Finding a maximal element of a convex set through its characteristic cone: An application to finding a strictly complementary solution

TL;DR

This paper introduces a convex programming framework using characteristic cones to find maximal elements and strictly complementary solutions in convex sets and polyhedra, extending Motzkin's polyhedral cone representation.

Contribution

It develops a novel convex programming approach based on characteristic cones for identifying maximal elements and strictly complementary solutions in convex sets.

Findings

Expresses convex sets as sums of extreme points and directions.

Provides a linear programming method for finding interior points.

Proposes procedures for identifying strictly complementary solutions.

Abstract

In order to express a polyhedron as the (Minkowski) sum of a polytope and a polyhedral cone, Motzkin (1936) made a transition from the polyhedron to a polyhedral cone. Based on his excellent idea, we represent a set by a characteristic cone. By using this representation, we then reach four main results: (i) expressing a closed convex set containing no line as the direct sum of the convex hull of its extreme points and conical hull of its extreme directions, (ii) establishing a convex programming (CP) based framework for determining a maximal element-an element with the maximum number of positive components-of a convex set, (iii) developing a linear programming problem for finding a relative interior point of a polyhedron, and (iv) proposing two procedures for the identification of a strictly complementary solution in linear programming.