The celebrated Motzkin-Straus formulation for the maximum clique problem provides a nontrivial characterization of the clique number of a graph in terms of the maximum value of a nonconvex quadratic function over a standard simplex. It was originally developed as a way of proving Tur\'{a}n's theorem in graph theory, but was later used to develop competitive algorithms for the maximum clique problem based on continuous optimization. Clique relaxations, such as $s$-defective clique and $s$-plex, emerged as attractive, more practical alternatives to cliques in network-based cluster detection models arising in numerous applications. This paper establishes continuous cubic formulations for the maximum $s$-defective clique problem and the maximum $s$-plex problem by generalizing the Motzkin-Straus formulation to the corresponding clique relaxations. The formulations are used to extend Tur\'{a}n's theorem and other known lower bounds on the clique number to the considered clique relaxations. Results of preliminary numerical experiments with the CONOPT solver demonstrate that the proposed formulations can be used to quickly compute high-quality solutions for the maximum $s$-defective clique problem and the maximum $s$-plex problem. The proposed formulations can also be used to generate interesting test instances for global optimization solvers.
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To appear at Mathematical Programming
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