Definition 1.1. Let C be a subset of R. We say C is convex if αx+ (1− α)y ∈ C, ∀x, y ∈ C, ∀α ∈ [0, 1]. Definition 1.2. Let C be a convex subset of R. A function f : C 7→ R is called convex if f(αx+ (1− α)y) ≤ αf(x) + (1− α)f(y), ∀x, y ∈ C, ∀α ∈ [0, 1]. The function f is called concave if −f is convex. The function f is called strictly convex if the above inequality is strict for all x, y ∈ C wi...

We prove that rank-.n 1/ convexity does not imply quasiconvexity with respect to divergence free fields (so-called S-quasiconvexity) in M n for m > n, by adapting the well-known Šverák’s counterexample to the solenoidal setting. On the other hand, we also remark that rank-.n 1/ convexity and S-quasiconvexity turn out to be equivalent in the space of n n diagonal matrices.

with n, m > 2, Ω ⊂ R, m < p <∞ and a compact set A ⊂ R with nonempty interior. In the case of a convex integrand f(s, ξ, · ) and a convex restriction set A = K, the global minimizers of (1.1) − (1.3) satisfy optimality conditions in the form of Pontryagin’s principle 01) even though the usual regularity condition for the equality operator (1.2) fails. 02) The question arises whether the Pontrya...

The consideration of barriers to travel plays an increasingly important role in the transportation and location literature. In one of the classical papers on location problems with barriers, Katz and Cooper (1981) considered the Weber problem (often also referred to as median problem) with one circular barrier region. Considering the same problem we develop new structural results showing that t...

with n, m > 2, Ω ⊂ R, m < p < ∞ and a compact set K ⊂ R with nonempty interior. In the case of a convex integrand f(s, ξ, · ) and a convex restriction set K, the global minimizers of (1.1) − (1.3) satisfy optimality conditions in the form of Pontryagin’s principle 01) even though the usual regularity condition for the equality operator (1.2) fails. 02) The question arises whether the Pontryagin...

Abstract. A set S ⊆ R is called to be Semidefinite (SDP) representable if S equals the projection of a set in higher dimensional space which is describable by some Linear Matrix Inequality (LMI). Clearly, if S is SDP representable, then S must be convex and semialgebraic (it is describable by conjunctions and disjunctions of polynomial equalities or inequalities). This paper proves sufficient c...

We focus on the problem of minimizing a convex function f over a convex set S given T queries to a stochastic first order oracle. We argue that the complexity of convex minimization is only determined by the rate of growth of the function around its minimizer xf,S , as quantified by a Tsybakov-like noise condition. Specifically, we prove that if f grows at least as fast as ‖x − xf,S‖ around its...

The infimal convolution of M-convex functions is M-convex. This is a fundamental fact in discrete convex analysis that is often useful in its application to mathematical economics and game theory. M-convexity and its variant called M-convexity are closely related to gross substitutability, and the infimal convolution operation corresponds to an aggregation. This note provides a succinct descrip...

Let S be a finite set with n elements in a real linear space. Let JS be a set of n intervals in R. We introduce a convex operator co(S,JS) which generalizes the familiar concepts of the convex hull conv S and the affine hull aff S of S. We establish basic properties of this operator. It is proved that each homothet of conv S that is contained in aff S can be obtained using this operator. A vari...

In this paper, we study properties of general closed convex sets that determine the closed-ness and polyhedrality of the convex hull of integer points contained in it. We first present necessary and sufficient conditions for the convex hull of integer points contained in a general convex set to be closed. This leads to useful results for special class of convex sets such as pointed cones, stric...