Abstract
This paper proposes a conic approximation algorithm for solving quadratic optimization problems with linear complementarity constraints.We provide a conic reformulation and its dual for the original problem such that these three problems share the same optimal objective value. Moreover, we show that the conic reformulation problem is attainable when the original problem has a nonempty and bounded feasible domain. Since the conic reformulation is in general a hard problem, some conic relaxations are further considered. We offer a condition under which both the semidefinite relaxation and its dual problem become strictly feasible for finding a lower bound in polynomial time. For more general cases, by adaptively refining the outer approximation of the feasible set, we propose a conic approximation algorithm to identify an optimal solution or an \(\epsilon \)-optimal solution of the original problem. A convergence proof is given under simple assumptions. Some computational results are included to illustrate the effectiveness of the proposed algorithm.
References
Arima, N., Kim, S., Kojima, M.: Simplified copositive and lagrangian relaxations for linearly constrained quadratic optimization problems in continuous and binary variables. Pac. J. Optim. 10(3), 437–451 (2013)
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.: On conic QPCCs, conic QCQPs and completely positive programs. Math. Program. Ser. A (2015)
Beasley, J.E.: Heuristic algorithms for the unconstrained binary quadratic programming problem. Technical report (1998)
Ben-Tal, A., Nemirovski, A.: Lectures on Modern Convex Optimization, Analysis, Algorithms and Engineering Applications. MPS/SIAM Series on Optimization, 1st edn. SIAM, Philadelphia (2001)
Billionnet, A., Elloumi, S.: Using a mixed integer quadratic programming solver for the unconstrained quadratic 0–1 problem. Math. Program. 109(1), 55–68 (2007)
Buchheim, C., Wiegele, A.: Semidefinite relaxations for non-convex quadratic mixed-integer programming. Math. Program. 141(1–2), 435–452 (2013)
Burer, S.: On the copositive representation of binary and continuous nonconvex quadratic programs. Math. Program. 120(2), 479–495 (2009)
Burer, S.: Optimizing a polyhedral-semidefinite relaxation of completely positive programs. Math. Prog. Comput. 2(1), 1–19 (2010)
Burer, S.: A gentle, geometric introduction to copositive optimization. Math. Program. Ser. B 151, 89–116 (2015)
Chen, X., Ye, J.J.: A class of quadratic programs with linear complementarity constraints. Set-Valued Var. Anal. 17(2), 113–133 (2009)
Engau, A., Anjos, M., Vannelli, A.: On handling cutting planes in interior-point methods for solving semidefinite relaxations of binary quadratic optimization problems. Optim. Methods Softw. 27(3), 539–559 (2012)
Fang, S.-C., Xing, W.: Linear Conic Optimization. Science Press, Beijing (2013)
Ferris, M.C., Pang, J.S.: Engineering and economic applications of complementarity problems. SIAM Rev. 39, 669–713 (1997)
Floudas, C.A., Pardalos, P.M.: Handbook of Test Problems in Local and Global Optimization, 1st edn. Springer, Dordrecht (1999)
Ghaddar, B., Anjos, M.F., Liers, F.: A branch-and-cut algorithm based on semidefinite programming for the minimum k-partition problem. Ann. Oper. Res. 188(1), 155–174 (2007)
Grant, M., Boyed, S.: CVX: Matlab software for disciplined convex programming, version 2.0(beta), http://cvxr.com/cvx (2013)
Hu, J., Mitchell, J.E., Pang, J.S., Bennett, K.P., Kunapuli, G.: On the global solution of linear programs with linear complementarity constraints. SIAM J. Optim. 19(1), 445–471 (2008)
Júdice, J.J., Faustino, A.M., Ribeiro, I.M.: On the solution of NP-hard linear complementarity problems. Sociedad de Estadística e Investigación Operativa TOP 10(1), 125–145 (2002)
Júdice, J.J.: Algorithms for linear programming with linear complementarity constraints. TOP 20(1), 4–25 (2012)
Kanzow, C., Schwartz, A.: A new regularization method for mathematical programs with complementarity constraints with strong convengence properities. SIAM J. Optim. 23(2), 770–798 (2013)
Kim, S., Kojima, M., Kanzow, C., Schwartz, A.: A Lagrangian-DNN relaxation: a fast method for computing tight lower bounds for a class of quadratic optimization problems. Math. Program. Ser. A 156, 161–187 (2016)
Leyffer, S., López-Calva, G., Nocedal, J.: Interior methods for mathematical programs with complementarity constraints. SIAM J. Optim. 17(1), 52–77 (2006)
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)
Lu, C., Jin, Q., Fang, S.-C., Wang, Z., Xing, W.: Adaptive computable approximation to cones of nonnegative quadratic functions. Optimization 64, 955–980 (2014)
Lu, C., Guo, X.: Convex reformulation for binary quadratic programming problems via average objective value maximization. Optim. Lett. 9(3), 523–535 (2014)
Luo, Z.Q., Ma, W.K., Mancho So, A., Ye, Y., Zhang, S.: Semidefinite relaxation of quadratic optimization problems. IEEE Signal Process. Mag. 27, 20–34 (2010)
Mitchell, J.E., Pang, J.S., Yu, B.: Obtaining Tighter Relaxations of Mathematical Programs with Complementarity Constraints, Modeling and Optimization: Theory and Applications. Springer, New York (2012)
Murty, K.G., Kabadi, S.N.: Some NP-complete problems in quadratic and nonlinear programming. Math. Program. 39, 117–129 (1987)
Rinaldi, G.: Rudy, http://www-user.tu-chemnitz.de/~helmberg/rudy.tar.gz (1998)
Saxena, A., Bonami, P., Lee, J.: Convex relaxations of non-convex mixed integer quadratically constrained programs: extended formulations. Math. Program. 124(1–2), 383–411 (2010)
Saxena, A., Bonami, P., Lee, J.: Convex relaxations of non-convex mixed integer quadratically constrained programs: projected formulations. Math. Program. 130(2), 359–413 (2011)
Sturm, J.F., Zhang, S.: On cones of nonnegative quadratic functions. Math. Oper. Res. 28, 246–267 (2003)
Tian, Y., Fang, S.-C., Deng, Z., Xing, W.: Computable representation of the cone of nonnegative quadratic forms over a general second-order cone and its application to completely positive programming. J. Ind. Manag. Optim. 9, 703–721 (2013)
Tuncel, L.: On the Slater condition for the SDP relaxations of nonconvex sets. Oper. Res. Lett. 29, 181–186 (2001)
Wiegele A.: Biq Mac Library, http://biqmac.uni-klu.ac.at/biqmaclib.html (2007)
Zhou, J., Chen, D., Wang, Z., Xing, W.: A conic approximation method for the 0–1 quadratic knapsack problem. J. Ind. Manag. Optim. 9, 531–547 (2013)
Zhou, J., Deng, Z., Fang, S.-C., Xing, W.: Detection of a copositive matrix over a \(p\)-th order cone. Pac. J. Optim. 10(3), 593–611 (2014)
Acknowledgments
This work has been partially supported by the National Natural Science Foundation of China under Grant Numbers 11171177, 11371216, 11526186 and 11571029, the Zhejiang Provincial Natural Science Foundation of China under Grant Number LQ16A010010, and US Army Research Office Grant Number W911NF-15-1-0223. The authors would also like to thank the editor and reviewers for their most valuable comments and suggestions.
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Zhou, J., Fang, SC. & Xing, W. Conic approximation to quadratic optimization with linear complementarity constraints. Comput Optim Appl 66, 97–122 (2017). https://doi.org/10.1007/s10589-016-9855-8
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DOI: https://doi.org/10.1007/s10589-016-9855-8