Convergence of new inertial proximal methods for dc programming

SIAM Journal on Optimization

Volumn 19, Issue 1, 2008, Pages 397-413

  Mainge, Paul Emile ^a   Moudafi, Abdellatif ^a  

^a Campus de Schoelcher   (Martinique)

Author keywords

Critical points; DC minimization; Proximal mappings; Subdifferentials

Indexed keywords

DYNAMICAL SYSTEMS; EQUATIONS OF STATE;

CONVERGENCE RESULTS; CRITICAL POINTS; D-C PROGRAMMING; DC MINIMIZATION; DIFFERENTIATE FUNCTIONS; DISCRETIZATION; FIRST ORDERS; LOCAL FEATURES; PROXIMAL MAPPINGS; PROXIMAL METHODS; REAL-VALUED FUNCTIONS; SEMI-CONTINUOUS; SUBDIFFERENTIALS; THEORETICAL BASIS;

ITERATIVE METHODS;

EID: 61349102509     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/060655183     Document Type: Article

References (29)

1. F. ALVAREZ, On the minimizing property of a second order dissipative dynamical system in Hilbert spaces, SIAM J. Control Optim., 38 (2000), pp. 1102-1119.
2. F. ALVAREZ AND H. ATTOUCH, An inertial proximal method for monotone operators via discretization of a nonlinear oscillator with damping, Set-Valued Anal., 9 (2001), pp. 3-31.
3. F. ALVAREZ, H. ATTOUCH, J. BOLTE, AND P. REDONT, A second-order gradient-like dissipative dynamical system with Hessian driven damping. Application to optimization and mechanics, J. Math. Pures Appl. (9), 81 (2002), pp. 747-779.
4. F. ALVAREZ AND J. M. PÉREZ, A dynamical system associated with Newton's method for parametric approximations of convex minimization problems, Appl. Math. Optim., 38 (1998), pp. 193-217.
5. H. ATTOUCH, J. BOLTE, AND P. REDONT, A second-order gradient-like dissipative dynamical system with Hessian driven damping. Link with the proximal algorithm, Control Cybernet., 31 (2002), pp. 643-657.
6. H. ATTOUCH, X. GOUDON, AND P. REDONT, The heavy ball system with friction method, I. The continuous dynamical system, Commun. Contemp. Math., 2 (2000), pp. 1-34.
7. H. ATTOUCH AND P. REDONT, The second-order in time continuous Newton method, in Approximation Optimization and Mathematical Economics, M. Lassonde ed., Physica, Heidelberg, 2001, pp. 25-36.
8. J.-P. CHEHAB AND M. RAYDAN, Implicit and adaptive inverse preconditioned gradient methods for nonlinear problems, Appl. Numer. Math., 55 (2005), pp. 32-47.
9. O. GÜLER, On the convergence of the proximal point algorithm for convex minimization, SIAM J. Control Optim., 29 (1991), pp. 403-419.
10. J. B. HIRIART-URRUTY, From convex optimization to nonconvex optimization, in Nonsmooth Optimization and Related Topics, F. H. Clarke, V. F. Demyanov, and F. Giannessi, eds., Plenum Press, New York, 1989, pp. 219-239.
11. J. B. HIRIART-URRUTY, Generalized differentiability, duality and optimization for problems dealing with difference of convex functions, in Convexity and Duality in Optimization, Lecture Notes in Econom. Math. Systems 256, Springer-Verlag, New York, 1986, pp. 37-70.
12. F. JULES AND P. E. MAINGÉ, Numerical approaches to stationary solution of a second order dissipative dynamical system, Optimization, 51 (2002), pp. 235-255.
13. K. C. KIWIEL, Proximity control in bundle methods for convex nondifferentiable minimization, Math. Programming, 46 (1990), pp. 105-122.
14. K. GOEBEL AND W. A. KIRK, Topics in Metric Fixed Points Theory, Cambridge Studies in Advanced Mathematics 28, Cambridge University Press, Cambridge, UK, 1990.
15. B. LEMAIRE, On the convergence of some iterative methods for convex minimization, in Recent Developments in Optimization, Lecture Notes in Econom. and Math. Systems 429, Springer-Verlag, Berlin, 1995, pp. 252-268.
16. B. MARTINET, Algorithmes pour la résolution des problèmes d'optimisation et de minmax, Thèse d'état Université de Grenoble, Grenoble, France, 1972.
17. J. E. MARTINEZ-LEGAZ AND A. SEEGER, A formula on the approximate subdifferential of the difference of convex functions, Bull. Austral. Math. Soc., 45 (1992), pp. 37-41.
18. PH. MICHEL, Problème d'optimisation défini par des fonctions qui sont somme de fonctions convexes et de fonctions dérivables, J. Math. Pures Appl. (9), 53 (1974), pp. 321-329.
19. A. MOUDAFI AND E. ELISABETH, An approximate inertial proximal method using enlargement of a maximal monotone operator, Int. J. Pure Appl. Math., 5 (2003), pp. 283-299.
20. A. MOUDAFI AND P.-E. MAINGÉ, On the convergence of an approximate proximal method for DC functions, 3. Comput. Math., 24 (2006), pp. 475-480.
21. A. MOUDAFI AND M. THÉRA, Finding the zero for the sum of two maximal monotone operators, J. Optim. Theory Appl., 94 (1997), pp. 425-448.
22. Z. OPIAL, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73 (1967), pp. 591-597.
23. J. P. PENOT, On the minimization of difference functions, J. Global Optim., 12 (1998), pp. 373-382.
24. D. T. PHAM AND E. B. SOUAD, Algorithms for solving a class of nonconvex optimization problems: Methods of subgradient, in Fermat Days 85: Mathematics for Optimization, North-Holland Math. Stud. 129, Elsevier, Amsterdam, 1986, pp. 249-270.
25. R. T. ROCKAFELLAR, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), pp. 877-898.
26. M. V. SOLODOV AND B. F. SVAITER, A hybrid projection-proximal point algorithm, J. Convex Anal., 6 (1999), pp. 59-70.
27. W. Y. SUN, R. J. B. SAMPAIO, AND M. A. B. CANDIDO, Proximal point algorithm for minimization of DC functions, J. Comput. Math., 21 (2003), pp. 451-462.
28. A. SZULKIN, Minimal principles for lower semi-continuous functions and applications to nonlinear boundary value problems, Annales Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), pp. 77-109.
29. J. F. TOLAND, Duality in nonconvex optimization, J. Math. Anal. Appl., 66 (1978), pp. 399-415.