The External Estimate of the Compact Set by Lebesgue Set of the Convex Function

Abstract

The finite-dimensional problem of embedding a given compact D ⊂ R p into the lower Lebesgue set G(α) = {y ∈ R p : f(y) 6 α} of the convex function f(·) with the smallest value of α due to the offset of D is considered. Its mathematical formalization leads to the problem of minimizing the function φ(x) = max y∈D f(y − x) on R p . The properties of the function φ(x) are researched, necessary and sufficient conditions and conditions for the uniqueness of the problem solution are obtained. As an important case for applications, the case when f(·) is the Minkowski gauge function of some convex body M is singled out. It is shown that if M is a polyhedron, then the problem reduces to a linear programming problem. The approach to get an approximate solution is proposed in which, having known the approximation of xi to obtain xi+1 it is necessary to solve the simpler problem of embedding the compact set D into the Lebesgue set of the gauge function of the set Mi = G(ai), where ai = f(xi). The rationale for the convergence for a sequence of approximations to the problem solution is given.

Keywords

• gauge function
• external estimate
• subdifferential
• quasiconvex function
• strongly convex set
• strongly convex function