Physarum Inspired Dynamics to Solve Semi-Definite Programs

TL;DR

This paper introduces Physarum-inspired dynamics and algorithms for solving positive semi-definite programs, extending previous work on linear programs and demonstrating theoretical and experimental convergence.

Contribution

It generalizes Physarum dynamics to matrix spaces for SDPs and proposes algorithms with proven convergence for solving complex optimization problems.

Findings

Proved soundness and convergence of the dynamics under mild conditions.

Extended Physarum dynamics from LPs to SDPs, handling non-commutative matrix products.

Demonstrated applications in combinatorial optimization problems like MaxCut and Lovasz theta number.

Abstract

Physarum Polycephalum is a slime mold that can solve shortest path problems. A mathematical model based on Physarum's behavior, known as the Physarum Directed Dynamics, can solve positive linear programs. In this paper, we present a family of Physarum-based dynamics extending the previous work and introduce a new algorithm to solve positive Semi-Definite Programs (SDP). The Physarum dynamics are governed by orthogonal projections (w.r.t. time-dependent scalar products) on the affine subspace defined by the linear constraints. We present a natural generalization of the scalar products used in the LP case to the matrix space for SDPs, which boils down to the linear case when all matrices in the SDP are diagonal, thus, representing an LP. We investigate the behavior of the induced dynamics theoretically and experimentally, highlight challenges arising from the non-commutative nature of matrix products, and prove soundness and convergence under mild conditions. Moreover, we consider a more abstract view on the dynamics that suggests a slight variation to guarantee unconditional soundness and convergence-to-optimality. By simulating these dynamics using suitable discretizations, one obtains numerical algorithms for solving positive SDPs, which have applications in discrete optimization, e.g., for computing the Goemans-Williamson approximation for MaxCut or the Lovasz theta number for determining the clique/chromatic number in perfect graphs.