Abstract
This paper proposes a new second-order cone programming (SOCP) relaxation for convex quadratic programs with linear complementarity constraints. The new SOCP relaxation is derived by exploiting the technique that two positive semidefinite matrices can be simultaneously diagonalizable, and is proved to be at least as tight as the classical SOCP relaxation and virtually it can be tighter. We also prove that the proposed SOCP relaxation is equivalent to the semidefinite programming (SDP) relaxation when the objective function is strictly convex. Then an effective branch-and-bound algorithm is designed to find a global optimal solution. Numerical experiments indicate that the proposed SOCP relaxation based branch-and-bound algorithm spends less computing time than the SDP relaxation based branch-and-bound algorithm on condition that the rank of the quadratic objective function is large. The superiority is highlighted when solving the strictly convex quadratic program with linear complementarity constraints.
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References
Bai, L., Mitchell, J.E., Pang, J.S.: On convex quadratic programs with linear complementarity constraints. Comput. Optim. Appl. 54(3), 517–554 (2013)
Bai, L., Mitchell, J.E., Pang, J.S.: Using quadratic convex reformulation to tighten the convex relaxation of a quadratic program with complementarity constraints. Optim. Lett. 8(3), 811–822 (2014)
Ben-Tal, A., Den Hertog, D.: Hidden conic quadratic representation of some nonconvex quadratic optimization problems. Math. Program. 143(1–2), 1–29 (2014)
Braun, S., Mitchell, J.E.: A semidefinite programming heuristic for quadratic programming problems with complementarity constraints. Comput. Optim. Appl. 31(1), 5–29 (2005)
Burer, S., Kim, S., Kojima, M.: Faster, but weaker, relaxations for quadratically constrained quadratic programs. Comput. Optim. Appl. 59(1–2), 27–45 (2014)
Deng, Z., Tian, Y., Lu, C., Xing, W.: Globally solving quadratic programs with convex objective and complementarity constraints via completely positive programming. J. Ind. Manag. Optim. 14(2), 625–636 (2018)
Jiang, H., Ralph, D.: QPECgen, a MATLAB generator for mathematical programs with quadratic objectives and affine variational inequality constraints. Comput. Optim. Appl. 13, 25–59 (1999)
Jiang, R., Li, D.: Simultaneous diagonalization of matrices and its applications in quadratically constrained quadratic programming. SIAM J. Optim. 26(3), 1649–1668 (2016)
Júdice, J.J., Faustino, A.: The linear-quadratic bilevel programming problem. INFOR Inf. Syst. Oper. Res. 32, 87–98 (1994)
Kim, S., Kojima, M.: Second order cone programming relaxation of nonconvex quadratic optimization problems. Optim. Methods Softw. 15, 201–224 (2001)
Kocuk, B., Dey, S.S., Sun, X.A.: Strong SOCP relaxations for the optimal power flow problem. Oper. Res. 64(6), 1177–1196 (2016)
Liu, G.S., Zhang, J.Z.: A new branch and bound algorithm for solving quadratic programs with linear complementarity constraints. J. Comput. Appl. Math. 146(1), 77–87 (2002)
Liu, G.S., Zhang, J.Z.: Semidefinite relaxation of quadratic optimization problems. IEEE Signal Process. Mag. 27(3), 20–34 (2010)
Muramatsu, M., Suzuki, T.: A new second-order cone programming relaxation for max-cut problems. J. Oper. Res. Soc. Jpn. 46(2), 164–177 (2003)
Newcomb, R.W.: On the simultaneous diagonalization of two semi-definite matrices. Q. Appl. Math. 19(2), 144–146 (1961)
Sturm, J.F.: Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim. Methods Softw. 11(1–4), 625–653 (1999)
Wang, J., Lu, J., Feng, Y.: Congruence diagonalization of two hermite matrices simultaneously. Int. J. Algebra 4(23), 1119–1125 (2010)
Zhou, J., Fang, S.-C., Xing, W.: Conic approximation to quadratic optimization with linear complementarity constraints. Comput. Optim. Appl. 66(1), 97–122 (2017)
Acknowledgements
Zhou’s research has been supported by the National Natural Science Foundation of China under Grant No. 11701512, the Zhejiang Provincial Natural Science Foundation of China under Grant No. LQ16A010010, and the Swiss Government Excellence Scholarship. Xu’s research has been supported by the the National Natural Science Foundation of China under Grant Nos. 11704336, 11647081, and the Zhejiang Provincial Natural Science Foundation of China under Grant No. LQ15A040003. The authors would like to thank Prof. Martin Jaggi at EPFL for his helpful comments.
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Zhou, J., Xu, Z. A simultaneous diagonalization based SOCP relaxation for convex quadratic programs with linear complementarity constraints. Optim Lett 13, 1615–1630 (2019). https://doi.org/10.1007/s11590-018-1337-8
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DOI: https://doi.org/10.1007/s11590-018-1337-8