Closedness Type Regularity Conditions in Convex Optimization and Beyond

Abstract

The closedness type regularity conditions have proven during the last decade to be viable alternatives to their more restrictive interiority type counterparts, in both convex optimization and different areas where it was successfully applied. In this review article we de-and reconstruct some closedness type regularity conditions formulated by means of epigraphs and subdifferentials, respectively, for general optimization problems in order to stress that they arise naturally when dealing with such problems. The results are then specialized for constrained and unconstrained convex optimization problems. We also hint toward other classes of optimization problems where closedness type regularity conditions were successfully employed and discuss other possible applications of them.

New duality results for evenly convex optimization problems

“…In this paper we present new results regarding evenly convex (e-convex) functions, in particular converse and total duality statements for e-convex problems that extend their counterparts from the (classical) convex case. Other results can be generalized to the current setting as well, for instance the ϵ -duality statements from [ 29 ], however the proofs work straightforwardly and present no difficulty so we opted not to include them here. On the other hand, some results known at the moment for proper, convex and lower semicontinuous functions, such as the maximal monotonicity of their subdifferentials or the fact that their proximal point operators are single valued, do not hold in general for e-convex functions – check for instance the function considered in [ 19 , Ex.…”

Characterizations of Robust and Stable Duality for Linearly Perturbed Uncertain Optimization Problems

“…Remark 6.1 Condition (ii) in Corollary 6.1 was quoted in[13, Theorem 4.3] for all (ε, x) ∈]0, +∞[×R, which is equivalent.…”