Stability and regular points of inequality systems

  • J. M. Borwein
Contributed Papers

Abstract

We undertake a general study of regular points of Lipschitz and strictly differentiable mappings with applications to tangent cone analysis, inversion theorems, perturbed optimization problems, and higher-order conditions.

Key Words

Stability inversion results Ekeland's theorem tangent cones value functions higher-order conditions 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alt, W.,Lipschitzian Perturbations of Infinite Optimization Problems, University of Bayreuth, Preprint, 1980.Google Scholar
  2. 2.
    Dmitruck, A. V., Milyutin, A. A., andOsmolovski, N. P.,Lysternik's Theorem and the Theory of Extrema, Russian Mathematics Surveys, Vol. 35, pp. 11–51, 1980.Google Scholar
  3. 3.
    Duong, P. C., andTuy, H.,Stability, Surjectivity, and Local Invertibility of Nondifferentiable Mappings, Acta Vietnamica, Vol. 3, pp. 89–103, 1978.Google Scholar
  4. 4.
    Ioffe, A. D.,Regular Points of Lipschitz Mappings, Transactions of the American Mathematics Society, Vol. 251, pp. 61–69, 1979.Google Scholar
  5. 5.
    Penot, J. P.,On Regularity Conditions in Mathematical Programming, Mathematical Programming Study, Vol. 19, pp. 167–199, 1982.Google Scholar
  6. 6.
    Robinson, S. M.,Stability Theory for Systems of Inequalities, Part II: Differentiable Nonlinear Systems, SIAM Journal on Numerical Analysis, Vol. 13, pp. 497–513, 1976.Google Scholar
  7. 7.
    Ekeland, I.,On the Variational Principle, Journal of Mathematical Analysis and Applications, Vol. 47, pp. 324–353, 1974.Google Scholar
  8. 8.
    Robinson, S. M.,Regularity and Stability for Convex Multi-Valued Functions, Mathematics of Operations Research, Vol. 1, pp. 130–143, 1976.Google Scholar
  9. 9.
    Dolecki, S.,A General Theory of Necessary Optimality Conditions, Journal of Mathematical Analysis and Applications, Vol. 78, pp. 267–308, 1980.Google Scholar
  10. 10.
    Ioffe, A. D.,Nonsmooth Analysis: Differential Calculus of Nondifferentiable Mappings, Transactions of the American Mathematical Society, Vol. 266, pp. 1056, 1981.Google Scholar
  11. 11.
    Ekeland, I.,Nonconvex Minimization Problems, Bulletin of the American Mathematical Society, Vol. 1, pp. 443–474, 1979.Google Scholar
  12. 12.
    Ekeland, I., andTemam, R.,Convex Analysis and Variational Problems, North-Holland, Amsterdam, Holland, 1976.Google Scholar
  13. 13.
    Caristi, J.,Fixed-Point Theorems for Mappings Satisfying Inwardness Conditions, Transactions of the American Mathematical Society, Vol. 215, pp. 241–251, 1976.Google Scholar
  14. 14.
    Sullivan, F.,A Characterization of Complete Metric Spaces, Proceedings of the American Mathematical Society, Vol. 83, pp. 345–356, 1981.Google Scholar
  15. 15.
    Borwein, J.,Completeness and the Contraction Principle, Proceedings of the American Mathematical Society, Vol. 87, pp. 246–250, 1983.Google Scholar
  16. 16.
    Tuy, H.,A Fixed-Point Theorem Involving a Hybrid Inwardness Condition, Mathematishe Nachritten, Vol. 102, pp. 271–275, 1981.Google Scholar
  17. 17.
    Hiriart-Urruty, J. B.,A Short Proof of the Variational Principle for Approximate Solutions of a Minimization Problem, American Mathematical Monthly, Vol. 91, pp. 206–207, 1983.Google Scholar
  18. 18.
    Clarke, F. H.,Generalized Gradients and Applications, Transactions of the American Mathematical Society, Vol. 205, pp. 247–262, 1974.Google Scholar
  19. 19.
    Clarke, F. H.,A New Approach to Lagrange Multipliers, Mathematics of Operations Research, Vol. 1, pp. 165–174, 1976.Google Scholar
  20. 20.
    Hiriart-Urruty, J. B.,Tangent Cones, Generalized Gradients, and Mathematical Programming in Banach Spaces, Mathematics of Operations Research, Vol. 4, pp. 79–97, 1979.Google Scholar
  21. 21.
    Rockafellar, R. T.,Conjugate Duality and Optimization, SIAM Publications, Philadelphia, Pennsylvania, 1974.Google Scholar
  22. 22.
    Rockafellar, R. T.,Directionally Lipschitzian Functions and Subdifferential Calculus, Proceedings of the London Mathematical Society, Vol. 39, pp. 331–355, 1979.Google Scholar
  23. 23.
    Nijenhuis, A.,Strong Derivatives and Inverse Mappings, American Mathematical Monthly, Vol. 81, pp. 969–980, 1974.Google Scholar
  24. 24.
    Rockafellar, R. T.,Generalized Directional Derivatives and Subgradients of Nonconvex Functions, Canadian Journal of Mathematics, Vol. 32, pp. 257–280, 1980.Google Scholar
  25. 25.
    Rockafellar, R. T.,The Theory of Subgradients and Its Applications to Problems of Optimization of Convex and Nonconvex Functions, Heldermann-Verlag, Berlin, Germany, 1981.Google Scholar
  26. 26.
    Clarke, F. H.,Optimization and Nonsmooth Analysis, Wiley, New York, New York, 1983.Google Scholar
  27. 27.
    Clarke, F. H.,On the Inverse Function Theorem, Pacific Journal of Mathematics, Vol. 64, pp. 97–102, 1976.Google Scholar
  28. 28.
    Hiriart-Urruty, J. B.,Refinements of Necessary Conditions for Optimality in Nondifferentiable Programming, I, Applied Mathematics and Optimization, Vol. 5, pp. 63–82, 1979.Google Scholar
  29. 29.
    Ioffe, A. D.,Necessary and Sufficient Conditions for Local Minimum, SIAM Journal on Control and Optimization, Vol. 17, pp. 245–288, 1979.Google Scholar
  30. 30.
    Borwein, J. M.,Convex Relations in Analysis and Optimization, Generalized Concavity in Optimization and Economics, Edited by S. Shaibk and W. Ziemba, Academic Press, New York, New York, pp. 335–377, 1981.Google Scholar
  31. 31.
    Penot, J. P.,Inversion à Droite d'Applications Nonlinéaires et Applications, Comptes Rendus, Academie de Science de Paris, Vol. 290, pp. 997–1000, 1980.Google Scholar
  32. 32.
    Pomerol, J. C.,Application de la Programmation Convexe à la Programmation Differèntiable, Comptes Rendus, Academie de Science de Paris, Vol. 279, pp. 1041–1044, 1979.Google Scholar
  33. 33.
    Rockafellar, R. T.,Clarke's Tangent Cone and the Boundaries of Sets in R n, Nonlinear Analysis: Theory, Methods, and Applications, Vol. 3, pp. 145–154, 1979.Google Scholar
  34. 34.
    Borwein, J. M., andStrowjas, H. M.,Directionally Lipschitzian Mappings on Baire Spaces, Canadian Journal of Mathematics, Vol. 34, pp. 95–130, 1984.Google Scholar
  35. 35.
    Penot, J. P.,A Characterization of Tangential Regularity, Nonlinear Analysis: Theory, Methods, and Applications, Vol. 5, pp. 625–643, 1981.Google Scholar
  36. 36.
    Halkin, H.,Implicit Functions and Optimization Problems without Continuous Differentiability of the Data, SIAM Journal on Control and Optimization, Vol. 12, pp. 229–236, 1974.Google Scholar
  37. 37.
    Ben-Tal, A., andZowe, J.,A Unified Theory of First-Order and Second-Order Conditions for Extremum Problems in Topological Vector Spaces, Mathematical Programming Study, Vol. 19, pp. 39–76, 1982.Google Scholar
  38. 38.
    Lempio, F., andZowe, J.,Higher-Order Optimality Conditions, Optimization and Operations Research in Modern Applied Mathematics, Edited by B. Korte, North-Holland, Amsterdam, Holland, 1982.Google Scholar
  39. 39.
    Maurer, H., andZowe, J.,Second-Order Necessary and Sufficient Conditions for Infinite-Dimensional Programming Problems, Mathematical Programming, Vol. 16, pp. 641–652, 1979.Google Scholar
  40. 40.
    Rudin, W.,Principles of Mathematical Analysis, 3rd Edition, McGraw-Hill, New York, New York, 1976.Google Scholar
  41. 41.
    Beltrami, E. J.,An Algorithmic Approach to Nonlinear Analysis and Optimization, Academic Press, New York, New York, 1970Google Scholar
  42. 42.
    Ward, D. E.,Nonsmooth Calculus in Finite Dimensions, Dalhousie University, Preprint, 1984.Google Scholar
  43. 43.
    Ursescu, C.,Multifunctions with Convex Closed Graph, Czechoslovak Mathematical Journal, Vol. 7, pp. 439–441, 1975.Google Scholar
  44. 44.
    Zowe, J.,A Remark on a Regularity Condition in Mathematical Programming, Journal of Optimization Theory and Applications, Vol. 25, pp. 375–382, 1978.Google Scholar
  45. 45.
    Zowe, J., andKurcyusz, S.,Regularity and Stability for the Mathematical Programming Problem in Banach Spaces, Applied Mathematics and Optimization, Vol. 5, pp. 49–62, 1979.Google Scholar
  46. 46.
    Luenberger, D. G.,Optimization by Vector Space Methods, John Wiley, New York, New York, 1969.Google Scholar
  47. 47.
    Borwein, J. M.,Continuity and Differentiability Properties of Convex Operators, Journal of the London Mathematical Society, Vol. 44, pp. 420–444, 1982.Google Scholar
  48. 48.
    Borwein, J. M.,Weak Tangent Cones and Optimization in Banach Spaces, SIAM Journal on Control and Optimization, Vol. 16, pp. 512–522, 1978.Google Scholar
  49. 49.
    Guignard, M.,Generalized Kuhn-Tucker Conditions for Mathematical Programming, SIAM Journal on Control and Optimization, Vol. 7, pp. 232–241, 1969.Google Scholar
  50. 50.
    Pomerol, J. C.,Application de la Programmation Convexe à la Programmation Nondiffèrentiable, Comptes Rendus, Academic de Science de Paris, Vol. 289, pp. 805–808, 1979.Google Scholar
  51. 51.
    Gauvin, J., andTolle, J. W.,Differentiable Stability in Nonlinear Programming, SIAM Journal on Control and Optimization, Vol. 15, pp. 294–311, 1977.Google Scholar
  52. 52.
    Lempio, F., andMaurer, H.,Differential Stability in Infinite-Dimensional Nonlinear Programming, Applied Mathematics and Optimization, Vol. 6, pp. 139–152, 1980.Google Scholar
  53. 53.
    Pomerol, J. C.,The Lagrange Multiplier Set and the Generalized Gradient Set of the Marginal Function of a Differentiable Program in a Banach Space, Journal of Optimization Theory and Applications, Vol. 38, pp. 307–317, 1982.Google Scholar
  54. 54.
    Rockafellar, R. T.,Lagrange Multipliers and Subderivatives of Optimal Value Functions in Nonlinear Programming, Mathematical Programming Study, Vol. 17, pp. 28–66, 1982.Google Scholar
  55. 55.
    Gollan, B.,Perturbed Theory for Abstract Optimization Problems, Journal of Optimization Theory and Applications, Vol. 35, pp. 417–441, 1981.Google Scholar
  56. 56.
    Gauvin, J., andDubeau, F.,Some Examples and Counterexamples for the Stability Analysis of Nonlinear Programming Problems, Ecole Polytechnique de Montreal, Preprint, 1982.Google Scholar
  57. 57.
    Ioffe, A. D.,Second-Order Conditions in Nonlinear Nonsmooth Problems of Semi-Infinite Programming, Semi-Infinite Programming and Applications, Edited by K. V. Kortanek and A. V. Fiacco, Springer-Verlag, Berlin, Germany, pp. 262–280, 1983.Google Scholar
  58. 58.
    Borwein, J. M.,Necessary and Sufficient Conditions for Quadratic Minimality, Numerical Functional Analysis and Optimization, Vol. 5, pp. 127–140, 1982.Google Scholar
  59. 59.
    Ben-Tal, A., andZowe, J.,Necessary and Sufficient Optimality Conditions for a Class of Nonsmooth Optimization Problems, University of Bayreuth, Preprint, 1980.Google Scholar

Copyright information

© Plenum Publishing Corporation 1986

Authors and Affiliations

  • J. M. Borwein
    • 1
  1. 1.Department of Mathematics, Statistics and Computing ScienceDalhousie UniversityHalifax

Personalised recommendations