Abstract

This article provides results guarateeing that the optimal value of a given convex infinite optimization problem and its corresponding surrogate Lagrangian dual coincide and the primal optimal value is attainable. The conditions ensuring converse strong Lagrangian (in short, minsup) duality involve the weakly-inf-(locally) compactness of suitable functions and the linearity or relative closedness of some sets depending on the data. Applications are given to different areas of convex optimization, including an extension of the Clark-Duffin Theorem for ordinary convex programs.

Author information

Contact details are reproduced from the original publication and may be historical.

Miguel A. Goberna

Dept. of Statistics and Operations Research, University of Alicante, Apt. de Correos 99, 03080 Alicante, Spain

mgoberna@ua.es

Marco Antonio López

Dept. of Statistics and Operations Research, University of Alicante, Apt. de Correos 99, 03080 Alicante, Spain

marco.antonio@ua.es

Keywords

  • Convex infinite programming
  • converse strong duality
  • minsup duality

Mathematics Subject Classification

90C25, 90C48, 49N15

References

  1. A. Auslender, M. Teboulle: Asymptotic Cones and Functions in Optimization and Variational Inequalities, Springer, New York (2003).Google Scholar
  2. J. F. Bonnans, A. Shapiro: Perturbation Analysis of Optimization Problems, Springer, New York (2000).https://doi.org/10.1007/978-1-4612-1394-9Google Scholar
  3. J. M. Borwein, A. S. Lewis: Partially finite convex programming. I: Quasi relative interiors and duality theory, Math. Program., Ser. B 57 (1992) 15–48.https://doi.org/10.1007/BF01581072Google Scholar
  4. J. M. Borwein, A. S. Lewis: Convex Analysis and Nonlinear Optimization. Theory and Examples, 2nd Ed., Springer, New York (2006).https://doi.org/10.1007/978-0-387-31256-9Google Scholar
  5. R. I. Boţ: Conjugate Duality in Convex Optimization, Springer, Berlin (2010).https://doi.org/10.1007/978-3-642-04900-2Google Scholar
  6. C. Castaing, M. Valadier: Convex Analysis and Measurable Multifunctions, Springer, Berlin (1977).https://doi.org/10.1007/BFb0087685Google Scholar
  7. N. Dinh, M. A. Goberna, M. A. López: From linear to convex systems: consistency, Farkas Lemma and applications, J. Convex Analysis 13 (2006) 279–290.https://doi.org/10.68381/jca13008Google Scholar
  8. N. Dinh, M. A. Goberna, M. A. López, T. Q. Son: New Farkas-type constraint qualifications in convex infinite programming, ESAIM, Control Optim. Calc. Var. 13 (2007) 580–597.https://doi.org/10.1051/cocv:2007027Google Scholar
  9. R. J. Duffin: Infinite programs, Ann. Math. Stud. 38 (1956) 157–170.https://doi.org/10.1515/9781400881987-007Google Scholar
  10. R. J. Duffin: Clark’s theorem on linear programs holds for convex programs, Proc. Natl. Acad. Sci. USA 75 (1978) 1624–1626.https://doi.org/10.1073/pnas.75.4.1624Google Scholar
  11. I. Ekeland, R. Temam: Analyse Convexe et Problèmes Variationnels, Dunod, Montreal (1974).Google Scholar
  12. E. Ernst, M. Volle: Zero duality gap for convex programs: a generalization of the Clark-Duffin Theorem, J. Optim. Theory Appl. 158 (2013) 668–686.https://doi.org/10.1007/s10957-013-0287-7Google Scholar
  13. D. H. Fang, C. Li, K. F. Ng: Constraint qualifications for extended Farkas’s lemmas and Lagrangian dualities in convex infinite programming, SIAM J. Optim. 20 (2009) 1311–1332.https://doi.org/10.1137/080739124Google Scholar
  14. D. H. Fang, C. Li, K. F. Ng: Constraint qualifications for optimality conditions and total Lagrange dualities in convex infinite programming, Nonlinear Anal. 73 (2010) 1143–1159.https://doi.org/10.1016/j.na.2010.04.020Google Scholar
  15. M. A. Goberna, V. Jeyakumar, M. A. López: Necessary and sufficient constraint qualifications for systems of infinite convex inequalities, Nonlinear Anal. 68 (2008) 1184–1194.https://doi.org/10.1016/j.na.2006.12.014Google Scholar
  16. M. A. Goberna, M. A. López: Linear Semi-Infinite Optimization, Wiley, Chichester (1998).Google Scholar
  17. V. Jeyakumar: Asymptotic dual conditions characterizing optimality for infinite convex programs, J. Optim. Theory Appl. 93 (1997) 153–165.https://doi.org/10.1023/A:1022606002804Google Scholar
  18. V. Jeyakumar, H. Wolkowicz: Zero duality gaps in infinite-dimensional programming, J. Optim. Theory Appl. 67 (1990) 87–107.https://doi.org/10.1007/BF00939737Google Scholar
  19. J. L. Joly: Une Famille de Topologies et de Convergences sur l’Ensemble des Fonctionnelles Convexes, PhD Thesis, Institut d’Informatique et de Mathématiques Appliqué es de Grenoble (1970).Google Scholar
  20. J. L. Joly, P.-J. Laurent: Stability and duality in convex minimization problems, Rev. Franc. Inform. Rech. Oper. 5(R–2) (1971) 3–42.https://doi.org/10.1051/m2an/197105R200031Google Scholar
  21. D. F. Karney: A duality theorem for semi-infinite convex programs and their finite subprograms, Math. Program. 27 (1983) 75–82.https://doi.org/10.1007/BF02591965Google Scholar
  22. P.-J. Laurent: Approximation et Optimization, Hermann, Paris (1972).Google Scholar
  23. M. Moussaoui, M. Volle: Sur la quasicontinuité et les fonctions unies en dualité convexe, C. R. Acad. Sci., Paris, Sér. I 322 (1996) 839–844.Google Scholar
  24. M. Moussaoui, M. Volle: Quasicontinuity and united functions in convex duality theory, Commun. Appl. Nonlinear Anal. 4 (1997) 73–89.Google Scholar
  25. J. Orihuela, M. A. Ruiz Galán: A coercive James’s weak compactness theorem and nonlinear variational problems, Nonlinear Anal. 75 (2012) 598–611.https://doi.org/10.1016/j.na.2011.08.062Google Scholar
  26. J.-C. Pomerol: Contribution à la Programmation Mathématique: Existence de Multiplicateurs de Lagrange et Stabilité, PhD Thesis, Université Pierre et Marie Curie, Paris (1980).Google Scholar
  27. T. Precupanu: Closedness conditions for the optimality of a family of non-convex optimization problems, Math. Operationsforsch. Stat., Ser. Optim. 15 (1984) 339–346.https://doi.org/10.1080/02331938408842946Google Scholar
  28. R. T. Rockafellar: Helly’s theorem and minima of convex functions, Duke Math. J. 32 (1965) 381–397.https://doi.org/10.1215/S0012-7094-65-03238-2Google Scholar
  29. R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173Google Scholar
  30. J. Saint Raymond: Weak compactness and variational characterization of the convexity, Mediterr. J. Math. 10 (2013) 927–940.https://doi.org/10.1007/s00009-012-0226-0Google Scholar
  31. A. Shapiro: On duality theory of convex semi-infinite programming, Optimization 54 (2005) 535–543.https://doi.org/10.1080/02331930500342823Google Scholar
  32. C. Zălinescu: A note on d-stability of convex programs and limiting Lagrangians, Math. Program. 53 (1992) 267–277.https://doi.org/10.1007/BF01585706Google Scholar
  33. C. Zălinescu: Convex Analysis in General Vector Spaces, World Scientific, Singapore (2002).https://doi.org/10.1142/5021Google Scholar

Suggested citation

M. A. Goberna, M. A. López, M. Volle. “Primal Attainment in Convex Infinite Optimization Duality.” Journal of Convex Analysis 21 (2014), No. 4, 1043–1064. https://doi.org/10.68381/jca21056

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Published by Heldermann Verlag, 2014. Rights now held by Banach Press LLC.