First and Second Order Optimality Conditions and Perturbation Analysis of Semi-Infinite Programming Problems

  • Alexander Shapiro
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 25)


In this paper we discuss finite dimensional optimization problems subject to an infinite number of inequality constraints (semi-infinite programming problems). We study such problems in a general framework of optimization problems subject to constraints formulated in a form of cone inclusions. General results on duality, and first and second order optimality conditions are presented and specified to considered semi-infinite programming problems. Finally some recent results on quantitative stability and sensitivity analysis of parameterized semi-infinite programming problems are discussed.


Lagrange Multiplier Perturbation Analysis Constraint Qualification Empty Interior Order Optimality Condition 
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.


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  1. [1]
    E. J. Anderson and P. Nash. Linear Programming in Infinite-Dimensional Spaces. Wiley, New York, 1987.zbMATHGoogle Scholar
  2. [2]
    A. Auslender and R. Cominetti. First and second order sensitivity analysis of nonlinear programs under directional constraint qualification conditions. Optimization, 21:351–363, 1990.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    A. Ben-Tal, M. Telboulle, and J. Zowe. Second order necessary optimality conditions for semi-infinite programming problems. In R. Hettich editor. Lecture Notes in Control and Information Sciences15. pages 17–30. Springer Verlag, Berlin, 1979.Google Scholar
  4. [4]
    A. Ben-Tal. Second order and related extremality conditions in nonlinear programming. Journal of Optimization Theory and Applications, 31:143–165, 1980.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    A. Ben-Tal and J. Zowe. A unified theory of first and second order conditions for extremum problems in topological vector spaces. Mathematical Programming Study, 19:39–76, 1982.CrossRefMathSciNetGoogle Scholar
  6. [6]
    J. F. Bonnans. Directional derivatives of optimal solutions in smooth nonlinear programming. Journal Optimization Theory and Applications, 73:27–45, 1992.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    J. F. Bonnans and R. Cominetti. Perturbed optimization in Banach spaces, part I: a general theory based on a weak directional constraint qualification. SIAM J. Control and Optimization, 34:1151–1171, 1996.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    J. F. Bonnans and R. Cominetti. Perturbed optimization in Banach spaces II: a theory based on a strong directional qualification. SIAM J. Control and Optimization, 34:1172–1189, 1996.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    J. F. Bonnans and R. Cominetti. Perturbed optimization in Banach spaces III: semi-infinite optimization. SIAM J. Control and Optimization, 34:1555–1567, 1996.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    J. F. Bonnans and A. Shapiro. Optimization problems with perturbations, a guided tour. SIAM Review, to appear.Google Scholar
  11. [11]
    J. F. Bonnans, R. Cominetti, and A. Shapiro. Second order necessary and sufficient optimality conditions under abstract constraints. Rapport de Recherche INRIA 2952.Google Scholar
  12. [12]
    J. F. Bonnans, R. Cominetti, and A. Shapiro. Sensitivity analysis of optimization problems under second order regular constraints. Rapport de Recherche INRIA 2989, 1996.Google Scholar
  13. [13]
    J. F. Bonnans and A. Shapiro. Perturbation Analysis of Optimization Problems. in preparation.Google Scholar
  14. [14]
    R. Cominetti. Metric regularity, tangent sets and second order optimality conditions. Applied Mathematics and Optimization, 21:265–287, 1990.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    R. Cominetti and J.-P. Penot. Tangent sets to unilateral convex sets. C. R. Acad. Sci. Paris Sér. I Math.321, 12:1631–1636, 1995.zbMATHMathSciNetGoogle Scholar
  16. [16]
    I. Ekeland and R. Temam. Analyse convexe et problèmes variationnels. Collection Etudes Mathématiques. Dunod; Gauthier-Villars, Paris-Brussels-Montreal, Que., 1974.zbMATHGoogle Scholar
  17. [17]
    A. V. Fiacco. Introduction to Sensitivity and Stability Analysis in Nonlinear Programming. New York: Academic Press, 1983.zbMATHGoogle Scholar
  18. [18]
    J. Gauvin. A necessary and sufficient regularity condition to have bounded multipliers in nonconvex programming. Mathematical Programming, 12:136–138, 1977.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    J. Gauvin and J. W. Tolle. Differential stability in nonlinear programming. SIAM J. Control and Optimization, 15:294–311, 1977.zbMATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    J. Gauvin and R. Janin. Directional behaviour of optimal solutions in nonlinear mathematical programming. Mathematics of Operations Research, 13:629–649, 1988.zbMATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    B. Gollan. On the marginal function in nonlinear programming. Mathematics of Operations Research, 9:208–221, 1984.zbMATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    E. G. Gol’shtein. Theory of Convex Programming. Translations of Mathematical Monographs 36, American Mathematical Society, Providence, RJ, 1972.Google Scholar
  23. [23]
    R. Hettich and H. Th. Jongen. Semi-infinite programming: conditions of optimality and applications. In J. Stoer, editor. Optimization Techniques2, pages 1–11. Springer, Berlin-Heidelberg-New York, 1978.CrossRefGoogle Scholar
  24. [24]
    R. Hettich and K. O. Kortanek. Semi-infinite programming: theory, methods and applications. SIAM Review, 35:380–429, 1993.zbMATHCrossRefMathSciNetGoogle Scholar
  25. [25]
    A.D. Ioffe. On some recent developments in the theory of second order optimality conditions. In S. Dolecki editor. Optimization, Lecture Notes in Mathematics, pages 55–68, vol. 1405. Springer Verlag, Berlin, 1989.CrossRefGoogle Scholar
  26. [26]
    A. D. Ioffe. Variational analysis of a composite function: A formula for the lower second order epi-derivative. J. of Mathematical Analysis and Applications, 160:379–405, 1990.CrossRefMathSciNetGoogle Scholar
  27. [27]
    H. Th. Jongen and G. Zwier. On the local structure of the feasible set in semi-infinite optimization. In B. Brosowski, F. Deutsch, editors. Parametric Optimization and Approximation, Int. Series of Num. Math., 72:185–202. Basel: Birkhäuser, 1985.Google Scholar
  28. [28]
    H. Kawasaki. An envelope-like effect of infinitely many inequality constraints on second-order necessary conditions for minimization problems. Mathematical Programming, 41:73–96, 1988.zbMATHCrossRefMathSciNetGoogle Scholar
  29. [29]
    H. Kawasaki. The upper and lower second order directional derivatives of a sup-type function. Mathematical Programming, 41:327–339, 1988.zbMATHCrossRefMathSciNetGoogle Scholar
  30. [30]
    H. Kawasaki. Second-order necessary and sufficient optimality conditions for minimizing a sup-type function. Appl. Math. Opt., 26:195–220, 1992.zbMATHCrossRefMathSciNetGoogle Scholar
  31. [31]
    D. Klatte. Stability of stationary solutions in semi-infinite optimization via the reduction approach. In W. Oettli, D. Pallaschke, editors. Advances in Optimization, pages 155–170. Springer, Berlin-Heidelberg-New York, 1992.Google Scholar
  32. [32]
    D. Klatte. Stable local minimizers in semi-infinite optimization: regularity and second-order conditions. J. Comp. Appl. Math., 56:137–157, 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  33. [33]
    D. Klatte. On regularity and stability in semi-infinite optimization. Set-Valued Analysis, 3:101–111, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  34. [34]
    D. Klatte and R. Henrion. Regularity and stability in nonlinear semi-infinite optimization. This volume.Google Scholar
  35. [35]
    S. Kurcyusz. On the existence and nonexistence of Lagrange multipliers in Ba-nach spaces. Journal of Optimization Theory and Applications, 20:81–110, 1976.zbMATHCrossRefMathSciNetGoogle Scholar
  36. [36]
    P.-J. Laurent. Approximation et Optimisation. Collection Enseignement des Sciences, No. 13. Hermann, Paris, 1972.zbMATHGoogle Scholar
  37. [37]
    F. Lempio and H. Maurer. Differential stability in infinite-dimensional nonlinear programming. Applied Mathematics and Optimization, 6:139–152, 1980.zbMATHCrossRefMathSciNetGoogle Scholar
  38. [38]
    E. S. Levitin. Differentiability with respect to a parameter of the optimal value in parametric problems of mathematical programming. Kibemetika, pp. 44–59, 1976.Google Scholar
  39. [39]
    E. S. Levitin. Perturbation Theory in Mathematical Programming. Wiley, Chichester, 1994.zbMATHGoogle Scholar
  40. [40]
    O. L. Mangasarian and S. Fromovitz. The Fritz John necessary optimality conditions in the presence of equality and inequality constraints. J. Math. Anal. Appl., 17:37–47, 1967.zbMATHCrossRefMathSciNetGoogle Scholar
  41. [41]
    J. P. Penot. Optimality conditions for minimax problems, semi-infinite programming problems and their relatives. Report 92/16, UPRA, Laboratoire de Math. Appl., Av. de l’Université, 64000 Pau, France.Google Scholar
  42. [42]
    J. P. Penot. Optimality conditions in mathematical programming and composite optimization. Mathematical Programming, 67:225–245, 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  43. [43]
    B. N. Pshenichnyi. Necessary Conditions for an Extremum. Marcel Dekker, New York, 1971.Google Scholar
  44. [44]
    S. M. Robinson. Regularity and stability for convex multivalued functions. Mathematics of Operations Research, 1:130–143, 1976.zbMATHCrossRefMathSciNetGoogle Scholar
  45. [45]
    S. M. Robinson. Stability theorems for systems of inequalities, Part II: differen-tiable nonlinear systems. SIAM J. Numerical Analysis, 13:497–513, 1976.zbMATHCrossRefGoogle Scholar
  46. [46]
    S. M. Robinson. First order conditions for general nonlinear optimization. SIAM Journal on Applied Mathematics, 30:597–607, 1976.zbMATHCrossRefMathSciNetGoogle Scholar
  47. [47]
    R. T. Rockafellar. Convex Analysis. Princeton University Press, Princeton, NJ, 1970.zbMATHGoogle Scholar
  48. [48]
    R. T. Rockafellar. Conjugate Duality and Optimization. Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1974.zbMATHCrossRefGoogle Scholar
  49. [49]
    R. T. Rockafellar. Second-order optimality conditions in nonlinear programming obtained by way of epi-derivatives. Mathematics of Operations Research, 14:462–484, 1989.zbMATHCrossRefMathSciNetGoogle Scholar
  50. [50]
    A. Shapiro. Second-order derivatives of extremal-value functions and optimality conditions for semi-infinite programs. Mathematics of Operations Research, 10:207–219, 1985.zbMATHCrossRefMathSciNetGoogle Scholar
  51. [51]
    A. Shapiro. Sensitivity analysis of nonlinear programs and differentiability properties of metric projections. SIAM J. Control and Optimization, 26:628–645, 1988.zbMATHCrossRefMathSciNetGoogle Scholar
  52. [52]
    A. Shapiro. Perturbation analysis of optimization problems in Banach spaces. Numerical Functional Analysis and Optimization, 13:97–116, 1992.zbMATHCrossRefMathSciNetGoogle Scholar
  53. [53]
    A. Shapiro. On Lipschitzian stability of optimal solutions of parametrized semi-infinite programs. Mathematics of Operations Research, 19:743–752, 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  54. [54]
    A. Shapiro. Directional differentiability of the optimal value function in convex semi-infinite programming. Mathematical Programming, Series A, 70:149–157, 1995.zbMATHMathSciNetGoogle Scholar
  55. [55]
    A. Shapiro. On uniqueness of Lagrange multipliers in optimization problems subject to cone constraints. SIAM J. Optimization, 7:508–518, 1997.zbMATHCrossRefGoogle Scholar
  56. [56]
    C. Ursescu. Multifunctions with convex closed graph. Czechoslovak Mathematical Journal, 25:438–441, 1975.MathSciNetGoogle Scholar
  57. [57]
    W. Wetterling. Definitheitsbedingungen für relative Extrema bei Optimierungs-und Approximationsaufgaben. Num. Math., 15:122–136, 1970.zbMATHCrossRefMathSciNetGoogle Scholar
  58. [58]
    J. Zowe and S. Kurcyusz. Regularity and stability for the mathematical programming problem in Banach spaces. Applied Mathematics and Optimization, 5:49–62, 1979.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1998

Authors and Affiliations

  • Alexander Shapiro
    • 1
  1. 1.Georgia Institute of TechnologySchool of Industrial and Systems EngineeringAtlantaUSA

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