Dual Bounded Generation: Polynomial, Second-order Cone and Positive Semidefinite Matrix Inequalities

TL;DR

This paper explores the dualization problem in monotone integer systems, establishing bounds and conditions for polynomial, second-order cone, and semidefinite inequalities, with applications in optimization.

Contribution

It extends the dualization framework to polynomial, second-order cone, and semidefinite inequalities, providing bounds and conditions for their enumeration.

Findings

Bounded the size of minimal infeasible vectors for certain inequality systems

Established sufficient conditions for polynomial, second-order cone, and semidefinite inequalities

Highlighted applications in optimization problems

Abstract

In the monotone integer dualization problem, we are given two sets of vectors in an integer box such that no vector in the first set is dominated by a vector in the second. The question is to check if the two sets of vectors cover the entire integer box by upward and downward domination, respectively. It is known that the problem is (quasi-)polynomially equivalent to that of enumerating all maximal feasible solutions of a given monotone system of linear/separable/supermodular inequalities over integer vectors. The equivalence is established via showing that the dual family of minimal infeasible vectors has size bounded by a (quasi-)polynomial in the sizes of the family to be generated and the input description. Continuing in this line of work, in this paper, we consider systems of polynomial, second-order cone, and semidefinite inequalities. We give sufficient conditions under which such bounds can be established and highlight some applications.