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Solution of boolean quadratic programming problems by two augmented Lagrangian algorithms based on a continuous relaxation

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  • Rupaj Kumar Nayak

    (International Institute of Information Technology)

  • Nirmalya Kumar Mohanty

    (International Institute of Information Technology)

Abstract

Many combinatorial optimization problems and engineering problems can be modeled as boolean quadratic programming (BQP) problems. In this paper, two augmented Lagrangian methods (ALM) are discussed for the solution of BQP problems based on a class of continuous functions. After convexification, the BQP is reformulated as an equivalent augmented Lagrangian function, and then solved by two ALM algorithms. Within this ALM algorithm, L-BFGS is called for the solution of unconstrained nonlinear programming problem. Experiments are performed on max-cut problem, 0–1 quadratic knapsack problem and image deconvolution, which indicate that ALM method is promising for solving large scale BQP by the quality of near optimal solution with low computational time.

Suggested Citation

  • Rupaj Kumar Nayak & Nirmalya Kumar Mohanty, 2020. "Solution of boolean quadratic programming problems by two augmented Lagrangian algorithms based on a continuous relaxation," Journal of Combinatorial Optimization, Springer, vol. 39(3), pages 792-825, April.
    • Handle: RePEc:spr:jcomop:v:39:y:2020:i:3:d:10.1007_s10878-019-00517-8
    DOI: 10.1007/s10878-019-00517-8
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    References listed on IDEAS

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    1. Svatopluk Poljak & Henry Wolkowicz, 1995. "Convex Relaxations of (0, 1)-Quadratic Programming," Mathematics of Operations Research, INFORMS, vol. 20(3), pages 550-561, August.
    2. Walter Murray & Kien-Ming Ng, 2010. "An algorithm for nonlinear optimization problems with binary variables," Computational Optimization and Applications, Springer, vol. 47(2), pages 257-288, October.
    3. Panos M Pardalos & Oleg A Prokopyev & Stanislav Busygin, 2006. "Continuous Approaches for Solving Discrete Optimization Problems," International Series in Operations Research & Management Science, in: Gautam Appa & Leonidas Pitsoulis & H. Paul Williams (ed.), Handbook on Modelling for Discrete Optimization, chapter 0, pages 39-60, Springer.
    4. Rupaj Kumar Nayak & Nirmalya Kumar Mohanty, 2019. "Improved row-by-row method for binary quadratic optimization problems," Annals of Operations Research, Springer, vol. 275(2), pages 587-605, April.
    5. Rupaj Kumar Nayak & Mahendra Prasad Biswal, 2018. "A low complexity semidefinite relaxation for large-scale MIMO detection," Journal of Combinatorial Optimization, Springer, vol. 35(2), pages 473-492, February.
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