Practical aspects of the Moreau-Yosida regularization: Theoretical preliminaries

SIAM Journal on Optimization

Volumn 7, Issue 2, 1997, Pages 367-385

  Lemaréchal, Claude ^a   Sagastizábal, Claudia ^a  

^a INRIA   (France)

Author keywords

Convex optimization; Mathematical programming; Proximal point; Second order differentiability

Indexed keywords

EID: 0040165092     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1052623494267127     Document Type: Article

References (28)

1. A. AUSLENDER, Numerical methods for nondifferentiable convex optimization, Math. Programming Study, 30 (1987), pp. 102-126.
2. R. BELLMAN, R. KALABA, AND J. LOCKETT, Numerical Inversion of the Laplace Transform, Elsevier, New York, 1966.
3. J. BONNANS, J. GILBERT, C. LEMARÉCHAL, AND C. SAGASTIZÁBAL, A family of variable metric proximal methods, Math. Programming, 68 (1995), pp. 15-48.
4. S. DOLECKI, G. SALINETTI, AND R. WETS, Convergence of functions: Equi-semi-continuity, Trans. Amer. Math. Soc., 276 (1983), pp. 409-429.
5. M. FUKUSHIMA, A descent algorithm for nonsmooth convex programming, Math. Programming, 30 (1984), pp. 163-175.
6. J.-B. HlRIART-URRUTY, The approximate first-order and second-order directional derivatives for a convex function, in Mathematical Theories of Optimization, J.-P. Cecconi and T. Zolezzi, eds., Lecture Notes in Mathematics 979, Springer-Verlag, Berlin, New York, 1983, pp. 154-166.
7. J.-B. HIRIART-URRUTY AND C. LEMARÉCHAL, Convex Analysis and Minimization Algorithms, Springer-Verlag, Berlin, New York, 1993.
8. C.-M. IP AND J. KYPARISIS, Local convergence of quasi-Newton methods for B-differentiable equations, Math. Programming, 56 (1992), pp. 71-89.
9. K. KIWIEL, Proximity control in bundle methods for convex nondifferentiable minimization, Math. Programming, 46 (1990), pp. 105-122.
10. C. LEMARÉCHAL AND C. SAGASTIZABAL, An approach to variable metric bundle methods, in Systems Modelling and Optimization, J. Henry and J.-P. Yvon, eds., Lecture Notes in Control and Inform. Sci. 197, Springer-Verlag, Berlin, New York, 1994, pp. 144-162.
11. C. LEMARÉCHAL AND C. SAGASTIZÁBAL, More than first-order developments of convex functions: Primal-dual relations, J. Convex Anal., (1996), pp. 1-14.
12. Y. LUCET, Formule explicite du hessien de la régularisée de Moreau-Yosida d'une fonction convexe f en fonction de l'épi-différentielle seconde de f, Report # LAO95-08, University of Toulouse, France, 1995.
13. B. MARTINET, Régularisation d'inéquations variationnelles par approximations successives, Revue Française d'Informatique et Recherche Opérationnelle, R-3 (1970), pp. 154-179.
14. R. MIFFLIN, A Quasi-Second-Order Proximal Bundle Algorithm, Technical report 93-3, University of Washington, Pullman, WA, 1993.
15. J. MOREAU, Proximité et dualité dans un espace hilbertien, Bull. Soc. Math. France, 93 (1965), pp. 273-299.
16. J. PANG AND L. QI, Nonsmooth equations: Motivation and algorithms, SIAM J. Control Optim., 3 (1993), pp. 443-465.
17. R. POLIQUIN AND R. ROCKAFELLAR, Generalized hessian properties of regularized nonsmooth functions, SIAM J. Optim., 6 (1996), pp. 1121-1137.
18. 1 optimization problems, Math. Programming, 64 (1994), pp. 277-294.
19. L. QI, Second-Order Analysis of the Moreau-Yosida Approximation of a Convex Function, Applied Mathematics Report AMR94/20, School of Mathematics, The University of New South Wales, Sydney, Australia, 1994.
20. L. QI AND J. SUN, A nonsmooth version of Newton's method, Math. Programming, 58 (1993), pp. 353-367.
21. M. QIAN, The Variable Metric Proximal Point Algorithm: Application to Optimization, Department of Mathematics, Manuscript GN-50, University of Washington, Seattle, WA 98195, 1992.
22. R. ROCKAFELLAR, Convex Analysis, Princeton University Press, Princeton, NJ, 1970.
23. R. ROCKAFELLAR, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), pp. 877-898.
24. R. ROCKAFELLAR, Maximal monotone relations and the second derivatives of nonsmooth functions, Annales de l'Institut Henri Poincaré, Analyse non linéaire, 2 (1985), pp. 167-186.
25. H. SCHRAMM AND J. ZOWE, A version of the bundle idea for minimizing a nonsmooth function: Conceptual idea, convergence analysis, numerical results, SIAM J. Optim., 2 (1992), pp. 121-152.
26. R. WETS, Convergence of convex functions, variational inequalities and convex optimization problems, in Variational Inequalities and Complementarity Problems, F. G. R. Cottle and J.-L. Lions, eds., Wiley, New York, 1980, pp. 405-419.
27. K. YOSIDA, Functional Analysis, Springer-Verlag, Berlin, New York, 1964.
28. C. LEMARÉCHAL AND C. SAGASTIZÁBAL, Variable metric bundle methods: From conceptual to implementable forms, Math. Programming, to appear.