An Extension of Duality-Stability Relations to Nonconvex Optimization Problems

Abstract

By an effective extension of the conjugate function concept a general framework for duality-stability relations in nonconvex optimization problems can be studied. The results obtained show strong correspondences with the duality theory for convex minimization problems. In specializations to mathematical programming problems the canonical Lagrangian of the model appears as the extended Lagrangian considered in exterior penalty function methods.

Autoconjugate functions and representations of monotone operators

“…His approach is connected with the following problem: given a monotone operator M : X =t X*, it is possible to get a closed convex function q on X x X* such that q*(x*,x) = q(x,x*) for any (x,x*) € X x X* and f M ^ q ^ p M , where f M is the Fitzpatrick representation of M and p M = f M . A positive answer is provided here in the broader framework of generalised convexity and generalised monotonicity (see [1,14,15,27,29]). For the study of maximal monotone operators and their representations by convex functions (on spaces which are larger than X x X*), we refer to the recent monograph by Simons [28].…”

“…Among dualities, a familiar class is formed by conjugacies (or conjugations), that is, dualities for which [14] and have been studied by a number of authors (see [1,6,11,16,19,18,15,22,23,26] and the references therein). It has also be shown by When W is a reflexive Banach space X, Y = X* and c is the evaluation (x,x*) >->• x*(x), this duality is close to the classical Fenchel conjugacy since it is composed of this conjugacy with the interchange of variables (x*,x) >-* (x,x*).…”

The bearing of duality on microeconomics

“…It suffices to prove relation (10). Using relations (6) and (4), it stems from the following equivalences…”

“…Here we adopt a more general framework since in microeconomics one usually set the problems in orthants or cones rather than in the whole spaces. The just defined conjugacy has the advantage of entering into the general framework of the Fenchel-Moreau conjugacy (see [53], [6] and many others references such as [27], [39], [38], [58], [66], [54], [77], [85]) for which the conjugate of f is given by relation (3) where c : X × Y → R := R ∪ {−∞, ∞} is an appropriate coupling function. In order to see that, it suffices to take for c the opposite of the indicator function ι F of (the graph of) F := (X × Y ) \E −1 , given by ι F (x, y) = 0 for (x, y) ∈ F, +∞ otherwise.…”

Unilateral Analysis and Duality

“…Although the notion of duality can be set in a more general framework ( [53,73,112]) we limit our approach to the case of conjugacies, in the line of Moreau [56,5,49,50,62,65,82,84,109,112,120]). In [80] the case Z is endowed with some preorder is considered.…”