On minimal representations by a family of sublinear functions

Abstract

This paper is a continuation of Grzybowski et al. (J Glob Optim 46:589-601, 2010) and is motivated by the study of exhausters i.e. families of closed convex sets. By Minkowski duality closed convex sets correspond to sublinear functions. Here we study the criteria of reducing representations of pointwise infimum of an infinite family of sublinear functions. A family { f i } i∈I of sublinear functions is by definition an exhaustive family of upper convex approximations of its pointwise infimum inf i∈I f i. A family of closed convex sets is by definition an exhauster of a pointwise infimum of a family of support functions of these convex sets. We establish codependence between infimum of a subset of Minkowski-Rådström-Hörmander cone and translation property of intersection of an exhauster (Sect. 4), between reducing an exhauster to single convex set and shadowing property of intersection of an exhauster (Theorem 5.1) and among all four of these properties (Theorem 6.1). In Grzybowski et al. (2010) the first example of two different minimal upper exhausters of the same function was presented. Here we give an example of infinitely many minimal exhausters of the same ph-function (Example 7.2). In Sect. 8, we give a criterion of minimality of an exhauster with the help of polars of sets belonging to the exhauster. We illustrate this criterion with two interesting examples (Example 8.3).

Saddle representations of positively homogeneous functions by linear functions

“…To this end we can use the methods of reducing exhaustive families which were developed in [24,25,26].…”

An extension of the Rådström cancellation theorem to cornets

“…This lemma turned out to be a basic tool in various fields and hundreds of papers have used it by now. For instance, in nonsmooth analysis [7][8][9]14,[18][19][20]34], optimization theory [15,36,38], theory of convex sets and functions [10,12,16,17,[23][24][25][26][27][28][29][30][31][32][33]35,40,59,60], set-valued analysis [2,13,37,39,41,[44][45][46]48], set-valued differential equations [3,4,11,22,43,49], set-valued functional equations [6,42,[51][52][53]…”

An extension of the Rådström Cancellation Theorem to Cornets

“…This lemma turned out to be a basic tool in various fields and hundreds of papers have used it by now. For instance, in nonsmooth analysis [7][8][9]14,[18][19][20]34], optimization theory [15,36,38], theory of convex sets and functions [10,12,16,17,[23][24][25][26][27][28][29][30][31][32][33]35,40,59,60], set-valued analysis [2,13,37,39,41,44,45,47,48], set-valued differential equations [3,4,11,22,43,49], set-valued functional equations [6,42,[51][52]…”