Characterizations of ɛ-duality gap statements for constrained optimization problems

Abstract

In this paper we present different regularity conditions that equivalently characterize various ε-duality gap statements (with ε ≥ 0) for constrained optimization problems and their Lagrange and Fenchel-Lagrange duals in separated locally convex spaces, respectively. These regularity conditions are formulated by using epigraphs and ε-subdifferentials. When ε = 0 we rediscover recent results on stable strong and total duality and zero duality gap from the literature. MSC: PreliminariesMotivated by recent results on stable strong and total duality for constrained convex optimization problems in [2,6,7,9,13,17] and the ones on zero duality gap in [15,16] we introduce in this paper several regularity conditions which characterize ε-duality gap statements (with ε ≥ 0) for a constrained optimization problem and its Lagrange and FenchelLagrange dual problems, respectively. The regularity conditions we provide in Section 2 are based on epigraphs, while the ones in Section 3 on ε-subdifferentials. In this way we extend many of the results in the mentioned papers, which are recovered as special cases when ε = 0, delivering thus generalizations of the classical Farkas-Minkowski and basic constraint qualifications. Moreover some statements in [6-8, 15, 16], which arise from our results in the special *

Closedness Type Regularity Conditions in Convex Optimization and Beyond

“…[26,27]). In this way some of our results from Grad [10], Boţ et al [18,20,29,38,39], Boncea and Grad [21,22], and Boţ and Grad [35] as well as different others from the literature (e.g., from [2-4, 6, 7, 19]) can be obtained as special cases of the general statements presented below.…”

“…which will not be mentioned further (see [22,38] for similar statements regarding this dual problem to (PC) that can be rediscovered as special cases of the main ones from this paper).…”

“…Moreover, the situation of total duality is closely related to some subdifferential formulae and the regularity conditions can be used to guarantee these, too. However, in some situations one can only show that the difference between the optimal objective values of the primal and dual problem is less than an ε ≥ 0, situation coined in Boncea and Grad [21,22] as ε-duality gap. Using as a basis our investigations in these articles, where the ε-duality gap for composed and constrained optimization problems, respectively, were characterized via epigraph and subdifferential inclusions, we provide in the following section characterizations via epigraph inclusions of stable ε-duality gap (for an ε ≥ 0) for very general optimization problems, with the involved functions not necessary convex, only proper.…”

“…Remark 11. Analogously, one can particularize the other statements regarding pairs of primal-dual problems (PG x * ) − (DG x * ), x * ∈ X * , for constrained optimization problems and their Lagrange duals, rediscovering or improving different statements from Boţ et al [20,39], Boncea and Grad [22], and Jeyakumar and Li [45,46]. Under additional assumptions which guarantee the convexity of the perturbation function L (e.g., take S and f convex and h C-convex), the strong duality statement for the problems (PC) and (DC L ) can be derived directly from Corollary 3.11 or Remark 8 by particularizing (RC G i ), i ∈ {1, 2, 3, 4} to (cf.…”

“…For being able to deliver by means of duality very general regularity conditions for the problem of guaranteeing the maximal monotonicity of the sum of two maximal monotone operators defined in reflexive Banach spaces, we introduced in Boţ et al [17] the notion of a set closed regarding a subspace (see also [18][19][20]). For the investigations in this study even more general closedness notions are necessary, that were first proposed in Boncea and Grad [21] (and when Z = X × R, in [22]; see also [10]). Definition 1.1.…”