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An efficient combined DCA and B&B using DC/SDP relaxation for globally solving binary quadratic programs

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Abstract

This paper addresses a new continuous approach based on the DC (Difference of Convex functions) programming and DC algorithms (DCA) to Binary quadratic programs (BQP) which play a key role in combinatorial optimization. DCA is completely different from other avalaible methods and featured by generating a convergent finite sequence of feasible binary solutions (obtained by solving linear programs with the same constraint set) with decreasing objective values. DCA is quite simple and inexpensive to handle large-scale problems. In particular DCA is explicit, requiring only matrix-vector products for Unconstrained Binary quadratic programs (UBQP), and can then exploit sparsity in the large-scale setting. To check globality of solutions computed by DCA, we introduce its combination with a customized Branch-and-Bound scheme using DC/SDP relaxation. The combined algorithm allows checking globality of solutions computed by DCA and restarting it if necessary and consequently accelerates the B&B approach. Numerical results on several series test problems provided in OR-Library (Beasley in J Global Optim, 8:429–433, 1996), show the robustness and efficiency of our algorithm with respect to standard methods. In particular DCA provides ϵ-optimal solutions in almost all cases after only one restarting and the combined DCA-B&B-SDP always provides (ϵ−)optimal solutions.

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Authors and Affiliations

  1. Laboratory of Modelling, Optimization & Operations Research (LMI), National Institute for Applied Sciences-Rouen, BP08-Avenue de l’Université, 76801, Saint-Etienne-du-Rouvray, France

    Tao Pham Dinh & Nam Nguyen Canh

  2. Laboratory of Theoretical and Applied Computer Science (LITA), Informatics Department UFR MIM, Metz University, Ile du Saulcy, 57045, Metz Cedex, France

    Hoai An Le Thi

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  1. Tao Pham Dinh
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  2. Nam Nguyen Canh
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  3. Hoai An Le Thi
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Correspondence to Tao Pham Dinh.

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Pham Dinh, T., Nguyen Canh, N. & Le Thi, H.A. An efficient combined DCA and B&B using DC/SDP relaxation for globally solving binary quadratic programs. J Glob Optim 48, 595–632 (2010). https://doi.org/10.1007/s10898-009-9507-y

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  • DOI: https://doi.org/10.1007/s10898-009-9507-y

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