×

zbMATH — the first resource for mathematics

Neural networks for solving second-order cone constrained variational inequality problem. (English) Zbl 1268.90107
Summary: 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 the effectiveness of the proposed neural networks.

MSC:
90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
68T05 Learning and adaptive systems in artificial intelligence
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aghassi, M., Bertsimas, D., Perakis, G.: Solving asymmetric variational inequalities via convex optimization. Oper. Res. Lett. 34, 481–490 (2006) · Zbl 1254.49003 · doi:10.1016/j.orl.2005.09.006
[2] Bertsekas, D.P.: Nonlinear Programming. Athena Scientific, Nashua (1995) · Zbl 0935.90037
[3] 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) · Zbl 1144.90493 · doi:10.1007/s10898-006-9027-y
[4] 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) · Zbl 1175.65070 · doi:10.1016/j.cam.2009.06.022
[5] 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) · Zbl 1187.90291 · doi:10.1016/j.ins.2009.11.014
[6] 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) · Zbl 1153.90542 · doi:10.1007/s10589-007-9086-0
[7] Chen, J.-S., Tseng, P.: An unconstrained smooth minimization reformulation of the second-order cone complementarity problem. Math. Program. 104, 293–327 (2005) · Zbl 1093.90063 · doi:10.1007/s10107-005-0617-0
[8] 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) · Zbl 0894.90143 · doi:10.1090/S0025-5718-98-00932-6
[9] 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) · Zbl 1074.68582 · doi:10.1016/j.neunet.2003.07.006
[10] Effati, S., Ghomashi, A., Nazemi, A.R.: Applocation of projection neural network in solving convex programming problems. Appl. Math. Comput. 188, 1103–1114 (2007) · Zbl 1121.65066 · doi:10.1016/j.amc.2006.10.088
[11] Effati, S., Nazemi, A.R.: Neural network and its application for solving linear and quadratic programming problems. Appl. Math. Comput. 172, 305–331 (2006) · Zbl 1093.65059 · doi:10.1016/j.amc.2005.02.005
[12] Fukushima, M.: Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Math. Program. 53(1), 99–110 (1992) · Zbl 0756.90081 · doi:10.1007/BF01585696
[13] Facchinei, F., Pang, J.: Finite-dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003) · Zbl 1062.90002
[14] Fukushima, M., Luo, Z.-Q., Tseng, P.: Smoothing functions for second-order-cone complimentarity problems. SIAM J. Optim. 12, 436–460 (2002) · Zbl 0995.90094 · doi:10.1137/S1052623400380365
[15] Golden, R.: Mathematical Methods for Neural Network Analysis and Design. MIT Press, Cambridge (1996)
[16] 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)
[17] 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) · Zbl 0979.90107 · doi:10.1023/A:1011245911067
[18] 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) · Zbl 0734.90098 · doi:10.1007/BF01582255
[19] Hartman, P., Stampacchia, G.: On some nonlinear elliptic differential functional equations. Acta Math. 115, 271–310 (1966) · Zbl 0142.38102 · doi:10.1007/BF02392210
[20] Hopfield, J.J., Tank, D.W.: Neural computation of decision in optimization problems. Biol. Cybern. 52, 141–152 (1985) · Zbl 0572.68041
[21] 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) · doi:10.1109/TNN.2006.879774
[22] 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) · doi:10.1109/TSMCB.2006.886166
[23] 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) · Zbl 1190.90239 · doi:10.1137/060657662
[24] Kennedy, M.P., Chua, L.O.: Neural network for nonlinear programming. IEEE Trans. Circuits Syst. 35, 554–562 (1988) · doi:10.1109/31.1783
[25] 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) · Zbl 1106.74044 · doi:10.1002/nme.1493
[26] Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, San Diego (1980) · Zbl 0457.35001
[27] Ko, C.-H., Chen, J.-S., Yang, C.-Y.: A recurrent neural network for solving nonlinear second-order cone programs. Submitted manuscript (2009)
[28] 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) · Zbl 0985.65072
[29] Lions, J., Stampacchia, G.: Variational inequalities. Commun. Pure Appl. Math. 20, 493–519 (1967) · Zbl 0152.34601 · doi:10.1002/cpa.3160200302
[30] Mancino, O., Stampacchia, G.: Convex programming and variational inequalities. J. Optim. Theory Appl. 9, 3–23 (1972) · Zbl 0223.90031 · doi:10.1007/BF00932801
[31] Miller, R.K., Michel, A.N.: Ordinary Differential Equations. Academic Press, San Diego (1982) · Zbl 0552.34001
[32] Ortega, J.M., Rheinboldt, W.C.: Iterative solution of nonlinear equations in several variables. Academic Press, San Diego (1970) · Zbl 0241.65046
[33] 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) · Zbl 1208.90170 · doi:10.1007/s10589-008-9166-9
[34] Patriksson, M.: Nonlinear Programming and Variational Inequality Problems: A Unified Approach. Applied Optimization, vol. 23. Dordrecht, Kluwer (1998) · Zbl 0912.90261
[35] Stampacchia, G.: Formes bilineares coercives sur les ensembles convexes. C. R. Acad. Sci. Paris 258, 4413–4416 (1964) · Zbl 0124.06401
[36] Stampacchia, G.: Variational inequalities. Theory and applications of monotone operators. In: Proceedings of the NATO Advanced Study Institute, pp. 101–192, Venice (1968)
[37] 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) · Zbl 1189.49013 · doi:10.1016/j.camwa.2009.07.084
[38] 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) · doi:10.1109/TCS.1986.1085953
[39] Wright, S.J.: An infeasible-interior-point algorithm for linear complementarity problems. Math. Program. 67, 29–51 (1994) · Zbl 0821.90119 · doi:10.1007/BF01582211
[40] 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) · Zbl 1368.92019 · doi:10.1109/81.995659
[41] 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) · doi:10.1109/TNN.2004.824252
[42] 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) · doi:10.1109/TNN.2004.841779
[43] Yashtini, M., Malek, A.: Solving complementarity and variational inequalities problems using neural networks. Appl. Math. Comput. 190, 216–230 (2007) · Zbl 1128.65052 · doi:10.1016/j.amc.2007.01.036
[44] 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) · doi:10.1109/72.363446
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.