Stable and total Fenchel duality for convex optimization problems in locally convex spaces

We consider the optimization problem (PA) infx∈X{f(x) + g(Ax)} where f and g are proper convex functions defined on locally convex Hausdorff topological vector spaces X and Y, respectively, and A is a linear operator from X to Y . By using the properties of the epigraph of the conjugated functions, some sufficient and necessary conditions for the strong Fenchel duality and the strong converse Fenchel duality of (PA) are provided. Sufficient and necessary conditions for the stable Fenchel duality and for the total Fenchel duality are also derived.National Natural Science Foundation of ChinaDirección General de Enseñanza SuperiorMinisterio de Ciencia e Innovació

Stable and total Fenchel duality for DC optimization problems in locally convex spaces

Author name used in this publication: X. Q. Yang.2011-2012 > Academic research: refereed > Publication in refereed journalVersion of RecordPublishe

Set optimization - a rather short introduction

Recent developments in set optimization are surveyed and extended including various set relations as well as fundamental constructions of a convex analysis for set- and vector-valued functions, and duality for set optimization problems. Extensive sections with bibliographical comments summarize the state of the art. Applications to vector optimization and financial risk measures are discussed along with algorithmic approaches to set optimization problems

Stable strong Fenchel and Lagrange duality for evenly convex optimization problems

By means of a conjugation scheme based on generalized convex conjugation theory instead of Fenchel conjugation, we build an alternative dual problem, using the perturbational approach, for a general optimization one defined on a separated locally convex topological space. Conditions guaranteeing strong duality for primal problems which are perturbed by continuous linear functionals and their respective dual problems, which is named stable strong duality, are established. In these conditions, the fact that the perturbation function is evenly convex will play a fundamental role. Stable strong duality will also be studied in particular for Fenchel and Lagrange primal–dual problems, obtaining a characterization for Fenchel case.This research was partially supported by MINECO of Spain and FEDER of EU, [grant numberMIM2014-59179-C2-1-P]; Conselleria de la Educacion de la Generalitat Valenciana, Spain, Pre-doc Program Vali+d, DOCV 6791/07.06.2012, [grant number ACIF-2013-156]

E′-Convex Sets and Functions: Properties and Characterizations

The main properties of evenly convex sets and functions have been deeply studied by different authors, and a duality theory for evenly convex optimization problems has been well developed as well. In this theory, the notion of e′-convexity appears as a necessary requirement for obtaining important results in strong and stable strong duality. This fact has motivated the authors to study possible properties of this kind of convexity in sets and functions, which is closely connected to even convexity

New duality results for evenly convex optimization problems

We present new results on optimization problems where the involved functions are evenly convex. By means of a generalized conjugation scheme and the perturbation theory introduced by Rockafellar, we propose an alternative dual problem for a general optimization one defined on a separated locally convex topological space. Sufficient conditions for converse and total duality involving the even convexity of the perturbation function and c-subdifferentials are given. Formulae for the c-subdifferential and biconjugate of the objective function of a general optimization problem are provided, too. We also characterize the total duality by means of the saddle-point theory for a notion of Lagrangian adapted to the considered framework.Research partially supported by MINECO of Spain and ERDF of EU, Grant MTM2014-59179-C2-1-P, Austrian Science Fund (FWF), Project M-2045, and German Research Foundation (DFG), Project GR3367/4-1