Dual certificates of primal cone membership

TL;DR

This paper introduces a new framework for efficiently certifying membership in convex cones using dual certificates, enabling practical verification and optimization in polynomial and cone programming.

Contribution

It develops verifiable primal membership certificates from dual cone vectors assuming a self-concordant barrier, applicable to various polynomial cones and low-dimensional optimization.

Findings

Certificates can be computed using interior-point methods.

Applicable to cones like SOS, SONC, SAGE, SDSOS in polynomial optimization.

Enables direct optimization over low-dimensional dual cones.

Abstract

We discuss optimization problems over convex cones in which membership is difficult to verify directly. In the standard theory of duality, vectors in the dual cone $K^*$ are associated with separating hyperplanes and interpreted as certificates of non-membership in the primal cone $K$. Complementing this perspective, we develop easily verifiable certificates of membership in $K$ given by vectors in $K^*$. Assuming that $K^*$ admits an efficiently computable logarithmically homogeneous self-concordant barrier, every vector in the interior of $K$ is associated with a full-dimensional cone of efficiently verifiable membership certificates. Consequently, rigorous certificates can be computed using numerical methods, including interior-point algorithms. The proposed framework is particularly well-suited to optimization over low-dimensional linear images of higher dimensional cones: we argue that these problems can be solved by optimizing directly over the (low-dimensional) dual cone, circumventing the customary lifting that introduces a large number of auxiliary variables. As an application, we derive a novel closed-form formula for computing exact primal feasible solutions from suitable dual feasible solutions; as the dual solutions approach optimality, the computed primal solutions do so as well. To illustrate the generality of our approach, we show that the new certification scheme is applicable to virtually every tractable subcone of nonnegative polynomials commonly used in polynomial optimization (such as SOS, SONC, SAGE and SDSOS polynomials, among others), facilitating the computation of rigorous nonnegativity certificates using numerical algorithms.