Theory and applications have shown that there are two important types of convergence for convex functions: pointwise convergence and convergence in a topology induced by the convergence of their epigraphs. We show that these two types of convergence are equivalent on the class of convex functions which are equi-lower semicontinuous. This turns out to be maximal classes of convex functions for w...

Convex hull of some given points is the intersection of all convex sets containing them. It is used as primary structure in many other problems in computational geometry and other areas like image processing, model identification, geographical data systems, and triangular computation of a set of points and so on. Computing the convex hull of a set of point is one of the most fundamental and imp...

A class of degenerate convex minimization problems allows for some adaptive finite element method (AFEM) to compute strongly converging stress approximations. The algorithm AFEM consists of successive loops of the form SOLVE→ ESTIMATE→ MARK→ REFINE and employs the bulk criterion. The convergence in L ′ (Ω;Rm×n) relies on new sharp strict convexity estimates of degenerate convex minimization pro...

We study functions whose directional derivatives can be written as a difference of two extended real valued sublinear function (ds functions). The pair of convex sets obtained as the support functions of these sublinear functions is provided by a familiar equivalence relation and is considered as the bisubdifferential of the function. We study calculus rules and give application to minimization...

Abstract. In this paper, we consider solving multiple-block separable convex minimization problems using alternating direction method of multipliers (ADMM). Motivated by the fact that the existing convergence theory for ADMM is mostly limited to the two-block case, we analyze in this paper, both theoretically and numerically, a new strategy that first transforms a multiblock problem into an equ...

The multiple load structural topology design problem is modeled as a minimization of the weight of the structure subject to equilibrium constraints and restrictions on the local stresses and nodal displacements. The problem involves a large number of discrete design variables and is modeled as a non-convex mixed 0–1 program. For this problem, several convex and mildly non-convex continuous rela...