Advertisement

Stability for a Class of Nonlinear Optimization Problems and Applications

  • C. Zălinescu
Part of the Ettore Majorana International Science Series book series (EMISS, volume 43)

Abstract

The aim of this chapter is to give a unified approach to some problems in nonlinear optimization using asymptotic cones, recession functions and asymptotically compact sets. Thus we establish a stability result for a class of nonconvex programming problems which turns out to be equivalent to Dedieu’s criterion for the closedeness of the image of a closed set by a multifunction. Also we obtain a formula for the recession function of the marginal function for the first time. This formula seems to be important and new also in the finite dimensional case. The convex version of the stability result is used to reobtain formulae for the conjugates, ϵ-subdifferentials and recession functions of some convex functions, results which are comparable with those of McLinden. It is also shown that in some cases one can perturbe the objective function of a family of convex problems such that the resulting problems have optimal solutions; the behaviour of the values of these perturbed problems and their solutions is also investigated. Another result establishes the relationship between conically compact sets introduced by Isac and Théra and asymptotic cones.

Keywords

Linear Subspace Pointed Cone Continuous Linear Operator Nonlinear Optimization Problem Asymptotic Cone 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    J.P. Aubin. “A Pareto minimum principle”, in `Differential Games and Related Topics’ (H.W. Kuhn and G.P. Szegö, Eds.), North-Holland, Amsterdam/London, (1971).Google Scholar
  2. [2]
    V. Barbu and T. Precupanu. “Convexity and Optimization in Banach Spaces”. Editura Academiei Bucaresti Si D. Reidel Publ. Co., Dordrecht/Boston/Lancaster (1986).Google Scholar
  3. [3]
    J.M. Borwein. “A note on perfect duality and limiting Lagrangeans”. Math. Program., 18 (1980), 330–337.CrossRefMATHGoogle Scholar
  4. [4]
    J.M. Borwein. “ Adjoint process duality”. Math.Oper.Res., 8 (1983), 403–434.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    B. Bourbaki. “ Espaces Vectoriels Topologiques”. Act. Sci. et Ind., Hermann, Paris (1966).Google Scholar
  6. [6]
    J.-P. Dedieu. “Cônes asymptote d’un ensemble non convexe. Application a l’optimisation”. C.R. Acad. Sci. Paris, 285 (1977), 501–503.MathSciNetMATHGoogle Scholar
  7. [7]
    J.-P. Dedieu. “Critères de fermeture pour l’image d’un fermé non convexe par une multiapplication”. C.R. Acad. Sci. Paris, 287 (1978), 941–943.MathSciNetMATHGoogle Scholar
  8. [8]
    J. Dieudonné. “Sur la séparation des ensembles convexes”. Math. Ann, 163 (1966), 1–3.MATHGoogle Scholar
  9. [9]
    S. Dolecki. “Lower semicontinuity of marginal functions”. In `Selected Topics in Operations Research and Mathematical Economics, Proceedings 1983’, G. Hammer and D. Pallaschke, eds., Lecture Notes in Economics and Mathematical Systems, 226, Springer Verlag, Berlin (1984).Google Scholar
  10. [10]
    I. Ekeland and R. Temam. “Analyse Convexe et Problemes Variationels”.Dunod, Gauthier-Villard, Paris (1974).Google Scholar
  11. [11]
    J. Gwinner. “Closed images of convex multivalued mappings in linear topological spaces with applications”. J.Math.Anal.Appl., 60 (1977), 75–86.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    J. Gwinner. “An extension lemma and homogeneous programming”. J.Optimization Theory Appl. 47 (1985), 321–336.MATHGoogle Scholar
  13. [13]
    J.-B. Hiriart-Urruty. “e-subdifferential calculus”. In `Convex Analysis and Optimization’, J.P. Aubin and R.B. Vinter eds., Research Notes in Mathematics, Pitman Advanced Publishing Program, Boston/London/Melbourne (1982).Google Scholar
  14. [14]
    R.B. Holmes. “Geometrical Functional Analysis and its Applications”. Springer, Berlin (1975).CrossRefGoogle Scholar
  15. [15]
    G. Isac and M. Théra. “Complementarity problem and the existence of the postcritical equilibrium state of the thin elastic plate”. Seminaire d’Analyse Numérique, Université Paul Sabatier, Toulouse III (1985–86), XI-1-XI-27.Google Scholar
  16. [16]
    D.F. Karney. “Duality theorem for semi-infinite convex programs and their finite subprograms”. Math.Program., (1983), 75–82.Google Scholar
  17. [17]
    D.F. Karney and T.D. Morley. “Limiting Lagrangeans: A primal approach”. J. Optimization Theory Appl., 48 (1986), 163–174.MathSciNetMATHGoogle Scholar
  18. [18]
    J. Kelley. “General Topology”. Springer Verlag, New-York (1975).MATHGoogle Scholar
  19. [19]
    G. Köthe. “Topological Vector Spaces, I”. Springer-Verlag, Berlin (1969).CrossRefMATHGoogle Scholar
  20. [20]
    L. McLinden. “Quasistable parametric optimization without compact level sets”. Technical Summary Report 2708, Mathematics Research Center, University of Wisconsin-Madison (1984).Google Scholar
  21. [21]
    J.-P. Penot. “Continuity properties of performence functions”. In “ Optimization Theory and Algortihms, J.B. Hiriart-Urruty, W. Oettli, J. Stoer (eds.), M. Dekker, New-York (1983).Google Scholar
  22. [22]
    J.-Ch. Pomerol. “Contribution à la programation mathématique: Existence de multiplicateur de Lagrange et stabilité”. Thesis, P. et M. Curie University, Paris (1980).Google Scholar
  23. [23]
    T. Precupanu. “On the stability in Fenchel-Rockafellar duality”. An. St. Univ. Iasi, s. Ia, 28 (1982), 19–24.MathSciNetGoogle Scholar
  24. [24]
    T. Precapanu. “Closedness conditions for the optimality of a family of nonconvex optimization problems”. Math. Operationsfors ch. Stat. Ser. Optimization, 15 (1984), 339–346.CrossRefGoogle Scholar
  25. [25]
    T. Precupanu. “Global sufficient optimality conditions for a family of nonconvex optimization problems”.4n. St. Univ. Iasi, s. Ia, 30, (1) (1984), 1–9.Google Scholar
  26. [26]
    S.M. Robinson. “Local epi-convergence and local optimization”. Math. Program., 37 (1987), 208–222.CrossRefMATHGoogle Scholar
  27. [27]
    R.T. Rockafellar. “Convex Analysis”. Princeton University Press, Princeton, New Jersey (1970).Google Scholar
  28. [28]
    R.T. Rockafellar. “Conjugate Duality and Optimization”. Regional conference series in applied mathematics, 16, SIAM, Philadelphia (1974).Google Scholar
  29. [29]
    C. Zàlinescu. “On an abstract control problem”. Numer. Fund. Anal. Optimiz., 2 (1980), 531–542.CrossRefMATHGoogle Scholar
  30. [30]
    C. Zàlinescu. “ Duality for vectorial nonconvex optimization by convexification and applications”. An. St. Univ. Iasi, s.Ia, 29 (3) (1983), 15–34.Google Scholar
  31. [31]
    C. Zàlinescu. “A note on d-stability of convex programs and limiting Lagrangians”. Submitted at Math. Program.Google Scholar
  32. [32]
    C. Zâlinescu. “On Borwein’s paper `adjoint process duality’ “. Math.Oper. Res., 11 (1986), 692–698.MathSciNetCrossRefGoogle Scholar
  33. [33]
    C. Zâlinescu. “Stabilité pour une classe de problémes d’optimisation non convexe”. To appear in C.R. Acad. Sci.,Paris.Google Scholar

Copyright information

© Springer Science+Business Media New York 1989

Authors and Affiliations

  • C. Zălinescu
    • 1
  1. 1.Faculty of MathematicsUniv. “Al.I.Cuza”Iaşi, R.S.Romania

Personalised recommendations