Set containment characterization and mathematical programming

Recently, many researchers studied set containment characterizations. In this paper, we introduce some set containment characterizations for quasiconvex programming. Furthermore, we show a duality theorem for quasiconvex programming by using set containment characterizations. Notions of quasiconjugate for quasiconvex functions, especially 1, -1-quasiconjugate, 1-semiconjugate, H-quasiconjugate and R-quasiconjugate, play important roles to derive characterizations of the set containments

On Subdifferentials Via a Generalized Conjugation Scheme: An Application to DC Problems and Optimality Conditions

This paper studies properties of a subdifferential defined using a generalized conjugation scheme. We relate this subdifferential together with the domain of an appropriate conjugate function and the ε-directional derivative. In addition, we also present necessary conditions for ε-optimality and global optimality in optimization problems involving the difference of two convex functions. These conditions will be written via this generalized notion of subdifferential studied in the first sections of the paper.Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. Research partially supported by MICIIN of Spain and ERDF of EU, Grant PGC2018 097960-B-C22

Analyzing linear systems containing strict inequalities via evenly convex hulls

The evenly convex hull of a given set is the intersection of all the open halfspaces which contain such set (hence the convex hull is contained in the evenly convex hull). This paper deals with finite dimensional linear systems containing strict inequalities and (possibly) weak inequalities as well as equalities. The number of inequalities and equalities in these systems is arbitrary (possibly infinite). For such kind of systems a consistency theorem is provided and those strict inequalities (weak inequalities, equalities) which are satisfied for every solution of a given system are characterized. Such results are formulated in terms of the evenly convex hull of certain sets which depend on the coefficients of the system.This work was supported by the MCYT of Spain and FEDER of UE, Grant BFM2002-04114-C0201