Formulae for the Conjugate and the Subdifferential of the Supremum Function

Abstract

The aim of this work is to provide formulae for the subdifferential and the conjungate function of the supremun function over an arbitrary family of functions. The work is principally motivated by the case when data functions are lower semicontinuous proper and convex. Nevertheless, we explore the case when the family of functions is arbitrary, but satisfying that the biconjugate of the supremum functions is equal to the supremum of the biconjugate of the data functions. The study focuses its attention on functions defined in finite-dimensional spaces, in this case the formulae can be simplified under certain qualification conditions. However, we show how to extend these results to arbitrary locally convex spaces without any qualification condition.

Necessary and Sufficient Optimality Conditions in DC Semi-infinite Programming

“…Next we provide characterizations of the subdifferential and the ε-subdifferential of the supremum function of a family of convex functions {g t , t ∈ T }. The first result, which is a Valadier-type formula (see [35,Theorem 2]), follows directly from [6,Corollary 3.13], and the second one, given in Theorem 4.3, is a corollary of [28,Theorem 5.4]. Proposition 4.2 ([6, Corollary 3.13]).…”

“…; however, the most elaborated results of that time were due to M. Valadier (1969) [35]." Recent contributions in this field are [4,5,6,7,8,15,16,26,28,30,31] and references therein. On the other hand, for the class of DC problems there also exist necessary and sufficient conditions for optimality, many times in terms of (exact/approximate) subdifferentials of the involved DC decomposition (see, e.g., [9,10,12,14,17,20,21,22] and references therein).…”

“…In the proof of the next theorem we need the following result established in [28,Theorem 5.1] (see [29] for a corrected proof). Downloaded 04/25/21 to 148.3.150.188.…”

Subdifferential Formulae for the Supremum of an Arbitrary Family of Functions

“…Now define w * i := P * (v * i ), λ * := z * +P * (q * ), w * := {i|αi =0} α i w * i , w n,i = P * (u * n,i ), it follows that w * n,i β −→ w * i , | w * n,i , u n,i | → 0 and w * n,i ∈ ∂(g + δ W )(u n,i ), u n,i ∈ ri L (dom h), g(u n,i ) → g(0), λ * ∈ N dom h∩L (0) and x * = w * + λ * , which concludes the proof of (39). Then, using (38) and (39) we conclude the first inclusion in (37).…”

“…More recent papers have studied the convex subdifferential of the supremum function when T is an arbitrary index set and {f t : t ∈ T } is an arbitrary family of (possibly non-smooth) convex functions (see, for example, [8,[12][13][14]23,37] and the reference therein). Due to the possible emptiness of the set of active indices at a given point x, the authors have considered the ε-active index set T ε (x) := {t ∈ T : f t (x) ≥ f (x) − ε}.…”