Solving a class of non-convex quadratic problems based on generalized KKT conditions and neurodynamic optimization technique

Malek, Alaeddin ; Hosseinipour-Mahani, Najmeh

Kybernetika, Tome 51 (2015) no. 5, pp. 890-908

Abstract (VO)

In this paper, based on a generalized Karush-Kuhn-Tucker (KKT) method a modified recurrent neural network model for a class of non-convex quadratic programming problems involving a so-called $Z$-matrix is proposed. The basic idea is to express the optimality condition as a mixed nonlinear complementarity problem. Then one may specify conditions for guaranteeing the global solutions of the original problem by using results from the S-lemma. This process is proved by building up a dynamic system from the optimality condition whose equilibrium point is exactly the solution of the mixed nonlinear complementarity problem. By the study of the resulting dynamic system it is shown that under given assumptions, steady states of the dynamic system are stable. Numerical simulations and comparisons with the other methods are presented to illustrate the efficiency of the practical technique that is proposed in this paper.

Détail

DOI : 10.14736/kyb-2015-5-0890

Classification : 37N40, 90C26
Keywords: non-convex quadratic optimization; recurrent neural network model; global optimality conditions; global convergence

Bibliographie

[1] Bazaraa, M. S., Shetty, C. M.: Nonlinear Programming Theory and Algorithms. Wiley and Sons, New York 1990. | MR | Zbl

[2] Beyer, D., Ogier, R.: Tabu learning: A neural network search method for solving nonconvex optimization problems. IEEE Int. Joint Conf. Neural Networks 2 (2000), 953-961.

[3] Bian, W., Xue, X.: Subgradient-based neural networks for nonsmooth nonconvex optimization problems. IEEE Trans. Neural Networks 20 (2009), 6, 1024-1038. | DOI | MR

[4] Chicone, C.: Ordinary Differential Equations with Applications. Second edition. Springer-Verlag, New York 2006. | MR

[5] Forti, M., Nistri, P., Quincampoix, M.: Convergence of neural networks for programming problems via a nonsmooth Lojasiewicz inequality. IEEE Trans. Neural Networka 17 (2006), 6, 1471-1486. | DOI

[6] Gao, X. B.: A novel neural network for nonlinear convex programming problems. IEEE Trans. Neural Network 15 (2004), 613-621. | DOI

[7] Hu, X.: Neurodynamic optimization: Towards nonconvexity. In: Recurrent Neural Networks ( X. Hu and P. Balasubramaniam, ed.), IN-TECH, 2008, pp. 289-308. | DOI

[8] Jeyakumar, V., Rubinov, A. M., Wu, Z. Y.: Non-convex quadratic minimization problems with quadratic constraints: global optimality conditions. Math. Program., Ser. A 110 (2007), 521-541. | DOI | MR | Zbl

[9] Jeyakumar, V., Lee, G. M., Li, G. Y.: Alternative theorems for quadratic inequality systems and global quadratic optimization. SIAM J. Optim 20 (2009), 2, 983-1001. | DOI | MR | Zbl

[10] Jeyakumar, V., Srisatkunarajah, S.: Lagrange multiplier necessary condition for global optimality for non-convex minimization over a quadratic constraint via S-lemma. Optim. Lett. 3 (2009), 23-33. | DOI | MR

[11] Khalil, H. K.: Nonlinear Systems. Third edition. Prentice Hall, 2002.

[12] Malek, A.: Application of recurrent neural networks to optimization problems. In: Recurrent Neural Networks ( X. Hu and P. Balasubramaniam, eds.), IN-TECH, 2008, pp. 255-288. | DOI

[13] Malek, A., Alipour, M.: Numerical solution for linear and quadratic programming problems using a recurrent neural network. Appl. Math. Comput 192 (2007), 27-39. | DOI | MR | Zbl

[14] Malek, A., Hosseinipour-Mahani, N., Ezazipour, S.: Efficient recurrent neural network model for the solution of general nonlinear optimization problems. Optimization Methods and Software 25 (2010), 489-506. | DOI | MR | Zbl

[15] Malek, A., Ezazipour, S., Hosseinipour-Mahani, N.: Double projection neural network for solving pseudomonotone variational inequalities. Fixed Point Theory 12 (2011), 2, 401-418. | MR

[16] Malek, A., Ezazipour, S., Hosseinipour-Mahani, N.: Projected dynamical systems and optimization problems. Bull. Iranian Math. Soc. 37 (2011), 2, 81-96. | MR | Zbl

[17] Malek, A., Yari, A.: Primal-dual solution for the linear programming problem using neural network. Appl. Math. Comput. 169 (2005), 198-211. | DOI | MR

[18] Miller, R. K., Michel, A. N.: Ordinary Differential Equations. Academic Press, 1982. | DOI | MR | Zbl

[19] Polik, I., Terlaky, T.: A survey of the S-Lemma. SIAM Rev. 49 (2007), 371-418. | DOI | MR | Zbl

[20] Sun, C. Y., Feng, C. B.: Neural networks for nonconvex nonlinear programming problems: A switching control approach. In: Lecture Notes in Computer Science 3495, Springer-Verlag, Berlin 2005, pp. 694-699. | DOI | Zbl

[21] Tao, Q., Liu, X., Xue, M. S.: A dynamic genetic algorithm based on continuous neural networks for a kind of non-convex optimization problems. Appl. Math. Comput. 150 (2004), 811-820. | DOI | MR | Zbl

[22] Tian, Y., Lu, Ch.: Nonconvex quadratic reformulations and solvable conditions for mixed integer quadratic programming problems. J. Industr. Managment Optim. 7 (2011), 1027-1039. | DOI | MR | Zbl

[23] Xia, Y., Feng, G., Wang, J.: A recurrent neural network with exponential convergence for solving convex quadratic program and related linear piecewise equation. Neural Networks 17 (2004), 1003-1015. | DOI

[24] Xia, Y. S., Feng, G., Wang, J.: A novel recurrent neural network for solving nonlinear optimization problems with inequality constraints. IEEE Trans. Neural Networks 19 (2008), 1340-1353. | DOI

[25] Xue, X., Bian, W.: A project neural network for solving degenerate convex quadratic program. Neurocomputing 70 (2007), 2449-2459. | DOI

[26] Yan, Z., Wang, J., Li, G.: A collective neurodynamic optimization approach to bound-constrained nonconvex optimization. Neural networks 55 (2014), 20-29. | DOI | Zbl

[27] Yashtini, M., Malek, A.: Solving complementarity and variational inequalities problems using neural networks. Appl. Math. Comput. 190 (2007), 216-230. | DOI | MR | Zbl

[28] Zhang, Y.: Computing a Celis-Dennis-Tapia trust-region step for equality constrained optimization. Math. Programming 55 (1992), 109-124. | DOI | MR | Zbl

[29] Zheng, X. J., Sun, X. L., Li, D., Xu, Y. F.: On zero duality gap in nonconvex quadratic programming problems. Global Optim. 52 (2012), 229-242. | DOI | MR | Zbl