GNCGCP - Graduated NonConvexity and Graduated Concavity Procedure

TL;DR

GNCGCP is a versatile optimization framework that simplifies solving complex combinatorial problems like graph matching and QAP by using graduated nonconvexity and concavity techniques.

Contribution

It introduces a new, simplified approach to approximate combinatorial optimization by combining graduated nonconvexity and concavity without explicit relaxations.

Findings

Achieves state-of-the-art performance on graph matching and QAP

Simplifies implementation by relying only on gradient information

Proves equivalence to a convex-concave relaxation procedure

Abstract

In this paper we propose the Graduated NonConvexity and Graduated Concavity Procedure (GNCGCP) as a general optimization framework to approximately solve the combinatorial optimization problems on the set of partial permutation matrices. GNCGCP comprises two sub-procedures, graduated nonconvexity (GNC) which realizes a convex relaxation and graduated concavity (GC) which realizes a concave relaxation. It is proved that GNCGCP realizes exactly a type of convex-concave relaxation procedure (CCRP), but with a much simpler formulation without needing convex or concave relaxation in an explicit way. Actually, GNCGCP involves only the gradient of the objective function and is therefore very easy to use in practical applications. Two typical NP-hard problems, (sub)graph matching and quadratic assignment problem (QAP), are employed to demonstrate its simplicity and state-of-the-art performance.