Necessary Conditions for an Extremum, Penalty Functions and Regularity

  • Boris N. Pshenichnyj
Part of the Progress in Systems and Control Theory book series (PSCT, volume 2)


The aim of this paper is to show that the three concepts mentioned in the title are closely related when considering the most general optimization problems. Each of these concepts has been studied in numerous works where different relations among them have been established. Here we give the most general statements and we show that in the presence of regularity the necessary conditions for an extremum in the form of the Lagrange multipliers rule are always fulfilled. It should be emphasized that for the most complete description of the point of extremum in problems with non-smooth constraints the optimality conditions must be formulated not in the form of a single rule of multipliers but in the form of the whole family of such rules. Only this makes it possible to avoid a situation when necessary conditions are fulfilled at points which trivially cannot be optimal.


Penalty Function Convex Analysis Tangent Cone Convex Approximation Continuous Convex Function 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Clarke F.H. A new approach to Lagrange multipliers, Math. of Operations Research. 1 (1976), 165–174.CrossRefGoogle Scholar
  2. 2.
    Pshenichnyj B. N. Convex analysis and extremal problems, Nauka, Moscow 1980 (in Russian).Google Scholar
  3. 3.
    Dem’yanov V. F., Rubinov A. M. Quasidifferential Calculus, Optimization Software, New York 1986.CrossRefGoogle Scholar
  4. 4.
    Dmitr’yuk A. W., Milyutin A. A., Osmolovskii, N. P. Lusternik’s theorem and extremum theory, Uspiekhy Matematichieskih Nauk 35 (1980), 11–46 (in Russian).Google Scholar
  5. 5.
    Aubin J.-P., Frankowska H. On inverse function theorems for self-valued maps. IIA SA, WP-84–68, 1984, 1–21.Google Scholar
  6. 6.
    Pshenichnyj B. N. Necessary conditions of extremum, M.N. 1982 (in Russian).Google Scholar
  7. 7.
    Neustadt L. Optimization: A Theory of Necessary Conditions. — Princeton University Press. Princeton, N.J., 1976.Google Scholar

Copyright information

© Springer Science+Business Media New York 1990

Authors and Affiliations

  • Boris N. Pshenichnyj
    • 1
  1. 1.V.M.Glushkov Institute of CyberneticsAcademy of Sciences of the Ukrainian SSRKiev 207USRR

Personalised recommendations