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Zeitschrift für Operations Research

, Volume 35, Issue 5, pp 401–424 | Cite as

Solving nonsmooth inclusions in the convex case

  • H. The Phung
  • P. Huy Dien
Theory
  • 23 Downloads

Abstract

We give a method for solving the inclusion 0εF(x), whereF is a set-valued map from a Hilbert space into a Banach space, whose graph is a closed convex set. The given algorithms are convergent if the problem is consistent and terminate after a finite number of iterations if otherwise. A convergence rate of geometrical progression is also obtained if a regularity condition of the Robinson's type is assumed.

Key words

Convex set-valued map support function inclusion convex inequality subgradient method 

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References

  1. [1]
    Dien PH, Phung HT (1989) An Algorithm for Finding a Solution to the Inclusion 0εF(x). J Optim Theor Appl (to appear)Google Scholar
  2. [2]
    Robinson SM (1975) A Subgradient Algorithm for SolvingK-convex Inequalities. Optimization and Operation Research, Lecture Notes in Economics and Mathematical Systems 117. Springer, Berlin, pp 237–245Google Scholar
  3. [3]
    Eremin II (1965) The Relaxation Method of Solving Systems of Inequalities with Convex Functions on the Left Side. Soviet Math Doklady 6:219–222Google Scholar
  4. [4]
    Dien PH (1983) Locally Lipschitzian set-valued maps and generalized extremal problems. Acta Math Viet 8:109–122Google Scholar
  5. [5]
    Lemarechal C (1986) Construction Bundle Methods for Convex Optimization. Hiriarty-Urruty JB (ed) Fermat Days 85. North-Holland, AmsterdamGoogle Scholar
  6. [6]
    Poljak BT (1969) Minimization of Unsmooth Functional. USSR Comp Math and Math Phys, 9(3): 14–29Google Scholar
  7. [7]
    Shor NZ (1968) The Rate of Convergence of the Generalized Gradient Descent Method. Cybernetics 4(3):79–80Google Scholar
  8. [8]
    Aubin IP, Ekeland IV (1984) Applied Nonlinear Analysis. Wiley, New YorkGoogle Scholar
  9. [9]
    Bazaraa MS, Shetty CM (1979) Nonlinear Programming Theory and Algorithms. Wiley, New YorkGoogle Scholar

Copyright information

© Physica-Verlag 1991

Authors and Affiliations

  • H. The Phung
    • 1
  • P. Huy Dien
    • 1
  1. 1.Institute of MathematicsBo Ho, HanoVietnam

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