# American Institute of Mathematical Sciences

October  2010, 6(4): 779-793. doi: 10.3934/jimo.2010.6.779

## Extended canonical duality and conic programming for solving 0-1 quadratic programming problems

 1 Department of Mathematical Sciences, Tsinghua University, Beijing, China 2 Department of Industrial and System Engineering, North Carolina State University, Raleigh, NC 27695, United States

Received  February 2010 Revised  April 2010 Published  September 2010

An extended canonical dual approach for solving 0-1 quadratic programming problems is introduced. We derive the relationship between the optimal solutions to the extended canonical dual problem and the original problem and prove that there exists no duality gap in-between. The extended canonical dual approach leads to a sufficient condition for global optimality, which is more general than known results of this kind. To solve the extended canonical dual problem, we construct corresponding conic programming problems and study their relationship to the extended canonical dual problem. Using this relationship, we design an algorithm for solving the extended canonical dual problem. Our work extends the known solvable sub-class of 0-1 quadratic programming problems.
Citation: Cheng Lu, Zhenbo Wang, Wenxun Xing, Shu-Cherng Fang. Extended canonical duality and conic programming for solving 0-1 quadratic programming problems. Journal of Industrial and Management Optimization, 2010, 6 (4) : 779-793. doi: 10.3934/jimo.2010.6.779
##### References:
 [1] K. Allemand, K. Fukuda, T. M. Liebling and E. Steiner, A polynomial case of unconstrained zero-one quadratic optimization, Mathematical Programming, 91 (2001), 49-52. [2] 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. [3] S.-C. Fang, D. Y. Gao, R.-L. Sheu and S.-Y. Wu, Canonical dual approach to solving 0-1 quadratic programming problem, J. Industrial and Management Optimization, 3 (2007), 125-142. [4] S.-C. Fang, D. Y. Gao, R.-L. Sheu and W. Xing, Global optimization for a class of fractional programming problems, J. Global Optimization, 45 (2009), 337-353. doi: 10.1007/s10898-008-9378-7. [5] D. Y. Gao, Canonical dual transformation method and generalized triality theory in nonsmooth global optimization, J. Global Optimization, 17 (2000), 127-160. doi: 10.1023/A:1026537630859. [6] D. Y. Gao, Advances in canonical duality theory with applications to global optimization, available at: http://www.math.vt.edu/people/gao/papers/focapo08.pdf. [7] M. R. Garey and D. S. Johnson, "Computers and Intractability: A Guide to the Theory of NP-Completeness," W. H. Freeman, San Francisco, CA, 1979. [8] M. X. Goemans and D. P. Williamson, Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming, J. ACM, 42 (1995), 1115-1145. doi: 10.1145/227683.227684. [9] V. Jeyakumar, A. M. Rubinov and Z. Y. Wu, Non-convex quadratic minimization problems with quadratic constraints: Global optimality conditions, Mathematical Programming, 110 (2007), 521-541. doi: 10.1007/s10107-006-0012-5. [10] Q. Jin, S.-C. Fang and W. Xing, On the global optimality of generalized trust region subproblems, Optimization. doi: DOI:10.1080/02331930902995236. [11] E. Klerk and D. V. Pasechnik, Approximation of the stability number of a graph via copositive programming, SIAM J. Optimization, 12 (2002), 875-892. doi: 10.1137/S1052623401383248. [12] C. Lu, Z. Wang and W. Xing, An improved lower bound and approximation algorithm for binary constrained quadratic programming problem, J. Global Optimization. doi: DOI: 10.1007/s10898-009-9504-1. [13] A. Ben-Israel and T. N. E. Greville, "Generalized Inverses," Springer-Verlag, 2003. [14] P. M. Pardalos and G. P. Rodgers, Computational aspects of a branch and bound algorithm for quadratic zero-one programming, Computing, 45 (1990), 131-144. doi: 10.1007/BF02247879. [15] J. F. Sturm and S. Z. Zhang, On cones of nonnegative quadratic functions, Mathematics of Operations Research, 28 (2003), 246-267. doi: 10.1287/moor.28.2.246.14485. [16] X. Sun, C. Liu, D. Li and J. Gao, On duality gap in binary quadratic programming, aviable at: http://www.optimization-online.org/DB_FILE/2010/01/2512.pdf. [17] Z. Wang, S.-C. Fang, D. Y. Gao and W. Xing, Global extremal conditions for multi-integer quadratic programming, J. Industrial and Management Optimization, 4 (2008), 213-225. [18] Z. Wang, S.-C. Fang, D. Y. Gao and W. Xing, Canonical dual approach to solving the maximum cut problem, Working Paper, to appear in J. Global Optimization. [19] L. A. Wolsey, "Integer Programming," Wiley-Interscience, 1998. [20] W. Xing, S.-C. Fang, D. Y. Gao and L. Zhang, Canonical duality solutions to quadratic programming over a quadratic constraint, Proceedings of ICOTA7, (2007), 35-36.

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##### References:
 [1] K. Allemand, K. Fukuda, T. M. Liebling and E. Steiner, A polynomial case of unconstrained zero-one quadratic optimization, Mathematical Programming, 91 (2001), 49-52. [2] 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. [3] S.-C. Fang, D. Y. Gao, R.-L. Sheu and S.-Y. Wu, Canonical dual approach to solving 0-1 quadratic programming problem, J. Industrial and Management Optimization, 3 (2007), 125-142. [4] S.-C. Fang, D. Y. Gao, R.-L. Sheu and W. Xing, Global optimization for a class of fractional programming problems, J. Global Optimization, 45 (2009), 337-353. doi: 10.1007/s10898-008-9378-7. [5] D. Y. Gao, Canonical dual transformation method and generalized triality theory in nonsmooth global optimization, J. Global Optimization, 17 (2000), 127-160. doi: 10.1023/A:1026537630859. [6] D. Y. Gao, Advances in canonical duality theory with applications to global optimization, available at: http://www.math.vt.edu/people/gao/papers/focapo08.pdf. [7] M. R. Garey and D. S. Johnson, "Computers and Intractability: A Guide to the Theory of NP-Completeness," W. H. Freeman, San Francisco, CA, 1979. [8] M. X. Goemans and D. P. Williamson, Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming, J. ACM, 42 (1995), 1115-1145. doi: 10.1145/227683.227684. [9] V. Jeyakumar, A. M. Rubinov and Z. Y. Wu, Non-convex quadratic minimization problems with quadratic constraints: Global optimality conditions, Mathematical Programming, 110 (2007), 521-541. doi: 10.1007/s10107-006-0012-5. [10] Q. Jin, S.-C. Fang and W. Xing, On the global optimality of generalized trust region subproblems, Optimization. doi: DOI:10.1080/02331930902995236. [11] E. Klerk and D. V. Pasechnik, Approximation of the stability number of a graph via copositive programming, SIAM J. Optimization, 12 (2002), 875-892. doi: 10.1137/S1052623401383248. [12] C. Lu, Z. Wang and W. Xing, An improved lower bound and approximation algorithm for binary constrained quadratic programming problem, J. Global Optimization. doi: DOI: 10.1007/s10898-009-9504-1. [13] A. Ben-Israel and T. N. E. Greville, "Generalized Inverses," Springer-Verlag, 2003. [14] P. M. Pardalos and G. P. Rodgers, Computational aspects of a branch and bound algorithm for quadratic zero-one programming, Computing, 45 (1990), 131-144. doi: 10.1007/BF02247879. [15] J. F. Sturm and S. Z. Zhang, On cones of nonnegative quadratic functions, Mathematics of Operations Research, 28 (2003), 246-267. doi: 10.1287/moor.28.2.246.14485. [16] X. Sun, C. Liu, D. Li and J. Gao, On duality gap in binary quadratic programming, aviable at: http://www.optimization-online.org/DB_FILE/2010/01/2512.pdf. [17] Z. Wang, S.-C. Fang, D. Y. Gao and W. Xing, Global extremal conditions for multi-integer quadratic programming, J. Industrial and Management Optimization, 4 (2008), 213-225. [18] Z. Wang, S.-C. Fang, D. Y. Gao and W. Xing, Canonical dual approach to solving the maximum cut problem, Working Paper, to appear in J. Global Optimization. [19] L. A. Wolsey, "Integer Programming," Wiley-Interscience, 1998. [20] W. Xing, S.-C. Fang, D. Y. Gao and L. Zhang, Canonical duality solutions to quadratic programming over a quadratic constraint, Proceedings of ICOTA7, (2007), 35-36.
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