 2.1 and 2.2, the set int M is nonempty and convex. On the other hand, by assumption, (O Y 1 , O y 2 )EM. Using a well-known separation theorem (see for instance

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Zeidler, Ref. 18), the sets { O y } and int M can be separated by a hyperplane; i.e., there exists y* = (y 1 *,y 2 *)e Y* x Y*= Y*, with y*=Oy*, such that

We show that the relation

holds; i.e., the same hyperplane separates the sets {OY} and M. Supposing the contrary, there exists u 0 eM such that y*(u0) Y by Since this map is continuous, then the set U := h - 1 ( V ) is an open neighborhood of uL. It is obvious that U R be a discontinuous linear functional and let g: X->Y 2 be the identity map. Then, F = ( f , g): X->R x Y2 is K-convexlike and there is no xeX such that

However, if y* e R+ and y* e Y*, with ( y * , y*) = (0, O y * ), are such that then

which taking into account the discontinuity of / leads to the contradiction

4. Optimality Conditions for Convexlike Programming

Let X, Yi, Ki, i= 1, 2, be as in Sections 2 and 3, and let

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Consider the problem

Let

Then, we have the following theorem. Theorem 4.1. Suppose that F=(f0, f, g): X->Y is K-convexlike and there exists u e R such that (u, O Y1 , O Y 2 ) e i n t ( F ( X ) + K). If x is an optimal solution of (P), then there exist y * e K ' and y*eK' such that

and the following complementarity condition holds:

Proof. The function F: X -> Y, given by

satisfies the assumptions of Theorem 3.1. By the optimality of x, there is no xeX such that

Therefore, there is no xeX such that

By Theorem 3.1(i), with R+ x K1 instead of K1 and ( f 0 - f 0 ( x ) , f ) /, there exist L*eR+, y * e K ' y * e K ' , not all zeros, such that

instead of

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It is easy to show that L * = 0 . Indeed, if A* = 0, then by (9) we have

where ( y * , y * ) = ( O Y * , O y * ), which implies that the sets are separated by a hyperplane. On the other hand, this separation is impossible, since (Oy1, O y2 )eint S. Dividing (9) by L*, we obtain (8) with The complementarity condition can be deduced easily also from (9), if we put x = x and take into account that f ( x ) e K 1 . D 5. Applications In this section, we present several consequences of Theorems 3.1 and 4.1.

Let X, Yi, i=1, 2, K1, be as in Section 3, and let f0: X->R, f : X - > Y1, g : X - > Y 2 . Consider the problem

and the system (S1)

there is a nonzero vector ( L * , y * , y * ) e R + x K* x Y* such that

Then, by Theorem 3.1, we obtain the following theorem of the alternative due to Tamminen (Ref. 9). Corollary 5.1. See Ref. 9, Proposition 4.1. Suppose that the following assumptions hold: (i)

for each x1, x2eX and Le[0, 1], there exists x3eX such that

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(ii) g(X) has a nonempty interior in Y2; (iii) there is an open set U < i n t g(X) and a point (r, y ) e R x Y1 such that, for every zeU, there is an xeX satisfying Then, we have that: (a) if Problem (P1) has no solution, then system (S1) has a solution; (b) if system (S1) has a solution (L*,y* ,y*) and L*>0, then Problem (P1) has no solution. Proof.

Take

It is clear that F is K-convexlike, where By the regularity conditions (ii) and (iii), it follows that the set F(X) + K has nonempty interior. Now, the result follows by Theorem 3.1. D Let X be a nonempty set, let Z be a topological vector space over the reals, and let C c Z be a convex cone with vertex Oz and such that int C=0. Consider a function f: X -> Z. Then, by Theorem 3.1, we obtain the following theorem of the alternative. Corollary 5.2. See Craven and Jeyakumar (Ref. 5). Suppose that / is C-convexlike. Then, exactly one of the following statements hold: (i)

there exists xeX such that f(x)e-int C;

(ii) there exists z*eC'\{O*} such that z*(f(x))>0, VxeX.

(10) (11)

Proof. Take Then, for g(x) = Oz, VxeX, the pair F= (f, g): X -> Y1 x y2 is K-convexlike, where K = K 1 x K 2 . Since the set f(X) + int C is open, we also have int(F(X) + K ) = P . The assertion follows by Theorem 3.1. D We mention that the results of Hayashi and Komiya (Ref. 1, Lemma 2.1), Zalinescu (Ref. 8, Lemma 3) and Craven (Ref. 12, Theorem 2.5.1) are particular cases of Corollary 5.2.

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Recall that, for A, B=P and L: A x B-> R, the pair (x, y) R and f: X -> Z. Next, we state the following strong duality result for convexlike programming problems. Corollary 5.3. Let Z be a locally convex space over the reals and let CcZ be a closed convex cone with vertex Oz and int C=P. Suppose that and that the Slater condition is satisfied, i.e., 3x0eX such that f(x0) e —int C. Let xeX. Then, the following assertions are equivalent: (i)

x is an optimal solution of (P2);

(12)

(ii)

3yeC' s.t. (x, y) is a saddle point of the Lagrangian L associated to (P2), (13)

Proof. (12) => (13). Apply Theorem 4.1 to

Note that the Slater condition implies the following regularity condition:

it can be seen that, in this case, Problem (P) reduces to (P2). (13) => (12). Since by (13) we have

taking into account that Z is locally convex space and that C is closed convex cone, it follows that f(x)e-C. Indeed, if f(x)E-C, then the convex sets {—f(x)} and C can be strictly separated by a hyperplane. Since C is

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closed, we have OzeC; thus, there exists y*eC'\{O*} such that y*(f(x))>0, which contradicts (14). Now, (12) follows by L(x, y)R2 be defined by

Consider the problem

It can be easily seen that the pair (cos x, | sin x| - R2/2) is K-convexlike for K = R 2 . The Slater condition is also satisfied. Hence, Corollary 5.3 can be applied for this problem. We have x = P solution of the problem and (P, 0) the saddle point of the Lagrangian. Duality results for convex programming problems can be found, for instance, in Ref. 21. By Corollary 5.1, we obtain the following result. Corollary 5.4. See Simons, Ref. 2, Theorem 1.3. Let X be a nonempty set, let T be a compact Hausdorff topological space, let f0: X ->R, h: T x X - > R be such that h ( . , x) is continuous for each xeX. Suppose further that, for each x1, x2eX, there exists x 3 eX such that

Let

and let x eA be such that

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Then, there exist Le[0, 1] and a measure v on T such that L+ ||v|| = 1 and

Proof.

Let

Take

By the assumption, there is no xeX such that

Now applying Corollary 5.2 to the function f = ( f 0 - f 0 ( x ) , obtain the assertion.

h): X->Z, we D

6. Extension of the Main Results In this section, we extend Theorems 3.1 and 4.1 to a broad new class of (t, OY2; K)-convexlike functions. As in Section 2, let X be a nonempty set, let Y1 and Y2 be topological vector spaces over the reals, let Ki c Yi i = 1, 2, be two convex cones with vertices OY1 and OY2 such that int K1=P, and let f: X->Y 1 , g: X-> Y2. Put

The function F is (t, Oy2; K)-convexlike if there exist t e K 1 and ae(0, 1) such that, for each x1, x2eX and e>0 real number, there exists x 3 eX with

The following basic convexity property holds.

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Lemma 6.1. Let us assume that F:X-> Y is a (t, OY2; K)-convexlike function. Then, the set

is nearly convex set and int M is a convex set. Proof. The nearly-convexity of M follows from the definition of the (t, OY2; K)-convexlike property of the function by elementary computations. Then, the interior of M is convex by Lemma 2.1. D Now, we are ready to generalize Theorem 3.1. Theorem 6.1. Suppose that F=(f,g): X - > Y is (t, OY2; K)-convexlike and the set F(X) + K has a nonempty interior. Then, the following assertions hold: (i)

If there is no xeX such that

then there exists y * € K ' and y * e K ' , with ( y * , y * ) = ( O Y * , such that

(ii)

Oy*),

If there exist y*eK', \{O y * } and y * e K ' such that (17) holds, then there is no xeX such that (15) and (16) hold.

Proof.

Based on the observation stated in the previous lemma, i.e.,

is a nearly convex set and its interior is convex, every step of the proof of Theorem 3.1 can be repeated for this more general case, as well. D Let us consider Problem (P) as in Section 4. Then, a result similar to Theorem 4.1 follows immediately from the previous theorem. If

then the triplet F=(f0, f, g ) : X - > Y is said to be (T, t, OY2; K)-convexlike if there exist T>0, teK 1 , ae(0, 1) such that, for each X1, x2eX and e>0 real

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number, there exists x 3 eX with

Now, we can state the following corollary. Corollary6.1. Suppose that F = ( f 0 , f, g ) : X - > Y is (T, t, OY2;K)-convexlike and there exists ueR such that (u, O Y1 , Oy2) e int(F(X) + K). If x is an optimal solution of Problem (P), then there exist y*eK' and y*EK' such that

and the following complementarity condition holds:

Theorem 6.1 and Corollary 6.1 generalize results due to Craven and Jeyakumar (Ref. 5) related to subconvexlike functions.

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