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A Lagrangian–DNN relaxation: a fast method for computing tight lower bounds for a class of quadratic optimization problems

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Abstract

We propose an efficient computational method for linearly constrained quadratic optimization problems (QOPs) with complementarity constraints based on their Lagrangian and doubly nonnegative (DNN) relaxation and first-order algorithms. The simplified Lagrangian–completely positive programming (CPP) relaxation of such QOPs proposed by Arima, Kim, and Kojima in 2012 takes one of the simplest forms, an unconstrained conic linear optimization problem with a single Lagrangian parameter in a CPP matrix variable with its upper-left element fixed to 1. Replacing the CPP matrix variable by a DNN matrix variable, we derive the Lagrangian–DNN relaxation, and establish the equivalence between the optimal value of the DNN relaxation of the original QOP and that of the Lagrangian–DNN relaxation. We then propose an efficient numerical method for the Lagrangian–DNN relaxation using a bisection method combined with the proximal alternating direction multiplier and the accelerated proximal gradient methods. Numerical results on binary QOPs, quadratic multiple knapsack problems, maximum stable set problems, and quadratic assignment problems illustrate the superior performance of the proposed method for attaining tight lower bounds in shorter computational time.

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Authors and Affiliations

  1. Department of Mathematics, Ewha W. University, 52 Ewhayeodae-gil, Sudaemoon-gu, Seoul, 120-750, Korea

    Sunyoung Kim

  2. Research and Development Initiative, Chuo University, 1-13-27, Kasuga, Bunkyo-ku, Tokyo, 112-8551, Japan

    Masakazu Kojima

  3. Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore, 119076, Singapore

    Kim-Chuan Toh

Authors
  1. Sunyoung Kim
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  2. Masakazu Kojima
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  3. Kim-Chuan Toh
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Correspondence to Sunyoung Kim.

Additional information

Sunyoung Kim’s research was supported by NRF 2012-R1A1A2-038982 and NRF 2014-R1A2A1A11049618. Masakazu Kojima’s research was supported by the Japan Science and Technology Agency (JST), the Core Research of Evolutionary Science and Technology (CREST) research project. Kim-Chuan Toh’s research was supported in part by Ministry of Education Academic Research Fund R-146-000-168-112.

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Kim, S., Kojima, M. & Toh, KC. A Lagrangian–DNN relaxation: a fast method for computing tight lower bounds for a class of quadratic optimization problems. Math. Program. 156, 161–187 (2016). https://doi.org/10.1007/s10107-015-0874-5

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  • DOI: https://doi.org/10.1007/s10107-015-0874-5

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