A New Steepest Descent Differential Inclusion-Based Method for Solving General Nonsmooth Convex Optimization Problems

Journal of Optimization Theory and Applications

Volumn 159, Issue 3, 2013, Pages 698-720

  Hosseini, Alireza ^a   Hosseini, S M ^a  

^a TARBIAT MODARES UNIVERSITY   (Iran)

Author keywords

Convergence of trajectories; Differential inclusion based methods; General nonsmooth convex optimization; Steepest descent neural network

Indexed keywords

DIFFERENTIAL EQUATIONS; NUMERICAL METHODS; STEEPEST DESCENT METHOD;

CONVERGENCE OF TRAJECTORIES; CONVEX OPTIMIZATION PROBLEMS; DIFFERENTIAL INCLUSION-BASED METHOD; DIFFERENTIAL INCLUSIONS; GENERAL NONSMOOTH CONVEX OPTIMIZATION; NEURAL-NETWORKS; NON-SMOOTH CONVEX OPTIMIZATIONS; NUMERICAL TESTS; OPTIMAL SOLUTION SETS; STEEP DESCENT NEURAL NETWORK;

CONVEX OPTIMIZATION;

EID: 84887320212     PISSN: 00223239     EISSN: 15732878     Source Type: Journal    
DOI: 10.1007/s10957-012-0258-4     Document Type: Article

References (31)

1. Mordukhovich, B.: Variations Analysis and Generalized Differentiation, I: Basic Theory. Springer, Berlin (2006).
2. Mordukhovich, B.: Variations Analysis and Generalized Differentiation, II: Applications. Springer, Berlin (2006).
3. Soleimani-damaneh, M.: Nonsmooth optimization using Mordukhovich's subdifferential. SIAM J. Control Optim. 48, 3403-3432 (2010).
4. Shor, N.: Minimization Methods for Non-differentiable Functions. Springer, Berlin (1985).
5. Kelley, J. E.: The cutting-plane method for solving convex programs. J. Soc. Ind. Appl. Math. 8, 703-712 (1960).
6. Goffin, J. L., Haurie, A., Vial, J. P.: Decomposition and nondifferentiable optimization with the projective algorithm. Manag. Sci. 38, 284-302 (1992).
7. Hiriart-Urruty, J. B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms: Part 2: Advanced Theory and Bundle Methods. Springer, Berlin (2010).
8. Solodov, M. V.: On approximations with finite precision in bundle methods for nonsmooth optimization. J. Optim. Theory Appl. 119, 151-165 (2003).
9. Zhao, X., Luh, P. B.: New bundle methods for solving lagrangian relaxation dual problems. J. Optim. Theory Appl. 113, 373-397 (2002).
10. Forti, M., Nistri, P., Quincampoix, M.: Generalized neural network for nonsmooth nonlinear programming problems. IEEE Trans. Circuits Syst. I 51, 1741-1754 (2004).
11. Bian, W., Xue, X.: Subgradient-based neural networks for nonsmooth nonconvex optimization problems. IEEE Trans. Neural Netw. 20, 1024-1038 (2009).
12. Li, G., Song, S., Guan, X.: Subgradient-based feedback neural networks for non-differentiable convex optimization problems. Sci. China Ser. F, Inf. Sci. 20, 421-435 (2006).
13. Li, G., Song, S., Wu, C.: Generalized gradient projection neural networks for nonsmooth optimization problems. Sci. China Ser. F, Inf. Sci. 53, 990-1004 (2010).
14. Lemaréchal, C., Nemirovskii, A., Nesterov, Y.: New variants of bundle methods. Math. Program. 69, 111-147 (1995).
15. Mäkelä, M. M.: Survey of bundle methods for nonsmooth optimization. Optim. Methods Softw. 17(1), 1-29 (2001).
16. Kiwiel, K. C.: Methods of Descent for Nondifferentiable Optimization. Lecture Notes in Mathematics. Springer, Berlin (1985).
17. Mäkelä, M. M., Neittaanmäki, P.: Nonsmooth Optimization. World Scientific, Singapore (1992).
18. Xia, Y., Leung, H., Wang, J.: A projection neural network and its application to constrained optimization problems. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 49, 442-457 (2002).
19. Xia, Y., Wang, J.: A general projection neural network for solving monotone variational inequalities and related optimization problems. IEEE Trans. Neural Netw. 15, 318-328 (2004).
20. Xia, Y., Wang, J.: Neural network for solving linear programming problems with bound variables. IEEE Trans. Neural Netw. 6, 515-519 (1995).
21. Xia, Y.: A new neural network for solving linear programming problems and its application. IEEE Trans. Neural Netw. 7, 525-529 (1996).
22. Wang, J., Hu, Q., Jiang, D.: A Lagrangian network for kinematic control of redundant robot manipulators. IEEE Trans. Neural Netw. 10, 1123-1132 (1999).
23. Aubin, J. P., Cellina, A.: Differential Inclusion: Set-Valued Maps and Viability Theory. Springer, Berlin (1984).
24. Mifflin, R.: Semismooth and semiconvex functions in constrained optimization. SIAM J. Control Optim. 15, 957-972 (1977).
25. Defeng, S., Womersley, R. S., Houduo, Q.: A feasible semismooth asymptotically Newton method for mixed complementarity problems. Math. Program. 94, 167-187 (2002).
26. Clarke, F. H.: Optimization and Nonsmooth Analysis. Society for Industrial and Applied Mathematics, Philadelphia (1983).
27. Filippov, A. F.: Differential Equations with Discontinuous Right-Hand Sides: Control Systems. Kluwer, Boston (1988).
28. Hosseini, A., Hosseini, S. M., Soleimani-damaneh, M.: A differential inclusion-based approach for solving nonsmooth convex optimization problems. Optimization (2011). doi: 10. 1080/02331934. 2011. 613993.
29. Borwein, J. M., Moors, W. B.: Essentially strictly differentiable Lipschitz functions. J. Funct. Anal. 149, 305-351 (1997).
30. Liu, Q., Wang, J.: A one-layer recurrent neural network for constrained nonsmooth optimization. IEEE Trans. Syst. Man Cybern., Part B 41, 1323-1333 (2011).
31. Lukšan, L., Vlček, J.: Test problems for nonsmooth unconstrained and linearly constrained optimization. Institute of Computer Science, Academy of Science of the Czech Republic, Technical report, 798 (2000).