A polyhedral homotopy algorithm for computing critical points of polynomial programs

TL;DR

This paper introduces a novel polyhedral homotopy algorithm that efficiently computes all critical points of polynomial programs, enabling global solutions to polynomial optimization problems with improved performance.

Contribution

It presents a new polyhedral homotopy method that constructs an optimal start system and leverages tropical geometry, improving over traditional algorithms in solving polynomial optimization problems.

Findings

Algorithm outperforms traditional homotopy methods in experiments.

Explicit construction of start systems reduces computational bottlenecks.

Method guarantees correctness through algebraic degree computations.

Abstract

In this paper we propose a method that uses Lagrange multipliers and numerical algebraic geometry to find all critical points, and therefore globally solve, polynomial optimization problems. We design a polyhedral homotopy algorithm that explicitly constructs an optimal start system, circumventing the typical bottleneck associated with polyhedral homotopy algorithms. The correctness of our algorithm follows from intersection theoretic computations of the algebraic degree of polynomial optimization programs and relies on explicitly solving the tropicalization of a corresponding Lagrange system. We present experiments that demonstrate the superiority of our algorithm over traditional homotopy continuation algorithms.