Abstract
In this work, we propose a new approach called “Sequential Linear Programming (SLP) algorithm” for finding an approximate global minimum of continuous and mixed-integer nonconvex quadratic programs (qps). In order to compare our algorithm with the existing approaches, we have developed an implementation with MATLAB and we presented some numerical experiments which compare the performance of our algorithm with the branch and cut algorithm implemented in CPLEX12.8 on 28 concave quadratic test problems, 64 nonconvex quadratic test problems and 12 mixed-integer nonconvex qps. The numerical results show that our algorithm has successfully found similar global objective values as CPLEX12.8 in almost all the considered test problems and it is competitive with CPLEX12.8, particularly in solving large problems (number of variables greater that 50 and less than 1000).
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References
An, L.T.H., Tao, P.D.: A branch and bound method via dc optimization algorithms and ellipsoidal technique for box constrained nonconvex quadratic problems. J. Global Optim. 13(2), 171–206 (1998)
Absil, P.-A., Tits, A.L.: Newton-KKT interior-point methods for indefinite quadratic programming. Comput. Optim. Appl. 36(1), 5–41 (2007)
Bentobache, M., Telli, M., Mokhtari, A.: A global minimization algorithm for concave quadratic programming. In: Proceedings of the 29th European Conference on Operational Research, EURO 2018, p. 329, University of Valencia, 08–11 July 2018
Bentobache, M., Telli, M., Mokhtari, A.: A simplex algorithm with the smallest index rule for concave quadratic programming. In: Proceedings of the Eighth International Conference on Advanced Communications and Computation, INFOCOMP 2018, pp. 88–93, Barcelona, Spain, 22–26 July 2018
Chinchuluun, A., Pardalos, P.M., Enkhbat, R.: Global minimization algorithms for concave quadratic programming problems. Optimization 54(6), 627–639 (2005)
CPLEX12.8, IBM Ilog. Inc., NY (2017)
Floudas, C.A., Pardalos, P.M., Adjiman, C., Esposito, W.R., Gumus, Z.H., Harding, S.T., Klepeis, J.L., Meyer, C.A., Schweiger, C.A.: Handbook of Test Problems in Local and Global Optimization. Nonconvex Optimization and its Applications. Springer, Boston (1999)
Globallib: Gamsworld global optimization library. http://www.gamsworld.org/global/globallib.htm. Accessed 15 Jan 2019
Hiriart-Urruty, J.B., Ledyaev, Y.S.: A note on the characterization of the global maxima of a (tangentially) convex function over a convex set. J. Convex Anal. 3, 55–62 (1996)
Horst, R.: An algorithm for nonconvex programming problems. Math. Program. 10, 312–321 (1976)
Matlab2018a. Mathworks, Inc., NY (2018)
Pardalos, P.M., Rodgers, G.: Computational aspects of a branch and bound algorithm for quadratic zero-one programming. Computing 45(2), 131–144 (1990)
Rusakov, A.I.: Concave programming under simplest linear constraints. Comput. Math. Math. Phys. 43(7), 908–917 (2003)
Strekalovsky, A.S.: Global optimality conditions for nonconvex optimization. J. Global Optim. 12(4), 415–434 (1998)
Sung, Y.Y., Rosen, J.B.: Global minimum test problem construction. Math. Program. 24(1), 353–355 (1982)
Tao, P.D., An, L.T.H.: Convex analysis approach to DC programming: theory, algorithms and applications. Acta Math. Vietnam. 22, 289–355 (1997)
Telli, M., Bentobache, M.: Mokhtari, A: A Successive Linear Approximations Approach for the Global Minimization of a Concave Quadratic Program, Submitted to Computational and Applied Mathematics. Springer (2019)
Tuy, H.: Concave programming under linear constraints. Doklady Akademii Nauk SSSR 159, 32–35 (1964)
Tuy, H.: DC optimization problems. In : Convex analysis and global optimization. Springer optimization and its applications, vol. 110, pp. 167–228, Second edn. Springer, Cham (2016)
Wang, F.: A new exact algorithm for concave knapsack problems with integer variables. Int. J. Comput. Math. 96(1), 126–134 (2019)
Xia, W., Vera, J., Zuluaga, L. F.: Globally solving non-convex quadratic programs via linear integer programming techniques. arXiv preprint, arXiv:1511.02423v3 (2018)
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Bentobache, M., Telli, M., Mokhtari, A. (2020). A Sequential Linear Programming Algorithm for Continuous and Mixed-Integer Nonconvex Quadratic Programming. In: Le Thi, H., Le, H., Pham Dinh, T. (eds) Optimization of Complex Systems: Theory, Models, Algorithms and Applications. WCGO 2019. Advances in Intelligent Systems and Computing, vol 991. Springer, Cham. https://doi.org/10.1007/978-3-030-21803-4_3
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