How to transform one convex function continuously into another

SIAM Review

Volumn 50, Issue 1, 2008, Pages 115-132

  Bauschke, Heinz H ^a   Lucet, Yves ^a   Trienis, Michael ^a  

^a UNIVERSITY OF BRITISH COLUMBIA   (Canada)

Author keywords

Attouch Wets topology; Continuous transformation; Convex function; Deformation; Epi convergence; Epi topology; Fenchel conjugate; Homotopy; Proximal average; Proximal mapping

Indexed keywords

COMPUTATIONAL METHODS; CONFORMAL MAPPING; CONVERGENCE OF NUMERICAL METHODS; MATHEMATICAL TRANSFORMATIONS; TOPOLOGY; VARIATIONAL TECHNIQUES;

CONVEX FUNCTION; EPI-CONVERGENCE; EPI-TOPOLOGY; FENCHEL CONJUGATE; HOMOTOPY;

FUNCTIONAL ANALYSIS;

EID: 40749098664     PISSN: 00361445     EISSN: None     Source Type: Journal    
DOI: 10.1137/060664513     Document Type: Article

References (24)

1. H. ATTOUCH, Variational Convergence for Functions and Operators, Appl. Math. Ser., Pitman, Boston, MA, 1984.
2. H. ATTOUCH AND R. J.-B. WETS, Isometries for the Legendre-Fenchel transform, Trans. Amer. Math. Soc., 296 (1986), pp. 33-60.
3. H. H. BAUSCHKE, Duality for Bregman projections onto translated cones and affine subspaces, J. Approx. Theory, 121 (2003), pp. 1-12.
4. H. H. BAUSCHKE, E. MATOUŠKOVÁ, AND S. REICH, Projection and proximal point methods: Convergence results and counterexamples, Nonlinear Anal., 56 (2004), pp. 715-738.
5. H. H. BAUSCHKE AND M. VON MOHRFNSCHILDT, Symbolic computation of Fenchel conjugates, SIGSAM Commun. Comput. Algebra, 40 (2006), pp. 18-28.
6. G. BEER, Topologies on Closed and Closed Convex Sets, Math. Appl. 268, Kluwer Academic, Dordrecht, The Netherlands, 1993.
7. J. M. BORWEIN AND C. H. HAMILTON, Symbolic Fenchel conjugation, Math. Prog., to appear.
8. J. M. BORWEIN AND A. S. LEWIS, Convex Analysis and Nonlinear Optimization: Theory and examples, CMS Books Math./Ouvrages Math. SMC 3, Springer, New York, 2000.
9. S. BOYD AND L. VANDENBERGHE, Convex Optimization, Cambridge University Press, Cambridge, UK, 2004.
10. H. BRÉZIS, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, North-Holland Math. Stud. 5, North-Holland, Amsterdam, The Netherlands, 1973.
11. H. BRÉZIS AND A. HARAUX, Image d'une somme d'opérateurs monotones et applications, Israel J. Math., 23 (1976), pp. 165-186.
12. C. H. HAMILTON, Symbolic Convex Analysis, Master's thesis, Simon Fraser University, Burnaby, Canada, 2005.
13. J.-B. HIRIART-URRUTY AND C. LEMARÉCHAL, Convex Analysis and Minimization Algorithms: Fundamentals, Grundlehren Math. Wiss. 305, Springer, Berlin, 1993.
14. J.-B. HIRIART-URRUTY AND C. LEMARÉCHAL, Convex Analysis and Minimization Algorithms: Advanced Theory and Bundle Methods, Grundlehren Math. Wiss. 306, Springer, Berlin, 1993.
15. A. J. JARVIS, A Brief Manual to Using Lucet's Code, Tech. report, University of Guelph, Guelph, Canada, 2004.
16. R. LUCCHETTI, Convexity and Well-posed Problems, CMS Books Math./Ouvrages Math. SMC 22, Springer, New York, 2006.
17. Y. LUCET, Fast Moreau envelope computation I: Numerical algorithms, Numer. Algorithms, 43 (2006), pp. 235-249.
18. Y. LUCET, H. H. BAUSCHKE, AND M. TRIENIS, The piecewise linear-quadratic model for computational convex analysis, Comput. Optim. Appl., to appear.
19. J.-J. MOREAU, Proximité et dualité dans un espace Hilbertien, Bull. Soc. Math. Rance, 93 (1965), pp. 273-299.
20. J.-P. PENOT AND C. ZǍLINESCU, Continuity of usual operations and variational convergences, Set-Valued Anal., 11 (2003), pp. 225-256.
21. R. T. ROCKAFELLAR, Convex Analysis, Princeton University Press, Princeton, NJ, 1970.
22. R. T. ROCKAFELLAR AND R. J.-B. WETS, Variational Analysis, Springer, Berlin, 1998.
23. S. SIMONS, Minimax and Monotonicity, Lecture Notes in Math. 1693, Springer, Berlin, 1998.
24. C. ZǍLINESCU, Convex Analysis in General Vector Spaces, World Scientific, River Edge, NJ, 2002.