Regularizations in abelian complete ordered groups

AbstractNotions about Φ-convexity are extended to abelian complete partially ordered group-valued mappings in an attempt to unify in a general theory notions of Φ-convex sets and Φ-convex mappings. We obtain some group specific results and particularly a characterization of support functions

ZERO DUALITY GAP FOR CONVEX PROGRAMS: A GENERAL RESULT

This article addresses a general criterion providing a zero duality gap for convex programs in the setting of the real locally convex spaces. The main theorem of our work is formulated only in terms of the constraints of the program, hence it holds true for any objective function fulfilling a very general qualification condition, implied for instance by standard qualification criteria of Moreau-Rockafellar or Attouch-Br ́ezis type. This result generalizes recent theorems by Champion, Ban & Song and Jeyakumar & Li

When is a convex cone the cone of all the half-lines contained in a convex set?

International audienceIn this article we prove that every convex cone V of a real vector space X possessing an uncountable Hamel basis may be expressed as the cone of all the half-lines contained within some convex subset C of X (in other words, V is the infinity cone to C). This property does not hold for lower-dimensional vector spaces; more precisely, a convex cone V in a vector space X with a denumerable basis is the infinity cone to some convex subset of X if and only if V is the union of a countable ascending sequence of linearly closed cones, while a convex cone V in a finite-dimensional vector space X is the infinity cone to some convex subset of X if and only if V is linearly closed

Even convexity, subdifferentiability, and Γ-regularization in general topological vector spaces

In this paper we provide new results on even convexity and extend some others to the framework of general topological vector spaces. We first present a characterization of the even convexity of an extended real-valued function at a point. We then establish the links between even convexity and subdifferentiability and the Γ-regularization of a given function. Consequently, we derive a sufficient condition for strong duality fulfillment in convex optimization problems.MICINN of Spain, Grant MTM2011-29064-C03-02

Duality for convex infinite optimization on linear spaces

This note establishes a limiting formula for the conic Lagrangian dual of a convex infinite optimization problem, correcting the classical version of Karney [Math. Programming 27 (1983) 75-82] for convex semi-infinite programs. A reformulation of the convex infinite optimization problem with a single constraint leads to a limiting formula for the corresponding Lagrangian dual, called sup-dual, and also for the primal problem in the case when strong Slater condition holds, which also entails strong sup-duality.This research was partially supported by Ministerio de Ciencia, Innovación y Universidades (MCIU), Agencia Estatal de Investigación (AEI), and European Regional Development Fund (ERDF), Project PGC2018-097960-B-C22

Functional Inequalities in the Absence of Convexity and Lower Semicontinuity with Applications to Optimization

In this paper we extend some results in [Dinh, Goberna, López, and Volle, Set-Valued Var. Anal., to appear] to the setting of functional inequalities when the standard assumptions of convexity and lower semicontinuity of the involved mappings are absent. This extension is achieved under certain condition relative to the second conjugate of the involved functions. The main result of this paper, Theorem 1, is applied to derive some subdifferential calculus rules and different generalizations of the Farkas lemma for nonconvex systems, as well as some optimality conditions and duality theory for infinite nonconvex optimization problems. Several examples are given to illustrate the significance of the main results and also to point out the potential of their applications to get various extensions of Farkas-type results and to the study of other classes of problems such as variational inequalities and equilibrium models.This research was partially supported by MICINN of Spain, grant MTM2008-06695-C03-01

Characterizations of robust and stable duality for linearly perturbed uncertain optimization problems

We introduce a robust optimization model consisting in a family of perturbation functions giving rise to certain pairs of dual optimization problems in which the dual variable depends on the uncertainty parameter. The interest of our approach is illustrated by some examples, including uncertain conic optimization and infinite optimization via discretization. The main results characterize desirable robust duality relations (as robust zero-duality gap) by formulas involving the epsilon-minima or the epsilon-subdifferentials of the objective function. The two extreme cases, namely, the usual perturbational duality (without uncertainty), and the duality for the supremum of functions (duality parameter vanishing) are analyzed in detail. © Springer Nature Switzerland AG 2020

New glimpses on convex infinite optimization duality

Given a convex optimization problem (P) in a locally convex topological vector space X with an arbitrary number of constraints, we consider three possible dual problems of (P), namely, the usual Lagrangian dual (D), the perturbational dual (Q), and the surrogate dual (Δ), the last one recently introduced in a previous paper of the authors (Goberna et al., J Convex Anal 21(4), 2014). As shown by simple examples, these dual problems may be all different. This paper provides conditions ensuring that inf(P)=max(D), inf(P)=max(Q), and inf(P)=max(Δ) (dual equality and existence of dual optimal solutions) in terms of the so-called closedness regarding to a set. Sufficient conditions guaranteeing min(P)=sup(Q) (dual equality and existence of primal optimal solutions) are also provided, for the nominal problems and also for their perturbational relatives. The particular cases of convex semi-infinite optimization problems (in which either the number of constraints or the dimension of X, but not both, is finite) and linear infinite optimization problems are analyzed. Finally, some applications to the feasibility of convex inequality systems are described.M. A. Goberna and M. A. López were partially supported by MINECO of Spain, Grant MTM2011-29064-C03-02

Relaxed Lagrangian duality in convex infinite optimization: reverse strong duality and optimality

We associate with each convex optimization problem posed on some locally convex space with an infinite index set T, and a given non-empty family H formed by finite subsets of T, a suitable Lagrangian-Haar dual problem. We provide reverse H-strong duality theorems, H-Farkas type lemmas and optimality theorems. Special attention is addressed to infinite and semi-infinite linear optimization problems.Comment: 19 page