Vector Optimization and Monotone Operators via Convex Duality

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.…”

“…They have proven thus to be viable alternatives to their more restrictive interiority type counterparts. In this review paper, that enhances and completes a similar study provided in the book [10], we provide a general look on the usage of the closedness type regularity conditions in the literature until now, showing that they arise naturally while dealing with optimization problems and pointing toward different assertions from the literature that can be rediscovered as special cases of the mentioned general results. To this end we deconstruct and then reconstruct closedness type regularity conditions formulated by means of epigraphs and subdifferentials, respectively, for general optimization problems, showing afterwards how to particularize them for constrained and unconstrained optimization problems.…”

“…Various such additional hypotheses were considered in the literature (some of which are mentioned later in Remark 8), the most relevant being the interiority type ones (mentioned in [19,27]) and the closedness type ones (cf. [3,4,6,7,10,19,29]). Moreover, the situation of total duality is closely related to some subdifferential formulae and the regularity conditions can be used to guarantee these, too.…”

“…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.…”

Scalarizations for Maximization with Respect to Polyhedral Cones

“…Examples are the scalarizations of Borwein [6] and Jahn [10] in terms of the dual cone and that of Soland [11] for Pareto maximization involving both the dual cone and aspiration levels for the individual objective functions. References [12][13][14][15][16][17][18] consolidate the theory and solution techniques for cone-ordered maximizations. However, specific results for polyhedral cones are scarce.…”