Regularity conditions for constrained extremum problems via image space

  • P. H. Dien
  • G. Mastroeni
  • M. Pappalardo
  • P. H. Quang
Contributed Papers


Exploiting the image-space approach, we give an overview of regularity conditions. A notion of regularity for the image of a constrained extremum problem is given. The relationship between image regularity and other concepts is also discussed. It turns out that image regularity is among the weakest conditions for the existence of normal Lagrange multipliers.

Key Words

Image space regularity conditions constraint qualifications Lagrange multipliers tangent cones calmness nondegeneracy penalty functions 


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  1. 1.
    Giannessi, F.,Theorems of the Alternative and Optimality Conditions, Journal of Optimization Theory and Applications, Vol. 42, pp. 331–365, 1984.Google Scholar
  2. 2.
    Hestenes, M. R.,Calculus of Variations and Optimal Control Theory, John Wiley, New York, New York, 1966.Google Scholar
  3. 3.
    Dien, P. H., Mastroeni, G., Pappalardo, M., andQuang, P. H.,Regularity Conditions for Constrained Extremum Problems via Image Space: the Linear Case (to appear).Google Scholar
  4. 4.
    Martein, L.,Regularity Conditions for Constrained Extremum Problems, Journal of Optimization Theory and Applications, Vol. 47, pp. 217–233, 1985.Google Scholar
  5. 5.
    Dien, P. H., andSach, P. H.,Further Properties of the Regularity of Inclusion Systems, Nonlinear Analysis: Theory, Methods and Applications, Vol. 3, pp. 1251–1267, 1989.Google Scholar
  6. 6.
    Dien, P. H.,On the Regularity Condition for the Extremal Problem under Locally Lipschitz Inclusion Constraints, Applied Mathematics and Optimization, Vol. 13, pp. 151–161, 1985.Google Scholar
  7. 7.
    Mangasarian, O. L., andFromovitz, S.,The Fritz-John Necessary Optimality Condition in the Presence of Equality and Inequality Constraints, Journal of Mathematical Analysis and Applications, Vol. 7, pp. 37–47, 1967.Google Scholar
  8. 8.
    Kurcyusz, S.,On the Existence and Nonexistence of Lagrange Multipliers in a Banach Space, Journal of Optimization Theory and Applications, Vol. 20, pp. 81–110, 1976.Google Scholar
  9. 9.
    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
  10. 10.
    Robinson, S. M.,Regularity and Stability for Convex Multivalued Functions, Mathematics of Operations Research, Vol. 1, pp. 130–143, 1976.Google Scholar
  11. 11.
    Penot, J. P.,A New Constraint Qualification Condition, Journal of Optimization Theory and Applications, Vol. 48, pp. 459–468, 1986.Google Scholar
  12. 12.
    Clarke, F.,Optimization and Nonsmooth Analysis, J. Wiley, New York, New York, 1984.Google Scholar
  13. 13.
    Cambini, A.,Nonlinear Separation Theorem, Duality, and Optimality Condition, Optimization and Related Fields, Edited by R. Conti, E. De Giorgi, and F. Giannessi, Springer-Verlag, Berlin, Germany, pp. 57–93, 1984.Google Scholar
  14. 14.
    Quang, P. H.,Lagrange Multiplier Rules via Image Space Analysis, Nonsmooth Optimization: Theory and Methods, Edited by F. Giannessi, Gordon and Breach, London, England, pp. 354–365, 1993.Google Scholar
  15. 15.
    Tardella, F.,On the Image of a Constrained Extremum Problem and Some Applications to the Existence of a Minimum, Journal of Optimization Theory and Applications, Vol. 60, pp. 93–104, 1989.Google Scholar

Copyright information

© Plenum Publishing Corporation 1994

Authors and Affiliations

  • P. H. Dien
    • 1
  • G. Mastroeni
    • 2
  • M. Pappalardo
    • 3
  • P. H. Quang
    • 1
  1. 1.Department of Dynamical Systems, Institute of MathematicsNCSR of VietnamHanoiVietnam
  2. 2.Department of MathematicsUniversity of MilanoMilanoItaly
  3. 3.Department of MathematicsUniversity of PisaPisaItaly

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