Characterizations of ε-duality gap statements for composed optimization problems

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.…”

“…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.…”

“…In order to deal with statements involving (η, ε)-saddle points, we generalize the classical notion of a saddle point (cf. [10,21]). Definition 3.1.…”

“…Remark 15. Analogously, one can particularize the other statements regarding pairs of primal-dual problems (PG x * ) − (DG x * ), x * ∈ X * , for unconstrained optimization problems and their Fenchel duals, rediscovering or improving different statements from Boţ and Wanka [6,7], Grad [10], Boţ et al [18,20], and Boncea and Grad [21]. Under additional assumptions which guarantee the convexity of the perturbation function U (e.g., take f and g convex), the strong duality statement for the problems (PU) and (DU) 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.…”