Solving Linear Semi-Infinite Programming Problems Using Recurrent Neural Networks

Document Type: بنیادی - نظری

Authors

1 Tarbiat Modarres university

2 Payame Noor university

Abstract

‎Linear semi-infinite programming problem is an important class of optimization problems which deals with infinite constraints‎. ‎In this paper‎, ‎to solve this problem‎, ‎we combine a discretization method and a neural network method‎. ‎By a simple discretization of the infinite constraints,we convert the linear semi-infinite programming problem into linear programming problem‎. ‎Then‎, ‎we use a recurrent neural network model‎, ‎with a simple structure based on a dynamical system to solve this problem‎. ‎The portfolio selection problem and some other numerical examples are solved to evaluate the effectiveness of the presented model.

Keywords

Main Subjects


[1] Abbe‎ ‎L. (2001) " ‎Two logarithmic barrier methods for convex semi-infinite programming ''‎, ‎In M‎. ‎A‎. ‎Goberna and M‎. ‎A‎. ‎Lopez‎, ‎(editors) Semi-infinite Programming Recent Advances‎, ‎Nonconvex Optimization and Its Applications 57‎, ‎169-195‎.

[2] Alipour ‎M.‎, ‎Rostamy‎ ‎D.‎, ‎Malek‎, ‎A. (2011) " ‎A recurrent neural network for nonlinear convex optimization with application to a class of variational inequalities problems ''‎, ‎Australian Journal of Basic and Applied Sciences‎, ‎5‎, ‎5‎, ‎894-909‎.

[3] Anderson ‎E‎. ‎J.‎, ‎Levis ‎A‎. ‎S. (1989) " ‎An extension of the simplex algorithm for semi-infinite linear programming ''‎, ‎Mathematical Programming 44‎, ‎247-269‎.

[4] Chen ‎Y‎. ‎H.‎, ‎Fang ‎S‎. ‎C. (1998) " ‎Solving convex programming with equality constraints by neural networks ''‎, ‎Computers & Mathematics with Applications‎, ‎36‎, ‎41-68‎.

[5] Ferris ‎M‎. ‎C.‎, ‎Philpott‎ ‎A.‎, ‎B‎. ‎(1989) " ‎An interior point algorithm for semi-infinite linear programming ''‎, ‎Mathematical Programming 43‎, ‎257-276‎.

[6] Floudas‎ ‎C‎. ‎A.‎, ‎Stein ‎O. (2007) " ‎The adaptive convexification algorithm‎: ‎A feasible point method for semi-infinite programming ''‎, ‎SIAM Journal on Optimization‎, ‎18‎, ‎4‎, ‎1187-1208‎.

[7] Goberna ‎M‎. ‎A. (2007) " ‎Semi-infinite programming ''‎, ‎European Journal of Operational Research‎, ‎180‎, ‎491-518‎.

[8] Gustafson ‎S. (1979) " ‎On numerical analysis in semi-infinite programming ''‎, ‎In Semi-Infinite Programming (Lecture Notes in Control and Information Sciences)‎, ‎15‎, ‎51-65‎.

[9] Hettich‎ ‎R.‎, ‎Kortanek ‎K‎. ‎O. (1993) " ‎Semi-infinite programming‎: ‎Theory‎, ‎methods and applications '', SIAM Review‎, ‎35‎, ‎3‎, ‎380-429‎.

[10] Lopez ‎M.‎, ‎Still ‎G.} (2007) " ‎Semi-infinite programming ''‎, ‎European Journal of Operational Research‎, ‎180‎, ‎491-518‎.

[11] Malek‎ ‎A.‎, ‎Yari ‎A.(2005) " ‎Primal dual solution for the linear programming problems using neural networks '', Applied Mathematics and Computation‎, ‎167‎, ‎198-211‎.

[12] Malek‎ ‎A.‎, ‎Yashtini‎ ‎M‎. ‎(2010) " ‎Image fusion algorithms for color and gray level images based on LCLS method and novel artificial neural network ''‎, ‎Neuro computing‎, ‎73‎, ‎937-943‎.

[13] Reemsten R.‎, ‎Gorner ‎S. (1998) " ‎Numerical methods for semi-infinite programming‎: ‎A survey In R‎. ‎Reemsten and J‎. ‎Rueckmann (editors)‎, ‎Semi-infinite Programming‎, ‎Non-convex Optimization and Its Applications‎, ‎25‎, ‎101‎, ‎195-275‎.

[14] Stein ‎O.‎, ‎Still ‎ ‎G.‎ ‎(2003) " ‎Solving semi-infinite optimization problems with interior point techniques '', SIAM Journal on Control and Optimization‎, ‎42‎, ‎3‎, ‎769-788‎.  

[15] Tank ‎D‎. ‎W.‎, ‎Hopfield‎ ‎J‎. ‎J. (1986) ‎" ‎Simple neural optimization network‎: ‎An A/D convertor‎, ‎signal decision circuit‎, ‎and a linear programming circuit '', IEEE Transactions on Circuits and Systems‎, ‎35‎, ‎533-541‎.

[16] ‎Vazquez ‎F‎. ‎G.‎, ‎Ruckmann‎ ‎J‎. ‎J.‎, ‎Stein ‎O.‎, ‎Still ‎G‎. ‎(2008) " ‎Generalized semi-infinite programming‎: ‎A tutorial ''‎, Journal of Computational and Applied Mathematics 217‎, ‎394-419‎.

[17] Vaz ‎F‎. ‎I‎. ‎A.‎, ‎Fernandes‎ ‎E‎. ‎M‎. ‎G‎. ‎P.‎, ‎Gomes‎ ‎M‎. ‎P‎. ‎S‎. ‎F. (2003) " ‎Robot trajectory planning with semi-infinite programming ''‎, ‎European Journal of Operational Research‎, ‎153‎, ‎607–617‎.  

[18] Vaz‎ ‎F‎. ‎I‎. ‎A.‎, ‎Ferreira ‎E‎. ‎C‎.  ‎(2009) "‎Air pollution control with semi-infinite programming ''‎, ‎Applied Mathematical Modelling 33‎, ‎1957-1969‎.

[19] Xia‎ ‎Y.‎, ‎Leung ‎H.‎, ‎Wang ‎J.} (2002) " ‎A Projection neural network and its application to constrained‎ ‎optimization problems ''‎, ‎IEEE Transactions on Circuits and Systems—I‎: ‎Fundamental theory and applications‎, ‎49‎, ‎4‎, ‎447-458‎.

[20] Xia ‎Y.‎, ‎Wang ‎J. (2004) "‎A recurrent neural network for nonlinear convex optimization subject to nonlinear inequality constraints'', IEEE Transactions on Circuits and Systems-I‎: ‎Regular Papers‎, ‎51‎, ‎7‎, ‎1385-1394‎.