# American Institute of Mathematical Sciences

April 2018, 14(2): 625-636. doi: 10.3934/jimo.2017064

## Globally solving quadratic programs with convex objective and complementarity constraints via completely positive programming

 1 School of Economics and Management, University of Chinese Academy of Sciences, Key Laboratory of Big Data Mining and Knowledge Management, Chinese Academy of Sciences, Beijing 100190, China 2 School of Business Administration, Southwestern University of Finance and Economics, Chengdu 611130, China 3 School of Economics and Management, North China Electric Power University, Beijing 102206, China 4 Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China

* Corresponding author: Cheng Lu

Received  August 2016 Revised  April 2017 Published  June 2017

Quadratic programs with complementarity constraints (QPCC) are NP-hard due to the nonconvexity of complementarity relation between the pairs of nonnegative variables. Most of the existing solvers are capable of solving QPCC by finding stationary solutions, which are not able to be verified as global or local optimal ones. In this paper, we aim to globally solve QPCC by a branch-and-bound algorithm, in which the doubly nonnegative (DNN) relaxation in each node is efficiently solved via an augmented Lagrangian method. The method is practically efficient due to the fact that the augmented Lagrangian function can be decomposed into two easy-to-solve subproblems. Computational results demonstrate the effectiveness of the proposed algorithm, with a particular highlight in only a few nodes for some instances.

Citation: Zhi-Bin Deng, Ye Tian, Cheng Lu, Wen-Xun Xing. Globally solving quadratic programs with convex objective and complementarity constraints via completely positive programming. Journal of Industrial & Management Optimization, 2018, 14 (2) : 625-636. doi: 10.3934/jimo.2017064
##### References:
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Schwartz, Theoretical and numerical comparison of relaxation methods for mathematical programs with complementarity constraints, Mathematical Programming, 137 (2013), 257-288. doi: 10.1007/s10107-011-0488-5. [11] J. Hu, J. E. Mitchell, J.-S. Pang, K. P. Bennett and G. Kunapuli, On the global solution of linear programs with linear complementarity constraints, SIAM Journal on Optimization, 19 (2008), 445-471. doi: 10.1137/07068463x. [12] X. X. Huang, D. Li and X. Q. Yang, Convergence of optimal values of quadratic penalty problems for mathematical programs with complementarity constraints, Journal of Industrial and Management Optimization, 2 (2006), 287-296. doi: 10.3934/jimo.2006.2.287. [13] H. Jiang and D. Ralph, QPECgen, a MATLAB generator for mathematical programs with quadratic objectives and affine variational inequality constraints, Computational Optimization and Applications, 13 (1999), 25-59. doi: 10.1023/A:1008696504163. [14] J. J. Júdice and A. Faustino, The linear-quadratic bilevel programming problem, Information Systems and Operational Research, 32 (1994), 87-98. [15] S. Kim, M. Kojima and K.-C. Toh, A lagrangian-dnn relaxation: A fast method for computing tight lower bounds for a class of quadratic optimization problems, Mathematical Programming, 156 (2016), 161-187. doi: 10.1007/s10107-015-0874-5. [16] S. Leyffer, MacMPEC: AMPL collection of mathematical problems with equilibrium constraints, 2015, URL http://wiki.mcs.anl.gov/leyffer/index.php/MacMPEC. [17] C. Lu, W. Xing, S. -C. Fang and Z. Deng, Doubly non-negative relaxation solution based branch-and-bound algorithms for mixed integer quadratic programs, Working paper. [18] C. Lu and X. Guo, Convex reformulation for binary quadratic programming problems via average objective value maximization, Optimization Letters, 9 (2015), 523-535. doi: 10.1007/s11590-014-0768-0. [19] O. Mangasarian and S. Fromovitz, The Fritz John necessary optimality conditions in the presence of equality and inequality constraints, Journal of Mathematical Analysis and Applications, 17 (1967), 37-47. doi: 10.1016/0022-247X(67)90163-1. [20] P. Pardalom and S. Jha, Complexity of uniqueness and local search in quadratic 0-1 programming, Operations Research Letters, 11 (1992), 119-123. doi: 10.1016/0167-6377(92)90043-3. [21] T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, N. J., 1970. [22] J. Sun and S. Zhang, A modified alternating direction method for convex quadratically constrained quadratic semidefinite programs, European Journal of Operational Research, 207 (2010), 1210-1220. doi: 10.1016/j.ejor.2010.07.020. [23] Z. Wen, D. Goldfarb and W. Yin, Alternating direction augmented lagrangian methods for semidefinite programming, Mathematical Programming Computation, 2 (2010), 203-230. doi: 10.1007/s12532-010-0017-1. [24] X.-Y. Zhao, D.-F. Sun and K.-C. Toh, A Newton-CG augmented Lagrangian method for semidefinite programming, SIAM Journal on Optimizaton, 20 (2010), 1737-1765. doi: 10.1137/080718206. [25] J. Zhou, S.-C. Fang and W. Xing, Conic approximation to quadratic optimization with linear complementarity constraints, Computational Optimization and Applications, 66 (2017), 92-122. doi: 10.1007/s10589-016-9855-8.

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##### References:
 [1] L. Bai, J. E. Mitchell and J.-S. Pang, On convex quadratic programs with linear complementarity constraints, Computational Optimization and Applications, 54 (2013), 517-554. doi: 10.1007/s10589-012-9497-4. [2] J. Beasley, OR-Library: distributing test problems by electronic mail, Journal of the Operational Research Society, 41 (1990), 1069-1072. [3] A. Billionnet, S. Elloumi and M.-C. Plateau, Improving the performance of standard solvers for quadratic 0-1 programs by a tight convex reformulation: The QCR method, Discrete Applied Mathematics, 157 (2009), 1185-1197. doi: 10.1016/j.dam.2007.12.007. [4] S. Burer, On the copositive representation of binary and continuous nonconvex quadratic programs, Mathematical Programming, 120 (2009), 479-495. doi: 10.1007/s10107-008-0223-z. [5] S. Burer, Optimizing a polyhedral-semidefinite relaxation of completely positive programs, Mathematical Programming Computation, 2 (2010), 1-19. doi: 10.1007/s12532-010-0010-8. [6] Y.-L. Chang, J.-S. Chen and J. Wu, Proximal point algorithm for nonlinear complementarity problem based on the generalized fischer-burmeister merit function, Journal of Industrial and Management Optimization, 9 (2013), 153-169. doi: 10.3934/jimo.2013.9.153. [7] P. J. Dickinson and L. Gijben, On the computational complexity of membership problems for the completely positive cone and its dual, Computational Optimization and Applications, 57 (2014), 403-415. doi: 10.1007/s10589-013-9594-z. [8] M. C. Ferris and J. S. Pang, Engineering and economic applications of complementarity problems, SIAM Review, 39 (1997), 669-713. doi: 10.1137/S0036144595285963. [9] C. Hao and X. Liu, A trust-region filter-sqp method for mathematical programs with linear complementarity constraints, Journal of Industrial and Management Optimization, 7 (2011), 1041-1055. doi: 10.3934/jimo.2011.7.1041. [10] T. Hoheisel, C. Kanzow and A. Schwartz, Theoretical and numerical comparison of relaxation methods for mathematical programs with complementarity constraints, Mathematical Programming, 137 (2013), 257-288. doi: 10.1007/s10107-011-0488-5. [11] J. Hu, J. E. Mitchell, J.-S. Pang, K. P. Bennett and G. Kunapuli, On the global solution of linear programs with linear complementarity constraints, SIAM Journal on Optimization, 19 (2008), 445-471. doi: 10.1137/07068463x. [12] X. X. Huang, D. Li and X. Q. Yang, Convergence of optimal values of quadratic penalty problems for mathematical programs with complementarity constraints, Journal of Industrial and Management Optimization, 2 (2006), 287-296. doi: 10.3934/jimo.2006.2.287. [13] H. Jiang and D. Ralph, QPECgen, a MATLAB generator for mathematical programs with quadratic objectives and affine variational inequality constraints, Computational Optimization and Applications, 13 (1999), 25-59. doi: 10.1023/A:1008696504163. [14] J. J. Júdice and A. Faustino, The linear-quadratic bilevel programming problem, Information Systems and Operational Research, 32 (1994), 87-98. [15] S. Kim, M. Kojima and K.-C. Toh, A lagrangian-dnn relaxation: A fast method for computing tight lower bounds for a class of quadratic optimization problems, Mathematical Programming, 156 (2016), 161-187. doi: 10.1007/s10107-015-0874-5. [16] S. Leyffer, MacMPEC: AMPL collection of mathematical problems with equilibrium constraints, 2015, URL http://wiki.mcs.anl.gov/leyffer/index.php/MacMPEC. [17] C. Lu, W. Xing, S. -C. Fang and Z. Deng, Doubly non-negative relaxation solution based branch-and-bound algorithms for mixed integer quadratic programs, Working paper. [18] C. Lu and X. Guo, Convex reformulation for binary quadratic programming problems via average objective value maximization, Optimization Letters, 9 (2015), 523-535. doi: 10.1007/s11590-014-0768-0. [19] O. Mangasarian and S. Fromovitz, The Fritz John necessary optimality conditions in the presence of equality and inequality constraints, Journal of Mathematical Analysis and Applications, 17 (1967), 37-47. doi: 10.1016/0022-247X(67)90163-1. [20] P. Pardalom and S. Jha, Complexity of uniqueness and local search in quadratic 0-1 programming, Operations Research Letters, 11 (1992), 119-123. doi: 10.1016/0167-6377(92)90043-3. [21] T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, N. J., 1970. [22] J. Sun and S. Zhang, A modified alternating direction method for convex quadratically constrained quadratic semidefinite programs, European Journal of Operational Research, 207 (2010), 1210-1220. doi: 10.1016/j.ejor.2010.07.020. [23] Z. Wen, D. Goldfarb and W. Yin, Alternating direction augmented lagrangian methods for semidefinite programming, Mathematical Programming Computation, 2 (2010), 203-230. doi: 10.1007/s12532-010-0017-1. [24] X.-Y. Zhao, D.-F. Sun and K.-C. Toh, A Newton-CG augmented Lagrangian method for semidefinite programming, SIAM Journal on Optimizaton, 20 (2010), 1737-1765. doi: 10.1137/080718206. [25] J. Zhou, S.-C. Fang and W. Xing, Conic approximation to quadratic optimization with linear complementarity constraints, Computational Optimization and Applications, 66 (2017), 92-122. doi: 10.1007/s10589-016-9855-8.
 Output: An approximate solution $Y$ and a valid lower bound $v$ of problem (DNP). 1. Set $(S, \sigma)=(S_0, \sigma_0)$ and $v=-\infty$. Initialize $Z=0$. 2: for $k=0, 1, ...$do 3:    Solve problem (Y-Block) to get Y. 4:    Solve problem (Z-Block to get Z. 5:    Update S according to (5). 6:    Update v by optimizing problem (Y-Block) with Z = 0 and σ = 0. 7:    Update σ if necessary. 8:    STOP, if termination criteria are met. 9: end for
 Output: An approximate solution $Y$ and a valid lower bound $v$ of problem (DNP). 1. Set $(S, \sigma)=(S_0, \sigma_0)$ and $v=-\infty$. Initialize $Z=0$. 2: for $k=0, 1, ...$do 3:    Solve problem (Y-Block) to get Y. 4:    Solve problem (Z-Block to get Z. 5:    Update S according to (5). 6:    Update v by optimizing problem (Y-Block) with Z = 0 and σ = 0. 7:    Update σ if necessary. 8:    STOP, if termination criteria are met. 9: end for
 Algorithm 2 A Branch-and-Bound Algorithm for Globally Solving Problem (QPCC) Input: Data (Q; q; A; b; E). Output: A global solution of (QPCC). 1: Preprocessing Step: Calculate the upper bound for variable x. 2: Initialization Step: The list $\mathcal{L}$ to be explored is initialized to contain the root node. Upper bound is set to +1, and lower bound is set to −1. 3: while $\mathcal{L}$ is not empty do 4:    Node Selection Step: Select and remove the best-first node from $\mathcal{L}$. 5:    Bounding Step: Solve the relaxation problem of the node by Algorithm 1. Update upper and lower bound if possible. If the node is not fathomed, go to the next step. 6:    Branching Step: Branch the node on the most violated complementarity constraint, generate two children nodes and add them to the list $\mathcal{L}$. 7: end while
 Algorithm 2 A Branch-and-Bound Algorithm for Globally Solving Problem (QPCC) Input: Data (Q; q; A; b; E). Output: A global solution of (QPCC). 1: Preprocessing Step: Calculate the upper bound for variable x. 2: Initialization Step: The list $\mathcal{L}$ to be explored is initialized to contain the root node. Upper bound is set to +1, and lower bound is set to −1. 3: while $\mathcal{L}$ is not empty do 4:    Node Selection Step: Select and remove the best-first node from $\mathcal{L}$. 5:    Bounding Step: Solve the relaxation problem of the node by Algorithm 1. Update upper and lower bound if possible. If the node is not fathomed, go to the next step. 6:    Branching Step: Branch the node on the most violated complementarity constraint, generate two children nodes and add them to the list $\mathcal{L}$. 7: end while
Computational results of six QPCC instances on MacMPEC
 Id $(m, n, |\mathcal{E}|)$ nodes time (sec.) bilevel2 (29, 13, 12) 13 4.86 bilevel2m (9, 21, 8) 5 1.74 flp4-1 (60,190, 80) 3 8.41 flp4-2 (110,270,110) 1 15.21 flp4-3 (170,380,140) 1 30.03 flp4-4 (250,550,200) 1 67.71
 Id $(m, n, |\mathcal{E}|)$ nodes time (sec.) bilevel2 (29, 13, 12) 13 4.86 bilevel2m (9, 21, 8) 5 1.74 flp4-1 (60,190, 80) 3 8.41 flp4-2 (110,270,110) 1 15.21 flp4-3 (170,380,140) 1 30.03 flp4-4 (250,550,200) 1 67.71
Computation results for randomly generated QPCC Instances
 $(m, n, |\mathcal{E}|)$ Ave. nodes Ave. CPU time (sec.) (4, 12, 3) 3 1.21 (15, 45, 10) 5 9.43 (25, 55, 20) 26 65.21 (30,100, 40) 121 310.66
 $(m, n, |\mathcal{E}|)$ Ave. nodes Ave. CPU time (sec.) (4, 12, 3) 3 1.21 (15, 45, 10) 5 9.43 (25, 55, 20) 26 65.21 (30,100, 40) 121 310.66
Computational results for binary quadratic programs from OR-Library
 $m=50, ~n=100, ~|\mathcal{E}|=50$ Id Optimal value Nodes {CPU time (sec.) bqp50-1 $\le-2160$ 3564 1800.00 bqp50-2 $-3702$ 3063 1570.22 bqp50-3 $-4626$ 69 19.53 bqp50-4 $-3544$ 767 330.86 bqp50-5 $-4012$ 531 226.87 bqp50-6 $-3693$ 107 49.31 bqp50-7 $-4520$ 605 247.22 bqp50-8 $-4216$ 771 363.41 bqp50-9 $-3780$ 3849 1692.21 bqp50-10 $\le-3507$ 3626 1800.00
 $m=50, ~n=100, ~|\mathcal{E}|=50$ Id Optimal value Nodes {CPU time (sec.) bqp50-1 $\le-2160$ 3564 1800.00 bqp50-2 $-3702$ 3063 1570.22 bqp50-3 $-4626$ 69 19.53 bqp50-4 $-3544$ 767 330.86 bqp50-5 $-4012$ 531 226.87 bqp50-6 $-3693$ 107 49.31 bqp50-7 $-4520$ 605 247.22 bqp50-8 $-4216$ 771 363.41 bqp50-9 $-3780$ 3849 1692.21 bqp50-10 $\le-3507$ 3626 1800.00
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