Stable strong Fenchel and Lagrange duality for evenly convex optimization problems

Abstract

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 disturbed primal problems by continuous linear functionals and their respective dual problems, which is named stable strong duality, are stablished. In these conditions, the evenly convexity of the perturbation function 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.

Closedness Type Regularity Conditions in Convex Optimization and Beyond

“…Closedness type regularity conditions were employed by different authors in other related research fields, too, like subdifferential calculus (e.g., by [10,18,29,[50][51][52]), DC programming (e.g., by [51,[53][54][55][56][57]), generalized convex optimization (e.g., by [58][59][60]), semiinfinite programming (e.g., by [15,16,50,53,[61][62][63][64]), semidefinite programming (e.g., by [43,45,46,65]), robust optimization (e.g., by [63,66,67]), location optimization (e.g., by [68]), vector optimization (e.g., by [10,29,63,64,69]), monotone operators ( [17,[70][71][72][73][74][75][76][77][78]), machine learning ( [79]) or variati...…”

New duality results for evenly convex optimization problems

“…Moreover, in [ 17 ] regularity conditions for strong duality between an e-convex problem and its Lagrange dual were established. In [ 18 ] we analysed the problem of stable strong duality, and deduced Fenchel and Lagrange type duality statements for unconstrained and constrained optimization problems, respectively. In [ 19 ] the Fenchel–Lagrange dual problem of a (primal) minimization problem, whose involved functions do not need to be e-convex a priori, was derived.…”

E′-Convex Sets and Functions: Properties and Characterizations

“…In [9] Fenchel-Lagrange duality is considered, where dual problems are expressed via the c-conjugates of the functions involved in the primal problem. [7] studies sufficient conditions and characterizations for stable strong duality in this generalized framework for Fenchel and Lagrange dualities. In addition, a comparison of the optimal values and solutions of the three alternative dual problems (Fenchel, Lagrange and Fenchel-Lagrange) is achieved in [8].…”