Generalized Convex Alternative Theorems and Cone Constrained Optimization Maria Caridad Natividad A thesis submitted for the degree of Master of Science in the School of Mathematics, University of New South Wales, Australia July 1992 Abstract This thesis studies nonlinear alternative theorems and their applications to cone constrained optimization problems. We examine alternative theo­ rems involving generalized cone convex functions. The different approaches in proving an alternative theorem and the relationships with other funda­ mental results in optimization and mathematical programming are studied. We also study the role of alternative theorems in the derivation of necessary optimality conditions and in the development of the duality theory. A new class of quasiconvex mappings, called *-quasiconvex, is defined and studied for which an alternative theorem is proved using a minimax theorem. This alternative theorem is applied to study cone constrained optimization problems involving *-quasiconvex mappings. Thus, we develop generalized Lagrangian optimality and duality results and therefore extend the funda­ mental Lagrangian results for convex programming problems to a new class of quasiconvex problems. Moreover, we also obtain necessary optimality con­ ditions for a class of nondifferentiable programs using upper approximations. Recently, Jeyakumar and Gwinner [30] established new versions of alter­ native theorems involving nonconvex inequality systems which use a new local closedness condition. From these theorems, approximate Lagrange multiplier results, zero duality gap property and certain stability results were derived. Here, we examine these versions of alternative theorems for generalized cone convex systems and apply the theorems to study the results that follow from these alternative theorems for cone constrained optimization problems. We study the relationship between the stability of the primal problem and the zero duality gap property and the characterization of the t-saddlepoints using t-subdifferentials of the value function of the primal problem at the point of no perturbation. Finally, we examine a stable alternative theorem which also gives a sufficient condition for the local closedness assumption. 11 Contents . Abstract 1 Contents iii Acknowledgements VI 1 Introduction 1 2 Preliminaries 5 2.1 Convexity of Sets and Functions . 6 2.2 Generalized Convexity . 8 2.3 Continuity and Subdifferentiability of Functions . 10 2.4 Linear Alternative Theorems and their Applications . 12 l1l 3 Approaches to Nonlinear Alternative Theorems 16 3.1 Introduction . 16 3.2 Nonlinear Alternative Theorems via Separation Theorems 17 3.3 An Alternative Theorem via a Minimax Theorem . 21 3.4 An Alternative Theorem via a Lagrangian Theorem . 26 3.5 Optimality Conditions and Duality . 28 4 An Alternative Theorem for *-quasiconvex Mappings and N ondifferentiable Optimization 33 4.1 Introduction ........................... 33 4 . 2 *-quas1convex . M appmgs. 34 4.3 An Alternative Theorem for *-quasiconvex Mappings . 39 4.4 Generalized Lagrangian Theorems for *-quasiconvex Cone Con­ strained Problems . ·. 42 4.5 Necessary Optimality Conditions for a Class of Nondifferen­ tiable Programs . 44 4.6 Upper Approximations and *-quasiconvexity . 46 IV 5 t-Alternative Theorems and Zero Duality Gaps 51 5.1 Introduction . 51 5.2 t-Alternative Theorems . 53 5.3 Approximate Lagrange Multipliers and Zero Duality Gaps . 57 5.4 Nearly Stable Problems . 61 5.5 A Stable Alternative Theorem . 64 Bibliography 67 V Acknowledgements I would like to express my deepest gratitude to my supervisor Dr. Vaithilingam Jeyakumar for his untiring guidance and support through­ out the course of my study and for teaching me not only mathematics but also what dedication is all about. Some of the results in Chapter 4 of this thesis were obtained in collaboration with him and Professor W. Oettli and appeared in [31]. I would also like to thank Esther Nababan and Xin Tian for helping me in the material preparations of the thesis. I am also indebted to my family for their support and patience and to my friends including Xiaojun Chen and Sadhana Subramani for making my stay in Sydney an enjoyable experience. The thesis would be impossible without the financial assistance of the Association for Educational Projects Ltd.. I am deeply grateful for their support. Finally, I would like to thank the School of Mathematics for providing excellent facilities and a stimulating working environment. Ad Dominam Nostram, Sedes Sapientiae. Vl Chapter 1 Introduction Alternative theorems play an important role in the derivation of necessary optimality conditions, in the development of the Lagrangian duality theory and in the scalarization of vector-valued optimization problems. These the­ orems have been extensively studied by various authors in the past three decades. One of the earliest linear alternative theorems was obtained by Farkas in 1902 (see (35]) and one of the first versions of the nonlinear case which involves convex functions was given by Fan, Glicksberg and Hoffman in 1957 (see (11]). Since then, different versions of alternative theorems (that is, in finite or infinite dimensional cases and involving convex or generalized con­ vex functions ) have been established in the literature (see (5], (7], (25), (28], (19] and the references therein). A standard way of proving an alternative theorem is via a separation theorem. Recently, it has been shown that under certain conditions, an alternative theorem can be proved using a minimax theorem (see (28), (8) and the references therein) and that under appropriate conditions, alternative, minimax and Lagrangian theorems are closely related to each other. This thesis studies nonlinear alternative theorems (also called solvability theorems or transposition theorems) involving generalized cone­ convex functions which include a new class of generalized convex functions and their applications to cone constrained optimization problems. 1 Alternative theorems which hold for systems of convex functions may not be generally extended to systems of quasiconvex functions (see [14]). There­ fore results that follow from an alternative theorem such as necessary opti­ mality conditions and the development of the standard Lagrangian duality theory for optimization problems may not be obtained for programming prob­ lems involving quasiconvex functions. In chapter 4 of this thesis, we study a new class of quasiconvex mappings, called *-quasiconvex, for which an al­ ternative theorem can be proved using a version of the minimax theorem for quasiconcav~onvex functions of Sion [41], and develop optimality and du­ ality theory for optimization problems involving *-quasiconvex mappings by applying the alternative theorem. Some of these results appear in the recent report by Jeyakumar, Oettli and Natividad [31]. On the other hand, it is known that the existence of a Lagrange multiplier plays a vital role in the development of the duality theory and the stability of optimization problems. Very recently, Jeyakumar and Gwinner [30] and Jeyakumar and Wolkowicz [32] studied nonconvex optimization problems in­ volving inequality systems for which no Lagrange multiplier exists yet duality gap is zero between the primal problem and the corresponding dual problem (that is, the optimal values of the primal and dual problems are equal) and the primal problem satisfies a certain approximate stability property. This was done by establishing new versions of alternative theorems using a new local closedness condition. In chapter 5, we establish these versions of alter­ native theorems for generalized cone-convex systems and apply the theorems to study zero duality gap results for cone constrained optimization problems. This development shows a new connection between alternative theorems and zero duality gap property. We now give an outline of the thesis. In the second chapter, basic results and definitions used throughout the thesis are presented. We also discuss how linear alternative theorems are used to study differentiable optimization problems. In passing, we also examine the role of linear alternative theo- 2 rems in the characterization of the weakened invexity of a finite dimensional programming problem. The third chapter examines various versions of alternative theorems and their relationship with other fundamental results in optimization and the dif­ ferent approaches in proving alternative theorems. The chapter begins by presenting versions of Gordan, Farkas and Motzkin alternative theorems for S-convex, S-convexlike and S-subconvexlike cone functions. Then we study the different ways of proving alternative theorems. We examine how a Basic (or Gordan type) Alternative Theorem is derived using a separation theorem (see [l], [4] [11], [25],[29], [19]), minimax theorem (see [5], [7],[8],[28],[24]), and Lagrangian theorem (see (28], (26]). We see that minimax, alternative and Lagrangian theorems are equivalent under appropriate conditions ( as shown in (28)). Thus, each of these can be derived directly from each other with­ out the use of a separation theorem. Then we shall see how an alternative theorem is used to derive necessary optimality conditions and to develop the Lagrangian duality theory. To complete this chapter we also show how alter­ native theorems are used in the derivation of necessary optimality conditions for nondifferentiable optimization problems. As we have mentioned earlier, an alternative theorem does not, in general, hold for quasiconvex