Advertisement

New applications of nonsmooth analysis to nonsmooth optimization

Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 979)

Keywords

Maximum Principle Optimal Control Problem Nonsmooth Analysis Nonsmooth Optimization mUltiplier Rule 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J.-P. Aubin, Lipschitz behavior of solutions to convex minimization problems, Working Paper A-2361, IIASA, Laxanburg 1981.Google Scholar
  2. 2.
    F.H. Clarke, A general control problem, in Calculus of Variations and Optimal Control, D.L. Russel, editor, Academic Press 1976, pp. 257–278.Google Scholar
  3. 3.
    _____, A new approach to Lagrange multipliers, Math. Operation Res., 1 (1976), 165–174.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    _____, The maximum principle under minimal hypotheses, SIAM J. Control Optimization 14 (1976), 1078–1091.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    _____, Generalized gradients of Lipschitz functionals, Adv. Math.Google Scholar
  6. 6.
    A.V. Dmitruk, A.A. Miljutin and N.P. Osmolovskii, The Ljusternik theorem and the theory of extremum, Uspehi Mat. Nauk 35:6 (1980), 11–46.MathSciNetGoogle Scholar
  7. 7.
    A.Ya. Dubovitskii and A.A. Miljutin, Translation of Euler equations, J. Computational Math. and Mathematical Physics, 9 (1969), 1263–1284.zbMATHGoogle Scholar
  8. 8.
    H. Halkin, Optimal Control as programming in infinite dimensional spaces, in Calculus of Variations, Classical and Modern, Edizioni Cremonese, Roma, 1966, 179–192.Google Scholar
  9. 9.
    _____, Mathematical programming without differentiability, in Calculus of Variations and Optimal Control, D.L. Russell, ed., Academic Press, 1976, 279–288.Google Scholar
  10. 10.
    J.-B. Hiriart-Urruty, Refinements of necessary optimality conditions in nondifferentiable programming, Appl. Math. Optim. 5 (1979), 63–82.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    A.D. Ioffe, Necessary and sufficient conditions for a local minimum, SIAM J. Control Optimization 17 (1979), 245–288.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    _____, Nonsmooth analysis: differential calculus of non-differentiable mappings, Trans. Amer. Math. Soc. 266 (1981), 1–56.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    _____, Sous-différentielles approchées de fonctions numériques, C.R. Acad. Sci. Paris (1981)Google Scholar
  14. 14.
    _____, Necessary conditions in nonsmooth optimization, Math. Operation Res., to appear.Google Scholar
  15. 15.
    _____, Approximate subdifferentials of nonconvex functions, Cahiers de Math. de la Decision, CEREMADE, Paris 1981Google Scholar
  16. 16.
    _____, Second order conditions in nonlinear nonsmooth semi-infinite programming, Intern. Symp. on Semi-Infinite Programming, Austi, Texas, September 1981.Google Scholar
  17. 17.
    A.D. Ioffe and V.M. Tikhomirov, Theory of Extremal Problems, Nauka, Moscow, 1974, North Holland, 1979.Google Scholar
  18. 18.
    A. Kruger, Calculus of generalized differentials, to appearGoogle Scholar
  19. 19.
    _____, Generalized differentials of nonsmooth functions and necessary conditions for an extremum, to appear.Google Scholar
  20. 20.
    A. Kruger and B. Mordukhovich, Extremal points and the Euler equation in nonsmooth optimization problems, Dokl. Acad. Nauk BSSR 24 (1980), 684–687.MathSciNetzbMATHGoogle Scholar
  21. 21.
    B. Mordukhovich, Maximum principle in the optimal time control problem with nonsmooth constraints, Appl. Math. Mech. 40 (1976), 1014–1023.MathSciNetGoogle Scholar
  22. 22.
    L.W. Neustadt, Optimization, Princeton Univ. Press 1976.Google Scholar
  23. 23.
    J. Warga, Necessary conditions without differentiability assumptions in optimal control, J. Diff. Eqs. 18 (1975), 41–62.MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    _____, Derivative containers, inverse functions and controllability, in Calculus of Variations and Control Theory, D.L. Russell, editor, Academic Press, 1976.Google Scholar
  25. 25.
    _____, Controllability and a multiplier rule for nondifferentiable optimization problems, SIAM J. Control Optimization 16 (1978), 803–812.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1983

Authors and Affiliations

There are no affiliations available

Personalised recommendations