A new constraint qualification for the formula of the subdifferential of composed convex functions in infinite dimensional spaces

Abstract

In this paper we work in separated locally convex spaces where we give equivalent statements for the formulae of the conjugate function of the sum of a convex lower-semicontinuous function and the precomposition of another convex lower-semicontinuous function which is also K-increasing with a K-convex K-epi-closed function, where K is a nonempty closed convex cone. These statements prove to be the weakest constraint qualifications given so far under which the formulae for the subdifferential of the mentioned sum of functions are valid. Then we deliver constraint qualifications inspired from them that guarantee some conjugate duality assertions. Two interesting special cases taken from the literature conclude the paper.

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.…”

“…Endowing then the functions with the classical convexity and topological properties, we derive new important equivalences as well as sufficient conditions, from which when ε = 0 closedness type regularity conditions are derived. These can then be employed, for instance, for subdifferential formulae, as done in Boţ et al [18], Hiriart-Urruty [34], and Boţ and Grad [35] or, like in Boţ and Grad [36] and Grad and Wanka [37], for providing formulae for biconjugates of combinations of functions.…”

“…One can thus notice that (RC G i ), i = 1, 2, 3, imply (RC G 4 ), which is equivalent to the stable strong duality for (PG) and (DG). An example to show that the closedness type regularity condition (RC G 4 ) is indeed weaker than its counterparts of continuity or interiority type follows, others being available for instance in Boţ et al [18,20].…”

“…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]).…”