The main outcomes of the paper are divided into two parts. First, we present a new dual for quadratic programs, in which, the dual variables are affine functions, and we prove strong duality. Since the new dual is intractable, we consider a modified version by restricting the feasible set. This leads to a new bound for quadratic programs. We demonstrate that the dual of the bound is a semi-definite relaxation of quadratic programs. In addition, we probe the relationship between this bound and the well-known bounds in the literature. In the second part, thanks to the new bound, we propose a branch and cut algorithm for concave quadratic programs. We establish that the algorithm enjoys global convergence. The effectiveness of the method is illustrated for numerical problem instances.
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Anstreicher, K.M.: Semidefinite programming versus the reformulation-linearization technique for nonconvex quadratically constrained quadratic programming. J. Glob. Optim. 43(2–3), 471–484 (2009)
Bao, X., Sahinidis, N.V., Tawarmalani, M.: Semidefinite relaxations for quadratically constrained quadratic programming: a review and comparisons. Math. Program. 129(1), 129 (2011)
Belotti, P.: Couenne: A User’s Manual. Tech. rep., Lehigh University (2009)
Ben-Tal, A., Nemirovski, A.: Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications, vol. 2. Siam, Philadelphia (2001)
Bomze, I.M.: Copositive relaxation beats lagrangian dual bounds in quadratically and linearly constrained quadratic optimization problems. SIAM J. Optim. 25(3), 1249–1275 (2015)
Bomze, I.M., Locatelli, M., Tardella, F.: New and old bounds for standard quadratic optimization: dominance, equivalence and incomparability. Math. Program. 115(1), 31 (2008)
Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)
Burer, S., Dong, H.: Separation and relaxation for cones of quadratic forms. Math. Program. 137(1–2), 343–370 (2013)
Burer, S., Vandenbussche, D.: A finite branch-and-bound algorithm for nonconvex quadratic programming via semidefinite relaxations. Math. Program. 113(2), 259–282 (2008)
Burkard, R.E., Cela, E., Pardalos, P.M., Pitsoulis, L.S.: The quadratic assignment problem. In: Du, D.-Z., Pardalos, P.M. (eds.) Handbook of Combinatorial Optimization, pp. 1713–1809. Springer, Berlin (1998)
Chen, J., Burer, S.: Globally solving nonconvex quadratic programming problems via completely positive programming. Math. Program. Comput. 4(1), 33–52 (2012)
Chuong, T., Jeyakumar, V.: Generalized lagrangian duality for nonconvex polynomial programs with polynomial multipliers. J. Glob. Optim. 72(4), 655–678 (2018)
Dür, M., Horst, R.: Lagrange duality and partitioning techniques in nonconvex global optimization. J. Optim. Theory Appl. 95(2), 347–369 (1997)
Floudas, C.A., Visweswaran, V.: Quadratic optimization. In: Horst, R., Pardalos, P.M. (eds.) Handbook of global optimization, pp. 217–269. Springer, Berlin (1995)
Fourer, R., Gay, D., Kernighan, B.: Ampl (vol. 117). Danvers, MA: Boyd & Fraser (1993)
Globallib: Gamsworld. http://www.gamsworld.org/global/globallib.htm (2013)
Gondzio, J., Yildirim, E.A.: Global Solutions of Nonconvex Standard Quadratic Programs Via Mixed Integer Linear Programming Reformulations. arXiv preprint arXiv:1810.02307 (2018)
Gorge, A., Lisser, A., Zorgati, R.: Generating cutting planes for the semidefinite relaxation of quadratic programs. Comput. Oper. Res. 55, 65–75 (2015)
Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches. Springer Science & Business Media, Berlin (1996)
ILOG, I.: Cplex 9.0 reference manual. ILOG CPLEX Division (2003)
Jiang, B., Li, Z., Zhang, S.: On cones of nonnegative quartic forms. Found. Comput. Math. 17(1), 161–197 (2017)
Kim, S., Kojima, M., Waki, H.: Generalized lagrangian duals and sums of squares relaxations of sparse polynomial optimization problems. SIAM J. Optim. 15(3), 697–719 (2005)
Konno, H.: Maximization of a convex quadratic function under linear constraints. Math. Program. 11(1), 117–127 (1976)
Konno, H., Thach, P.T., Tuy, H.: Optimization on Low Rank Nonconvex Structures, vol. 15. Springer Science & Business Media, Berlin (2013)
Lasserre, J.B.: An Introduction to Polynomial and Semi-Algebraic Optimization, vol. 52. Cambridge University Press, Cambridge (2015)
Lasserre, J.B., Toh, K.C., Yang, S.: A bounded degree sos hierarchy for polynomial optimization. EURO J. Comput. Optim. 5(1–2), 87–117 (2017)
Laurent, M., Sun, Z.: Handelman’s hierarchy for the maximum stable set problem. J. Glob. Optim. 60(3), 393–423 (2014)
Locatelli, M., Schoen, F.: Global Optimization: Theory, Algorithms, and Applications, vol. 15. Siam, Philadelphia (2013)
Lovász, L., Schrijver, A.: Cones of matrices and set-functions and 0–1 optimization. SIAM J. Optim. 1(2), 166–190 (1991)
Luenberger, D.G.: A double look at duality. IEEE Trans. Autom. Control 37(10), 1474–1482 (1992)
Luo, Z.Q., Ma, W.K., So, A.M.C., Ye, Y., Zhang, S.: Semidefinite relaxation of quadratic optimization problems. IEEE Signal Process. Mag. 27(3), 20–34 (2010)
Mangasarian, O.L.: Nonlinear Programming. Siam, Philadelphia (1974)
MOSEK, A.: The mosek optimization toolbox for matlab manual. version 8.1. http://docs.mosek.com/8.1/toolbox/index.html (2017)
Nesterov, Y., Wolkowicz, H., Ye, Y.: Semidefinite programming relaxations of nonconvex quadratic optimization. In: Wolkowicz, H., Saigal, R., Vandenberghe, L. (eds.) Handbook of Semidefinite Programming, pp. 361–419. Springer, Berlin (2000)
Nie, J.: Optimality conditions and finite convergence of lasserre’s hierarchy. Math. Program. 146(1–2), 97–121 (2014)
Pardalos, P.M., Vavasis, S.A.: Quadratic programming with one negative eigenvalue is np-hard. J. Glob. Optim. 1(1), 15–22 (1991)
Pólik, I., Terlaky, T.: A survey of the s-lemma. SIAM Rev. 49(3), 371–418 (2007)
Renegar, J.: A Mathematical View of Interior-Point Methods in Convex Optimization, vol. 3. Siam, Philadelphia (2001)
Sahinidis, N.V., Tawarmalani, M.: Baron 18.11.12: Global optimization of mixed-integer nonlinear programs. User’s manual (2017)
Sahni, S.: Computationally related problems. SIAM J. Comput. 3(4), 262–279 (1974)
Sherali, H.D., Tuncbilek, C.H.: A global optimization algorithm for polynomial programming problems using a reformulation-linearization technique. J. Glob. Optim. 2(1), 101–112 (1992)
Sherali, H.D., Tuncbilek, C.H.: A reformulation-convexification approach for solving nonconvex quadratic programming problems. J. Glob. Optim. 7(1), 1–31 (1995)
Shor, N.Z.: Dual quadratic estimates in polynomial and boolean programming. Ann. Oper. Res. 25(1), 163–168 (1990)
Sponsel, J., Bundfuss, S., Dür, M.: An improved algorithm to test copositivity. J. Glob. Optim. 52(3), 537–551 (2012)
Sturm, J.F., Zhang, S.: On cones of nonnegative quadratic functions. Math. Oper. Res. 28(2), 246–267 (2003)
Tuy, H.: Convex Analysis and Global Optimization (Springer Optimization and Its Applications Book 110). Springer, Berlin (2016)
Xia, W., Vera, J., Zuluaga, L.F.: Globally solving non-convex quadratic programs via linear integer programming techniques. arXiv preprint arXiv:1511.02423 (2015)
Zamani, M.: New bounds for nonconvex quadratically constrained quadratic programming. arXiv preprint arXiv:1902.08861 (2019)
Zheng, X., Sun, X., Li, D., Xu, Y.: On zero duality gap in nonconvex quadratic programming problems. J. Glob. Optim. 52(2), 229–242 (2012)
I am very grateful to two anonymous referees for their valuable comments and suggestions which help to improve the paper considerably. The author would like to thank associate editor for the very thoughtful comments provided on the first version of this manuscript. This research was in part supported by a grant from the Iran National Science Foundation (No. 96010653, Principal investigator: Dr. Majid Soleimani-damaneh).
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Zamani, M. A new algorithm for concave quadratic programming. J Glob Optim 75, 655–681 (2019). https://doi.org/10.1007/s10898-019-00787-w
- Non-convex quadratic programming
- Semi-definite relaxation
- Branch and cut method
- Concave quadratic programming