A General System for Heuristic Solution of Convex Problems over Nonconvex Sets

TL;DR

This paper introduces a flexible heuristic framework for approximately solving convex problems constrained by nonconvex sets, utilizing convex relaxations, restrictions, local search, and ADMM, implemented in the NCVX package.

Contribution

It presents a general heuristic approach for nonconvex problems with convex objectives, implemented in an extendable Python package, without extensive problem-specific tuning.

Findings

Effective in solving various well-known nonconvex problems

Requires solving only a few convex subproblems

Demonstrates broad applicability and effectiveness

Abstract

We describe general heuristics to approximately solve a wide variety of problems with convex objective and decision variables from a nonconvex set. The heuristics, which employ convex relaxations, convex restrictions, local neighbor search methods, and the alternating direction method of multipliers (ADMM), require the solution of a modest number of convex problems, and are meant to apply to general problems, without much tuning. We describe an implementation of these methods in a package called NCVX, as an extension of CVXPY, a Python package for formulating and solving convex optimization problems. We study several examples of well known nonconvex problems, and show that our general purpose heuristics are effective in finding approximate solutions to a wide variety of problems.