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Journal of Combinatorial Optimization

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

Rupaj Kumar Nayak, Nirmalya Kumar Mohanty

Published Jan 2, 2020• 2 citations
Publisher
DOI
10.1007/s10878-019-00517-8

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This paper addresses the solution of a non-convex 0-1 QCQP using two faster convergent augmented Lagrangian methods (ALM) based on a class of continuous functions and a parameter free convexification method. The binary constraints are converted into a class of continuous functions and the non-convex constraints are convexified by a parameter free method. The proposed algorithm is then tested on a set of QCQP problems available in the research of Zheng and co-workers. We also present the comparison results among relaxed versions of SDPs and some state-of-the-art ALMs and obtained an advantage over their counterparts.
Boolean quadratic optimization problems occur in a number of applications. Their mixed integer-continuous nature is challenging, since it is inherently NP-hard. For this motivation, semidefinite programming relaxations (SDR’s) are proposed in the literature to approximate the solution, which recasts the problem into convex optimization. Nevertheless, SDR’s do not guarantee the extraction of the correct binary minimizer. In this paper, we present a novel approach to enhance the binary solution recovery. The key of the proposed method is the exploitation of known information on the eigenvalues of the desired solution. As the proposed approach yields a non-convex program, we develop and analyze an iterative descent strategy, whose practical effectiveness is shown via numerical results.