Duality in reverse convex optimization

SIAM Journal on Optimization

Volumn 8, Issue 4, 1998, Pages 1029-1037

  Lemaire, B ^a  

^a UNIV MONTPELLIER   (France)

Author keywords

D.C. optimization; Duality; Reverse convex

Indexed keywords

EID: 0032343083     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1052623495295857     Document Type: Article

References (26)

1. W. BRIEC, Une nouvelle approche de la mesure de l'efficacité productive, Thèse de Doctorat en Sciences Economiques, Ecole des Hautes Etudes en Sciences Sociales, Aix-Marseille, 1994.
2. W. BRIEC AND B. LEMAIRE, Technical efficiency and distance to a reverse convex set, European J. Oper. Res., to appear.
3. I. EKELAND AND R. TEMAM, Convex Analysis and Variational Problems, North-Holland American Elsevier, New York, 1976.
4. W. FENCHEL, Convex Cones, Sets and Functions, Multilith Lecture Notes, Princeton University, Princeton, NJ, 1953.
5. C. FRANCHETTI AND I. SINGER, Best approximation by elements of caverns in normed linear spaces, Boll. Un. Mat. Ital. B (5), 17 (1980), pp. 33-43.
6. J.-B. HIRIART-URRUTY, Generalized differentiability, duality and optimization for problems dealing with differences of convex functions, Lecture Notes in Econom. and Math. Systems 256, J. Ponstein, ed., Springer-Verlag, Berlin, 1985, pp. 37-70.
7. J.-B. HIRIART-URRUTY, A general formula on the conjugate of the difference of functions, Canad. Math. Bull., 29 (1986), pp. 482-485.
8. J.-B. HIRIART-URRUTY, From convex optimization to nonconvex optimization. Necessary and sufficient conditions for global optimality, in Nonsmooth Optimization and Related Topics (Erice 1988), F. H. Clarke, V. F. Demyanov, and F. Gianessi, eds., Plenum Press, New York, 1989, pp. 219-239.
9. J.-B. HIRIART-URRUTY, How to regularize a difference of convex functions, J. Math. Anal. Appl., 162 (1991), pp. 196-209.
10. R. HORST AND H. TUY, Global Optimization (Deterministic Approach), 3rd ed., Springer-Verlag, New York, 1996.
11. J. J. MOREAU, Théorèmes inf-sup, C. R. Acad. Sci. Paris, 258 (1964), pp. 2720-2722.
12. J. J. MOREAU, Fonctionnelles convexes, Séminaire sur les e.d.p., Collège de France, 1966.
13. L. NIRENBERG, Functional Analysis, New York University, New York, 1961.
14. P. D. TAO AND E. B. SOUAD, Duality in D.C. optimization. Subgradient methods, in Trends in Mathematical Optimization, Internat. Ser. Numer. Math. 84(c), Birkhäuser Verlag, Basel, 1988, pp. 277-293.
15. R. T. ROCKAFELLAR, Duality theorems for convex functions, Bull. Amer. Math. Soc., 70 (1964), pp. 189-192.
16. R. T. ROCKAFELLAR, Extension of Fenchel's duality theorem for convex functions, Duke Math. J., 33 (1966), pp. 81-90.
17. I. SINGER, A Fenchel-Rockafellar type duality theorem for maximization, Bull. Austral. Math. Soc., 20 (1979), pp. 193-198.
18. I. SINGER, Maximization of lower semicontinuous convex functionals on bounded subsets of locally convex spaces. II: Quasi-Lagrangian duality theorems, Results. Math., 3 (1980), pp. 235-248.
19. I. SINGER, Minimization of continuous convex functionals on complements of convex subsets of locally convex spaces, Math. Oper. Statist. Ser. Optim., 11 (1980), pp. 221-234.
20. I. SINGER, Optimization by level set methods, VI: Generalization of surrogate type reverse convex duality, Optimization, 18 (1987), pp. 485-499.
21. I. SINGER, Some further duality theorems for optimization problems with reverse convex constraint sets, J. Math. Anal. Appl., 171 (1992), pp. 205-219.
22. P. T. THACH, Quasiconjugates of functions. Duality relationship between quasiconvex minimization under a reverse convex constraint and quasiconvex maximization under a convex constraint, and applications, J. Math. Anal. Appl., 159 (1991), pp. 299-322.
23. P. T. THACH, R. E. BURKARD, AND W. OETTLI, Mathematical programs with a two-dimensional reverse convex constraint, J. Global Optim., 1 (1991), pp. 145-154.
24. P. T. THACH, A nonconvex duality with zero gap and applications, SIAM J. Optim., 4 (1994), pp. 44-64.
25. J. F. TOLAND, A duality principle for nonconvex optimization and the calculus of variation, Arch. Rational Mech. Anal., 71 (1979), pp. 41-61.
26. H. TUY, D.C. optimization: Theory, methods and algorithms, in Handbook of Global Optimization, R. Horst and P. M. Pardalos, eds., Kluwer Academic Publishers, Norwell, MA, 1995, pp. 149-216.