ABSTRACT
The characterizations and the properties of the solution set related to an optimization problem having multiple optimal solutions are of fundamental importance in understanding the behavior of solution methods. Mangasarian [177] has presented simple and elegant characterizations for the solution set of convex extremum problems with one solution known. These results have been further extended to various classes of optimization problems such as infinite dimensional convex optimization problems (Jeyakumar and Wolkowicz [117], Jeyakumar [114]), generalized convex optimization problems (Burke and Ferris [32], Penot [220], Jeyakumar et al. [115]), and convex vector optimization problems (Jeyakumar et al. [116]). Jeyakumar and Yang [119] have obtained the characterizations for the solution set of a differentiable pseudolinear programming problem by using the basic properties of pseudolinear functions. Lu and Zhu [171] have established some characterizations for locally Lipschitz pseudolinear functions and the solution set of a pseudolinear programming problem using the Clarke subdifferential on Banach spaces. Recently, Lalitha and Mehta [155, 156] introduced the notion of h-pseudolinear functions and derived the characterizations for the solution set of h-pseudolinear programming problems. Very recently, Smietanski [257] has derived some characterizations of directionally differentiable pseudolinear functions, which are not necessarily differentiable. Using these characterizations, Smietanski [257] obtained several characterizations for the solution sets of constrained non-smooth optimization problems involving directionally differentiable pseudolinear functions.
