Subsmooth semi-infinite and infinite optimization problems

Abstract

We first consider subsmoothness for a function family and provide formulas of the subdifferential of the pointwise supremum of a family of subsmooth functions. Next, we consider subsmooth infinite and semi-infinite optimization problems. In particular, we provide several dual and primal characterizations for a point to be a sharp minimum or a weak sharp minimum for such optimization problems.

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

“…The next corollary of Theorem 4.1, inspired by the corresponding result of [30], provides precise calculations of the generalized gradient of the supremum of semismooth functions as a direct consequence of Corollary 4.2 and Corollary 4.3. Subsmooth sets were introduced and comprehensively studied in [1].…”

“…; see, e.g., [6,30,31] and the references therein. However, we are not familiar with any results in the literature concerning counterparts of (1.5) with no topological requirements on T .…”

“…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.…”

Equivalent properties of global weak sharp minima with applications

“…In addition, the condition imposed in the foregoing theorem only needs the function to be lower semi-continuous, a rather weak condition in optimization. Hence, our result is applicable even for the case where the subgradient of f does not exist, while in [2][3][4][5][6], f is required, at least, to be subdifferentiable.…”

“…Of particular note in this fields is the paper by Burke and Ferris [1], which gave an extensive exposition of the notation and its impacted on convex programming and convergence analysis. Since then, this notion was extensively studied by many authors, for example, necessary or sufficient conditions of weak sharp minima for nonconvex programming [2,3], and necessary and sufficient conditions of local weak sharp minima for sup-type (or lower-C 1 ) functions [4,5]. Recent development of weak sharp minima and its related to other issues can be found in [5][6][7][8].…”