Subdifferential Calculus Rules in Convex Analysis: A Unifying Approach Via Pointwise Supremum Functions

Abstract

We provide a rule to calculate the subdifferential set of the pointwise supremum of an arbitrary family of convex functions defined on a real locally convex topological vector space. Our formula is given exclusively in terms of the data functions and does not require any assumption either on the index set on which the supremum is taken or on the involved functions. Some other calculus rules, namely chain rule formulas of standard type, are obtained from our main result via new and direct proofs.

Subdifferentials of Nonconvex Supremum Functions and Their Applications to Semi-infinite and Infinite Programs with Lipschitzian Data

“…Several counterparts of (1.3) in the inclusion and equality forms are obtained for convex functions without any assumptions imposed on the topological structure of the index set T and on the behavior of f t with respect to t by using the perturbation T ε (x) := t ∈ T f t (x) ≥ f (x) − ε , ε ≥ 0, (1.4) of the active index set first used in [27]. To the best of our knowledge, the most powerful results in this direction are obtained in [12,17] via the approximate subdifferentials of convex analysis for the functions f t atx with no (semi)continuity requirements on f t (·). The functions f t (·) are not even assumed to be convex in [17] but the situation is actually reduced to convexity under the relaxation assumption f * * (x) = sup t∈T f * * t (x) via the biconjugate functions imposed in both papers [12,17].…”

“…However, till recent years the vast majority of publications on SIP and infinite programs have concerned problems (1.1) with compact index sets under certain continuity assumptions imposed on f t with respect to the index variable; both these requirements seem to be very essential for the methods employed in the aforementioned publications. Such compactness and continuity assumptions are not imposed in [4,5,8,12,17,20,21,30] among other recent publications.…”

Subsmooth semi-infinite and infinite optimization problems

“…Recently, under the convexity assumption, Hantoute et al [13] and Lopez and Volle [21] further provided some formulas for the subdifferential of pointwise supremum functions.…”

New Formulas for the Fenchel Subdifferential of the Conjugate Function

“…The last conclusion immediately fellows from the …rst one in view of (7). Now, in view of the relationship Argmin f = @f ( ); we derive from Theorem 4 and Corollaries 6 and 7 new formulas for the set Argmin(cof ) by means of Argmin f .…”

“…At this point, the purpose of this paper is to provide such formulas which are valid in more general settings. Let us mention that this study enters in the intensive research on subdi¤erential calculus rules for pointwise supremum functions, see [3,5,6,7,10,11,13,14,18] [13]. The most general one, given in Theorem 2 of [13] in terms of the approximate subdi¤erentials of the initial function, applies for any extended real-valued function de…ned on a locally convex space X and gives us @f (x ) = \ ">0 y 2dom f co (@ " f )…”