Optimality Conditions and Characterizations of the Solution Sets in Generalized Convex Problems and Variational Inequali
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Optimality Conditions and Characterizations of the Solution Sets in Generalized Convex Problems and Variational Inequalities Vsevolod I. Ivanov
Received: 5 February 2012 / Accepted: 1 December 2012 / Published online: 13 December 2012 © Springer Science+Business Media New York 2012
Abstract We derive necessary and sufficient conditions for optimality of a problem with a pseudoconvex objective function, provided that a finite number of solutions are known. In particular, we see that the gradient of the objective function at every minimizer is a product of some positive function and the gradient of the objective function at another fixed minimizer. We apply this condition to provide several complete characterizations of the solution sets of set-constrained and inequality-constrained nonlinear programming problems with pseudoconvex and second-order pseudoconvex objective functions in terms of a known solution. Additionally, we characterize the solution sets of the Stampacchia and Minty variational inequalities with a pseudomonotone-star map, provided that some solution is known. Keywords Global optimization · Pseudoconvex function · Variational inequality · Pseudomonotone-star map
1 Introduction Characterizations of the solution sets of the nonlinear programming problems with multiple solutions, provided that a fixed minimizer is known, play important role in optimization. Such results were first obtained for convex problems by Mangasarian [1]. Further investigation has been done by Burke and Ferris [2], where the objective function is extended real-valued, proper and convex, by Jeyakumar [3] (infinite dimensional convex program), by Jeyakumar and Yang [4] (convex composite multiobjective problem). In 1995, Jeyakumar and Yang [5] gave new characterizations for Communicated by Vaithilingam Jeyakumar. This research is partially supported by the TU Varna Grant No. 18/2012. V.I. Ivanov () Department of Mathematics, Technical University of Varna, 9010 Varna, Bulgaria e-mail: [email protected]
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J Optim Theory Appl (2013) 158:65–84
pseudolinear problems, which were extended to the nondifferentiable case by Lalitha and Mehta [6], and Smietanski [7]. Some results of this type in the case where the function is pseudoconvex, have been derived independently by Ivanov [8] and Wu [9]. Bianchi and Schaible [10] extended the results of Jeyakumar and Yang [5] to obtain characterizations of the solution set of PPM variational inequality problem. Similar characterizations for the nondifferentiable case were given by Lalitha and Mehta [6]. Jeyakumar and Yang’s type results involving higher-order Dini directional derivatives were obtained by Ivanov [11] for higher-order pseudoconvex problems. In some more papers, Mangasarian’s type characterizations were derived for convex problems (see Wu and Wu [12], Jeyakumar, Lee and Dinh [13]), but to our knowledge nobody has extended the Mangasarian results to nonconvex functions. On the other hand, some similar results concerning variational inequality problems appeared in the papers
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