optimality conditions

Weak Stationarity: Eliminating the Gap between Necessary and Sufficient Conditions

Categories Nonsmooth Optimization Tags , , , , ,

Starting from known necessary extremality conditions in terms of strict subdifferentials and normals the notion of weak stationarity is introduced. It is defined in terms of initial space elements. The necessary conditions become necessary and sufficient (for stationarity). CitationSchool of Information Technology and Mathematical Sciences, Centre of Information and Applied Optimization, University of Ballarat, POB … Read more

On a class of nonsmooth composite functions

Categories Convex and Nonsmooth Optimization, Nonsmooth Optimization Tags , , , , ,

We discuss in this paper a class of nonsmooth functions which can be represented, in a neighborhood of a considered point, as a composition of a positively homogeneous convex function and a smooth mapping which maps the considered point into the null vector. We argue that this is a sufficiently rich class of functions and … Read more

Optimality Conditions for Vector Optimization with Set-Valued Maps

Categories Generalized Convexity/Monoticity, Multi-Criteria Optimization, Nonlinear Optimization Tags , , , , ,

Based on near convexity, we introduce the concepts of nearly convexlike set-valued maps and nearly semiconvexlike set-valued maps, give some charaterizations of them, and investigate the relationships between them. Then a Farkas-Minkowski type alternative theorem is shown under the assumption of near semiconvexlikeness. By using the alternative theorem and some other lemmas, we establish necessary … Read more