http://repub.eur.nl/res/aut/392/ List of Publications en http://repub.eur.nl/static-eur/img/logo.png http://repub.eur.nl http://repub.eur.nl/res/pub/19268/ 2007-08-01T00:00:00Z In this paper, we consider first the most important classes of cone convexlike vector-valued functions and give a dual characterization for some of these classes. It turns out that these characterizations are strongly related to the closely convexlike and Ky Fan convex bifunctions occurring within minimax problems. Applying the Lagrangian perturbation approach, we show that some of these classes of cone convexlike vector-valued functions show up naturally in verifying strong Lagrangian duality for finite-dimensional optimization problems. This is achieved by extending classical convexity results for biconjugate functions to the class of so-called almost convex functions. In particular, for a general class of finite-dimensional optimization problems, strong Lagrangian duality holds if some vector-valued function related to this optimization problem is closely K-convexlike and satisfies some additional regularity assumptions. For K a full-dimensional convex cone, it turns out that the conditions for strong Lagrangian duality simplify. Finally, we compare the results obtained by the Lagrangian perturbation approach worked out in this paper with the results achieved by the so-called image space approach initiated by Giannessi. http://repub.eur.nl/res/pub/19261/ 2007-03-01T00:00:00Z In this paper we discuss necessary and sufficient conditions for different minimax results to hold using only linear programming duality and the finite intersection property for compact sets. It turns out that these necessary and sufficient conditions have a clear interpretation within zero-sum game theory. We apply these results to derive necessary and sufficient conditions for strong duality for a general class of optimization problems. http://repub.eur.nl/res/pub/7809/ 2006-06-10T00:00:00Z In this chapter we give an overview on the theory of noncooperative games. In the first part we consider in detail for zero-sum (and constant-sum) noncooperative games under which necessary and sufficient conditions on the payoff function and different (extended) strategy sets for both players an equilibrium saddlepoint exists. This is done by using the most elementary proofs. One proof uses the separation result for disjoint convex sets, while the other proof uses linear programming duality and some elementary properties of compact sets. Also, for the most famous saddlepoint result given by Sion's minmax theorem an elementary proof using only the definition of connectedness is given. In the final part we consider n-person nonzero-sum noncooperative games and show by a simple application of the KKM lemma that a so-called Nash equilibrium point exists for compact strategy sets and concavity conditions on the payoff functions. http://repub.eur.nl/res/pub/7756/ 2006-05-11T00:00:00Z In this chapter we give an overview on the theory of noncooperative games. In the first part we consider in detail for zero-sum (and constant-sum) noncooperative games under which necessary and sufficient conditions on the payoff function and different (extended) strategy sets for both players an equilibrium saddlepoint exists. This is done by using the most elementary proofs. One proof uses the separation result for disjoint convex sets, while the other proof uses linear programming duality and some elementary properties of compact sets. Also, for the most famous saddlepoint result given by Sion's minmax theorem an elementary proof using only the definition of connectedness is given. In the final part we consider n-person nonzero-sum noncooperative games and show by a simple application of the KKM lemma that a so-called Nash equilibrium point exists for compact strategy sets and concavity conditions on the payoff functions. http://repub.eur.nl/res/pub/19273/ 2006-01-01T00:00:00Z In this paper we discuss the level set method of Joó and how to use it to give an elementary proof of the well-known minimax theorem of Sion. Although this proof technique was initiated by Joó and based on the intersection of upper level sets and a clever use of the topological notion of connectedness, it is not very well known and accessible for researchers in optimization. At the same time we simplify the original proof of Joó and give a more elementary proof of the celebrated minimax theorem of Sion. http://repub.eur.nl/res/pub/19414/ 2005-06-01T00:00:00Z In this note we review some known minimax theorems with applications in game theory and show that these results form an equivalent chain which includes the strong separation result in finite dimensional spaces between two disjoint closed convex sets of which one is compact. By simplifying the proofs we intend to make the results more accessible to researchers not familiar with minimax or noncooperative game theory. http://repub.eur.nl/res/pub/6558/ 2005-06-01T00:00:00Z In this note we review some known minimax theorems with applications in game theory and show that these results form an equivalent chain which includes the strong separation result in finite dimensional spaces between two disjoint closed convex sets of which one is compact. By simplifying the proofs we intend to make the results more accessible to researchers not familiar with minimax or noncooperative game theory. http://repub.eur.nl/res/pub/1931/ 2005-04-03T00:00:00Z In this paper we will show that the closely K-convexlike vector-valued functions with K Rm a nonempty convex cone and related classes of vector-valued functions discussed in the literature arise naturally within the theory of biconjugate functions applied to the Lagrangian perturbation scheme in finite dimensional optimization. For these classes of vectorvalued functions an equivalent characterization of the dual objective function associated with the Lagrangian is derived by means of a dual representation of the relative interior of a convex cone. It turns out that these characterizations are strongly related to the closely convexlike and Ky-Fan convex bifunctions occurring within minimax problems. Also it is shown for a general class of finite dimensional optimization problems that strong Lagrangian duality holds in case a vector-valued function related to the functions in this optimization problem is closely K-convexlike and satisfies some additional regularity condition. http://repub.eur.nl/res/pub/1611/ 2004-09-17T00:00:00Z In the first chapter of this book the basic results within convex and quasiconvex analysis are presented. In Section 2 we consider in detail the algebraic and topological properties of convex sets within Rn together with their primal and dual representations. In Section 3 we apply the results for convex sets to convex and quasiconvex functions and show how these results can be used to give primal and dual representations of the functions considered in this field. As such, most of the results are well-known with the exception of Subsection 3.4 dealing with dual representations of quasiconvex functions. In Section 3 we consider applications of convex analysis to noncooperative game and minimax theory, Lagrangian duality in optimization and the properties of positively homogeneous evenly quasiconvex functions. Among these result an elementary proof of the well-known Sion’s minimax theorem concerningquasiconvex-quasiconcave bifunctions is presented, thereby avoiding the less elementary fixed point arguments. Most of the results are proved in detail and the authors have tried to make these proofs as transparent as possible. Remember that convex analysis deals with the study of convex cones and convex sets and these objects are generalizations of linear subspaces and affine sets, thereby extending the field of linear algebra. Although some of the proofs are technical, it is possible to give a clear geometrical interpretation of the main ideas of convex analysis. Finally in Section 5 we list a short and probably incomplete overview on the history of convex and quasiconvex analysis. http://repub.eur.nl/res/pub/1537/ 2004-08-30T00:00:00Z In this paper we discuss the level set method of Joó and how to use it to give an elementary proof of the well-known Sion’s minimax result. Although this proof technique is initiated by Joó and based on the inter-section of upper level sets and a clever use of the topological notion of connectedness, it is not very well known and accessible for researchers in optimization. At the same time we simplified the original proof of Joó and give a more elementary proof of the celebrated Sion’s minimax theorem. http://repub.eur.nl/res/pub/1219/ 2004-04-14T00:00:00Z In this paper we discuss necessary and sufficient conditions for different minimax results to hold using only linear programming duality and the finite intersection property of compact sets. It turns out that these necessary and sufficient conditions have a clear interpretation within zero-sum game theory. In the last section we apply these results to derive necessary and sufficient conditions for strong Lagrangean duality for a large class of optimization problems. http://repub.eur.nl/res/pub/240/ 2002-10-22T00:00:00Z In this paper the well-known minimax theorems of Wald, Ville and Von Neumann are generalized under weaker topological conditions onthe payoff function ƒ and/or extended to the larger set of the Borel probabilitymeasures instead of the set of mixed strategies. http://repub.eur.nl/res/pub/158/ 2002-01-24T00:00:00Z In this paper we review known minimax results with applications in game theory and show that these results are easy consequences of the first minimax result for a two person zero sum game with finite strategy sets published by von Neumann in 1928: Among these results are the well known minimax theorems of Wald, Ville and Kneser and their generalizations due to Kakutani, Ky-Fan, König, Neumann and Gwinner-Oettli. Actually it is shown that these results form an equivalent chain and this chain includes the strong separation result in finite dimensional spaces between two disjoint closed convex sets of which one is compact. To show these implications the authors only use simple properties of compact sets and the well-known Weierstrass Lebesgue lemma. http://repub.eur.nl/res/pub/6845/ 2001-08-27T00:00:00Z In this paper which will appear as a chapter in the Handbook of Generalized Convexity we discuss the basic ideas of convex and quasiconvex analysis in finite dimensional Euclidean spaces. To illustrate the usefulness of this branch of mathematics also applications to optimization theory and noncooperative game theory are considered. http://repub.eur.nl/res/pub/11536/ 1999-08-01T00:00:00Z In this paper, we introduce several classes of generalized convex functions already discussed in the literature and show the relation between these classes. Moreover, a Gordan–Farkas type theorem is proved for all these classes and it is shown how these theorems can be used to verify strong Lagrangian duality results in finite-dimensional optimization. http://repub.eur.nl/res/pub/1528/ 1998-11-26T00:00:00Z In this paper we review and unify some of the classes of generalized convex functions introduced by different authors to prove minimax results in infinite dimensional spaces and show the relations between those classes. We also list for the most general class already introduced by Jeyakumar an elementary proof of a minimax result. The proof of this result only uses a finite dimensional separation theorem and although this minimax result is already presented by Neumann and independently Jeyakumar we believe that the present proof is shorter and more transparent. http://repub.eur.nl/res/pub/1401/ 1997-01-01T00:00:00Z In this paper we introduce several classes of generalized convex functions already discussed in the literature and show the relation between those function classes. Moreover, for some of those function classes a Farkas-type theorem is proved. As such this paper unifies and extends results existing in the literature and shows how these results can be used to verify Farkas-type theorems and strong Lagrangian duality results in finite dimensional optimization.