Special Pair of Dual Parametric Nonlinear Optimization Problems with Common Set of Optimal Points

Abstract

On the base of a given strictly convex function defined on the Euclidean space E n ( n S 2) we can-without the assumption that it is differentiable - introduce some manifolds in topologic sense. Such manifolds are sets of all optimal points of a certain parametric non-linear optimization problem. This paper presents above all certain generalization of some results of [F. No − i ) ka and L. Grygarová (1991). Some topological questions connected with strictly convex functions. Optimization , 22 , 177-191. Akademie Verlag, Berlin] and [L. Grygarová (1988). Über Lösungsmengen spezieller konvexer parametrischer Optimierungsaufgaben . Optimization 19 , 215-228. Akademie Verlag Berlin], under less strict assumptions. The main results are presented in Sections 3 and 4, in Section 3 the geometrical characterization of the set of optimal points of a certain parametric minimization problem is presented; in Section 4 we study a maximization non-linear parametric problem assigned to it. It seems that it is a certain pair of parametric optimization problems with the same set of their optimal points, so that this pair of problems can be denoted as a pair of dual parametric non-linear optimization problems. This paper presents, most of all in Section 2, a number of interesting geometric facts about strictly convex functions. From the point of view of non-smooth analysis the present article is a certain complement to Chapter 4.3 of the book [B. Bank, J. Guddat, D. Klatte, B. Kummer and K. Tammer (1982). Nonlinear Parametric Optimization . Akademie Verlag, Berlin] where a convex parametric minimization problem is considered under more general and stronger conditions (but without any assumptions concerning strict convexity and without geometrical aspects).