The Gradient Projection Method for a Supporting Function on the Unit Sphere and Its Applications

Abstract

We consider minimization of the supporting function of a convex compact set on the unit sphere. In essence, this is the problem of projecting zero onto a compact convex set. We consider sufficient conditions for solving this problem with a linear rate using a first order algorithm—the gradient projection method with a fixed step-size and with Armijo’s step-size. We consider some applications for problems with set-valued mappings. The mappings in the work basically are given through the set-valued integral of a set-valued mapping with convex and compact images or as the Minkowski sum of finite number of convex compact sets, e.g., ellipsoids. Unlike another solution ways, e.g., with approximation in a certain sense of the mapping, the considered algorithm much weaker depends on the dimension of the space and other parameters of the problem. It also allows efficient error estimation. Numerical experiments confirm the effectiveness of the considered approach.