Abstract
In this paper, we consider using the neural networks to efficiently solve the second-order cone constrained variational inequality (SOCCVI) problem. More specifically, two kinds of neural networks are proposed to deal with the Karush-Kuhn-Tucker (KKT) conditions of the SOCCVI problem. The first neural network uses the Fischer-Burmeister (FB) function to achieve an unconstrained minimization which is a merit function of the Karush-Kuhn-Tucker equation. We show that the merit function is a Lyapunov function and this neural network is asymptotically stable. The second neural network is introduced for solving a projection formulation whose solutions coincide with the KKT triples of SOCCVI problem. Its Lyapunov stability and global convergence are proved under some conditions. Simulations are provided to show effectiveness of the proposed neural networks.
Similar content being viewed by others
References
Aghassi, M., Bertsimas, D., Perakis, G.: Solving asymmetric variational inequalities via convex optimization. Oper. Res. Lett. 34, 481–490 (2006)
Bertsekas, D.P.: Nonlinear Programming. Athena Scientific, Nashua (1995)
Chen, J.-S.: The semismooth-related properties of a merit function and a decent method for the nonlinear complementarity problem. J. Glob. Optim. 36, 565–580 (2006)
Chen, J.-S., Gao, H.-T., Pan, S.-H.: An R-linearly convergent derivative-free algorithm for nonlinear complementarity problems based on the generalized Fischer-Burmeister merit function. J. Comput. Appl. Math. 232, 455–471 (2009)
Chen, J.-S., Ko, C.-H., Pan, S.-H.: A neural network based on the generalized Fischer-Burmeister function for nonlinear complementarity problems. Inf. Sci. 180, 697–711 (2010)
Chen, J.-S., Pan, S.-H.: A family of NCP functions and a descent method for the nonlinear complementarity problem. Comput. Optim. Appl. 40, 389–404 (2008)
Chen, J.-S., Tseng, P.: An unconstrained smooth minimization reformulation of the second-order cone complementarity problem. Math. Program. 104, 293–327 (2005)
Chen, X., Qi, L., Sun, D.: Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities. Math. Comput. 67(222), 519–540 (1998)
Dang, C., Leung, Y., Gao, X., Chen, K.: Neural networks for nonlinear and mixed complementarity problems and their applications. Neural Netw. 17, 271–283 (2004)
Effati, S., Ghomashi, A., Nazemi, A.R.: Applocation of projection neural network in solving convex programming problems. Appl. Math. Comput. 188, 1103–1114 (2007)
Effati, S., Nazemi, A.R.: Neural network and its application for solving linear and quadratic programming problems. Appl. Math. Comput. 172, 305–331 (2006)
Fukushima, M.: Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Math. Program. 53(1), 99–110 (1992)
Facchinei, F., Pang, J.: Finite-dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003)
Fukushima, M., Luo, Z.-Q., Tseng, P.: Smoothing functions for second-order-cone complimentarity problems. SIAM J. Optim. 12, 436–460 (2002)
Golden, R.: Mathematical Methods for Neural Network Analysis and Design. MIT Press, Cambridge (1996)
Hammond, J.: Solving asymmetric variational inequality problems and systems of equations with generalized nonlinear programming algorithms. Ph.D. Dissertation, Department of Mathematics, MIT, Cambridge (1984)
Han, Q., Liao, L.-Z., Qi, H., Qi, L.: Stability analysis of gradient-based neural networks for optimization problems. J. Glob. Optim. 19, 363–381 (2001)
Harker, P., Pang, J.-S.: Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications. Math. Program. 48, 161–220 (1990)
Hartman, P., Stampacchia, G.: On some nonlinear elliptic differential functional equations. Acta Math. 115, 271–310 (1966)
Hopfield, J.J., Tank, D.W.: Neural computation of decision in optimization problems. Biol. Cybern. 52, 141–152 (1985)
Hu, X., Wang, J.: Solving pseudomonotone variational inequalities and pseudoconvex optimization problems using the projection neural network. IEEE Trans. Neural Netw. 17, 1487–1499 (2006)
Hu, X., Wang, J.: A recurrent neural network for solving a class of general variational inequalities. IEEE Trans. Syst. Man Cybern. B 37, 528–539 (2007)
Kanzow, C., Ferenczi, I., Fukushima, M.: On the local convergence of semismooth Newton methods for linear and nonlinear second-order cone programs without strict complementarity. SIAM J. Optim. 20, 297–320 (2009)
Kennedy, M.P., Chua, L.O.: Neural network for nonlinear programming. IEEE Trans. Circuits Syst. 35, 554–562 (1988)
Kanno, Y., Martins, J.A.C., Pinto da Costa, A.: Three-dimensional quasi-static frictional contact by using second-order cone linear complementarity problem. Int. J. Numer. Methods Eng. 65, 62–83 (2006)
Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, San Diego (1980)
Ko, C.-H., Chen, J.-S., Yang, C.-Y.: A recurrent neural network for solving nonlinear second-order cone programs. Submitted manuscript (2009)
Liao, L.-Z., Qi, H., Qi, L.: Solving nonlinear complementarity problems with neural networks: a reformulation method approach. J. Comput. Appl. Math. 131, 342–359 (2001)
Lions, J., Stampacchia, G.: Variational inequalities. Commun. Pure Appl. Math. 20, 493–519 (1967)
Mancino, O., Stampacchia, G.: Convex programming and variational inequalities. J. Optim. Theory Appl. 9, 3–23 (1972)
Miller, R.K., Michel, A.N.: Ordinary Differential Equations. Academic Press, San Diego (1982)
Ortega, J.M., Rheinboldt, W.C.: Iterative solution of nonlinear equations in several variables. Academic Press, San Diego (1970)
Pan, S.-H., Chen, J.-S.: A semismooth Newton method for the SOCCP based on a one-parametric class of SOC complementarity functions. Comput. Optim. Appl. 45, 59–88 (2010)
Patriksson, M.: Nonlinear Programming and Variational Inequality Problems: A Unified Approach. Applied Optimization, vol. 23. Dordrecht, Kluwer (1998)
Stampacchia, G.: Formes bilineares coercives sur les ensembles convexes. C. R. Acad. Sci. Paris 258, 4413–4416 (1964)
Stampacchia, G.: Variational inequalities. Theory and applications of monotone operators. In: Proceedings of the NATO Advanced Study Institute, pp. 101–192, Venice (1968)
Sun, J.-H., Zhang, L.-W.: A globally convergent method based on Fischer-Burmeister operators for solving second-order-cone constrained variational inequality problems. Comput. Math. Appl. 58, 1936–1946 (2009)
Tank, D.W., Hopfield, J.J.: Simple neural optimization network: an A/D converter, signal decision circuit, and a linear programming circuit. IEEE Trans. Circuits Syst. 33, 533–541 (1986)
Wright, S.J.: An infeasible-interior-point algorithm for linear complementarity problems. Math. Program. 67, 29–51 (1994)
Xia, Y., Leung, H., Wang, J.: A projection neural network and its application to constrained optimization problems. IEEE Trans. Circuits Syst. I 49, 447–458 (2002)
Xia, Y., Leung, H., Wang, J.: A general projection neural network for solving monotone variational inequalities and related optimization problems. IEEE Trans. Neural Netw. 15, 318–328 (2004)
Xia, Y., Wang, J.: A recurrent neural network for solving nonlinear convex programs subject to linear constraints. IEEE Trans. Neural Netw. 16, 379–386 (2005)
Yashtini, M., Malek, A.: Solving complementarity and variational inequalities problems using neural networks. Appl. Math. Comput. 190, 216–230 (2007)
Zak, S.H., Upatising, V., Hui, S.: Solving linear programming problems with neural networks: a comparative study. IEEE Trans. Neural Netw. 6, 94–104 (1995)
Author information
Authors and Affiliations
Corresponding author
Additional information
J. Sun is also affiliated with Department of Mathematics, National Taiwan Normal University.
J.-S. Chen is member of Mathematics Division, National Center for Theoretical Sciences, Taipei Office. The author’s work is partially supported by National Science Council of Taiwan.
Rights and permissions
About this article
Cite this article
Sun, J., Chen, JS. & Ko, CH. Neural networks for solving second-order cone constrained variational inequality problem. Comput Optim Appl 51, 623–648 (2012). https://doi.org/10.1007/s10589-010-9359-x
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10589-010-9359-x