Computing proximal points of nonconvex functions

Mathematical Programming

Volumn 116, Issue 1-2, 2009, Pages 221-258

  Hare, Warren ^a   Sagastizábal, Claudia ^b  

^a MCMASTER UNIVERSITY   (Canada)

^b IMPA   (Brazil)

Author keywords

Lower C2; Nonconvex optimization; Nonsmooth optimization; Prox regular; Proximal point

Indexed keywords

AEROSPACE APPLICATIONS; ARCHITECTURAL DESIGN; C (PROGRAMMING LANGUAGE); COMPUTATIONAL METHODS; CONFORMAL MAPPING; CONVERGENCE OF NUMERICAL METHODS; EQUATIONS OF STATE; FUNCTIONS; MATHEMATICAL OPERATORS; OPTIMIZATION; VARIATIONAL TECHNIQUES;

BLACK BOXES; CONVERGENCE (MATHEMATICS); CONVEX (CONJUGATE , RECESSION , AFFINE , POLYHEDRAL , SUPPORT) FUNCTION; CONVEX OPTIMIZATION; CRITICAL POINTS (CP); DESIGN AND ANALYSIS; EFFICIENT METHODS; FIXED POINTS (FP); FUNCTION VALUES; IN ORDER; LIPSCHITZ CONTINUOUS; NATURAL EXTENSION; NON CONVEX FUNCTIONS; NON-CONVEX OPTIMIZATION; NONCONVEX; OPTIMIZATION TECHNIQUES; PIECEWISE AFFINE MODELS; PRACTICAL IMPLEMENTATION; PROXIMAL POINT METHODS; PROXIMAL POINTS; SUBGRADIENT; VARIATIONAL ANALYSIS; VARIATIONAL INEQUALITIES;

SET THEORY;

EID: 45749096678     PISSN: 00255610     EISSN: 14364646     Source Type: Journal    
DOI: 10.1007/s10107-007-0124-6     Document Type: Article

References (38)

1. Auslender A. (1987). Numerical methods for nondifferentiable convex optimization. Math. Program. Stud. 30: 102-126
2. Auslender A., Crouzeix J.P. and Fedit P. (1987). Penalty-proximal methods in convex programming. J. Optim. Theory Appl. 55(1): 1-21
3. Auslender A. and Haddou M. (1995). An interior-proximal method for convex linearly constrained problems and its extension to variational inequalities. Math. Program. 71(1, Ser. A): 77-100
4. Bellman R., Kalaba R. and Lockett J. (1966). Numerical Inversion of the Laplace Transform. Elsevier, Amsterdam
5. Bernard F. and Thibault L. (2004). Prox-regularity of functions and sets in Banach spaces. Set Valued Anal. 12(1-2): 25-47
6. Bernard F. and Thibault L. (2005). Prox-regular functions in Hilbert spaces. J. Math Anal. Appl. 303(1): 1-14
7. Bonnans J.F., Gilbert J., Lemaréchal C. and Sagastizábal C. (1995). A family of variable metric proximal methods. Math. Program. Ser. A 68: 15-47
8. Cominetti R. and Courdurier M. (2003). Coupling general penalty schemes for convex programming with the steepest descent and the proximal point algorithm. SIAM J. Optim. 13(3): 745-765 (electronic)
9. Correa R. and Lemaréchal C. (1993). Convergence of some algorithms for convex minimization. Math. Program. 62(2): 261-275
10. Dolan E. and Moré J. (2002). Benchmarking optimization software with performance profiles. Math. Program. 91(2, Ser. A): 201-213
11. Eckstein J. (1993). Nonlinear proximal point algorithms using Bregman functions, with applications to convex programming. Math. Oper. Res. 18: 202-226
12. Frangioni A. (1996). Solving semidefinite quadratic problems within nonsmooth optimization algorithms. Comput. Oper. Res. 23(11): 1099-1118
13. Fuduli A., Gaudioso M. and Giallombardo G. (2003). Minimizing nonconvex nonsmooth functions via cutting planes and proximity control. SIAM J. Optim. 14(3): 743-756 (electronic)
14. Güler O. (1992). New proximal point algorithms for convex minimization. SIAM J. Optim. 2: 649-664
15. Hare, W., Lewis, A.: Identifying active contraints via partial smoothness and prox-regularity. J. Convex Anal. 11, 251-266
16. Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms. No. 305-306 in Grund. der math. Wiss. Springer, Heidelberg (two volumes, 1993)
17. Kiwiel K. (1986). A method for solving certain quadratic programming problems arising in nonsmooth optimization. IMA J. Numer. Anal. 6: 137-152
18. Kiwiel K. (1991). Exact penalty functions in proximal bundle methods for constrained convex nondifferentiable minimization. Math. Program. 52(2, Ser. B): 285-302
19. Lemaréchal C. and Sagastizábal C. (1997). Variable metric bundle methods: from conceptual to implementable forms. Math. Program. Ser. A 76: 393-410
20. Lemaréchal C., Strodiot J.J. and Bihain A. (1981). On a bundle method for nonsmooth optimization. In: Mangasarian, O., Meyer, R., and Robinson, S. (eds) Nonlinear Programming, vol. 4, pp 245-282. Academic, New York
21. Lukšan L. and Vlček J. (1998). A bundle-Newton method for nonsmooth unconstrained minimization. Math. Program. 83(3, Ser. A): 373-391
22. Lukšan L. and Vlček J. (2001). Globally convergent variable metric method for nonconvex nondifferentiable unconstrained minimization. J. Optim. Theory Appl. 2: 407-430
23. Makela, M., Neittaanmaki, P.: Nonsmooth Optimization: Analysis and Algorithms with Applications to Optimal Control. World Scientific, Singapore (1992)
24. Martinet B. (1970). Régularisation d'inéquations variationnelles par approximations successives. Rev. Française Informat. Recherche Opérationnelle 4(Ser. R-3): 154-158
25. Mifflin R. (1977). Semi-smooth and semi-convex functions in constrained optimization. SIAM J. Control Optim. 15: 959-972
26. Mifflin, R.: Convergence of a modification of lemaréchal's algorithm for nonsmooth optimization. In: Progress in Nondifferentiable Optimization, IIASA Collaborative Proc., vol. 8, pp. 85-95. (Ser. CP-82) (1982)
27. Mifflin R. (1982). A modification and extension of Lemaréchal's algorithm for nonsmooth minimization. Math. Program. Stud. 17: 77-90
28. Mifflin R. and Sagastizábal C. (2003). Primal-dual gradient structured functions: second-order results; links to epi-derivatives and partly smooth functions. SIAM J. Optim. 13(4): 1174-1194
29. Mifflin R., Sagastizábal C. (2004) VU -smoothness and proximal point results for some nonconvex functions. Optim. Meth. Softw. 19(5): 463-478
30. Mordukhovich B.S. (1976). Maximum principle in the problem of time optimal response with nonsmooth constraints. Prikl. Mat. Meh. 40(6): 1014-1023
31. Moreau J. (1965). Proximité et dualité dans un espace Hilbertien. Bulletin de la Société Mathématique de France 93: 273-299
32. Poliquin R.A. and Rockafellar R.T. (1996). Generalized Hessian properties of regularized nonsmooth functions. SIAM J. Optim. 6(4): 1121-1137
33. Poliquin R.A. and Rockafellar R.T. (1996). Prox-regular functions in Variational Analysis. Trans. Am. Math. Soc. 348(5): 1805-1838
34. Popova, N., Tarasov, V.: A modification of the cutting-plane method with accelerated convergence. In: Nondifferentiaible Optimization: Motivations and application, Lecture Notes in Economics and Mathematical Systems, pp. 284-290 (1984)
35. Rockafellar R. (1976). Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Math. Oper. Res. 1: 97-116
36. Rockafellar R. (1976). Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14: 877-898
37. Rockafellar, R., Wets, R.B.: Variational Analysis. No. 317 in Grund. der math. Wiss. Springer, Heidelberg (1998)
38. Yosida K. (1964) Functional Analysis. Springer, Heidelberg