A General Regularized Continuous Formulation for the Maximum Clique Problem

Abstract

In this paper, we develop a general regularization-based continuous optimization framework for the maximum clique problem. In particular, we consider a broad class of regularization terms that can be included in the classic Motzkin-Strauss formulation and we develop conditions that guarantee the equivalence between the continuous regularized problem and the original one in both a global and a local sense. We further analyze, from a computational point of view, two different regularizers that satisfy the general conditions.2010 Mathematics Subject Classification. Primary: 90C30; Secondary: 90C35, 90C20.

Finite Convergence of Sum-of-Squares Hierarchies for the Stability Number of a Graph

“…Clearly \scrM \ast (G, w) = \scrM (G, w) if there is no w-critical edge in G. In the unweighted case, one can, for instance, select B = I + 2A G \in \scrM \ast (G, e) as a perturbation of the adjacency matrix, as already observed earlier, e.g., in [6,42]. Recent work, e.g., in [10,20] uses such perturbed (also called regularized) formulations to approximate the maximum stable problem by applying first order methods.…”

Frank–Wolfe and friends: a journey into projection-free first-order optimization methods

“…The goal in finding a clique C such that it is maximal (i.e., it is not contained in any strictly larger clique). This corresponds to find a local solution to the following equivalent (this time non-convex) StQP (see, e.g., Bomze 1997;Bomze et al 1999;Hungerford and Rinaldi 2019 for further details):…”

Frank-Wolfe and friends: a journey into projection-free first-order optimization methods

“…The goal in finding a clique C such that |C| is maximal (i.e., it is not contained in any strictly larger clique). This corresponds to find a local minimum for the following equivalent (this time non-convex) StQP (see, e.g., [10,11,51] for further details):…”

“…where A G is the adjacency matrix of G. Due to the peculiar structure of the problem, FW methods can be fruitfully used to find maximal cliques (see, e.g., [51]).…”