GNCGCP - Graduated NonConvexity and Graduated Concavity Procedure

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. Index Terms Combinatorial optimization, Graduated optimization, Deterministic annealing, Subgraph matching, Quadratic assignment problem