Lipschitzian Stability in Optimization: The Role of Nonsmooth Analysis

  • R. T. Rockafellar
Part of the Lecture Notes in Economics and Mathematical Systems book series (LNE, volume 255)


The motivations of nonsmooth analysis are discussed. Applications are given to the sensitivity of optimal values, the interpretation of Lagrange multipliers, and the stability of constraint systems under perturbation.


Tangent Cone Constraint Qualification Lipschitz Continuity Nonsmooth Analysis Lower Semicontinuous Function 
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  1. [1]
    F.H. Clarke, Generalized gradients and applications, Trans. Amer. Soc. 205 (1975), pp. 247–262.CrossRefGoogle Scholar
  2. [2]
    F.H. Clarke, Optimization and Nonsmooth Analysis,Wiley-Interscience, New York, 1983.Google Scholar
  3. [3]
    R.T. Rockafellar, Convex Analysis, Princeton University Press, Princeton NJ, 1970.Google Scholar
  4. [4]
    R.T. Rockafellar, Favorable classes of Lipschitz continuous functions in subgradient optimization, Processes in Nondi,fferentiable Optimization, E. Nurminski (ed.), IIASA Collaborative Proceeding Series, International Institute for Applied Systems Analysis, Laxenburg, Austria, 1982, pp. 125 - 143.Google Scholar
  5. [5]
    S. Saks, Theory of the Integral, Monografie Matematyczne Ser., no. 7, 1937; 2nd rev.ed. Dover Press, New York, 1964.Google Scholar
  6. [6]
    R.T. Rockafellar, Extensions of subgradient calculus with applications to optimization, J. Nonlinear Anal., to appear in 1985.Google Scholar
  7. [7]
    R.W. Chaney, Math. Oper. Res. 9 (1984).Google Scholar
  8. [8]
    R.T. Rockafellar, Generalized directional derivatives and subgradients of non-convex functions, Canadian J. Math. 32 (1980), pp. 157 - 180.Google Scholar
  9. [9]
    R.T. Rockafellar, The Theory of Subgradients and its Applications to Problems of Optimization: Convex and Nonconvex Functions, Heldermann Verlag, West Berlin, 1981.Google Scholar
  10. [10]
    R.T. Rockafellar, Clarke’s tangent cones and the boundaries of closed sets in R a, J. Nonlin. Anal. 3 (1970), pp. 145 –154.CrossRefGoogle Scholar
  11. [11]
    F.H. Clarke, A new approach to Lagrange multipliers, Math. Oper. Res. 1 (1976), pp. 165 - 174.CrossRefGoogle Scholar
  12. [12]
    J-B Hiriart-Urruty, Refinements of necessary optimality conditions in nondifferentiable programming, I, Appl. Math. Opt. 5 (1979), pp. 63 - 82.CrossRefGoogle Scholar
  13. [13]
    R.T. Rockafellar, Lagrange multipliers and subderivatives of optimal value functions in nonlinear programming, Math. Prog. Study 17 (1982), 28 - 66.CrossRefGoogle Scholar
  14. [14]
    J. Gauvin, The generalized gradient of a marginal function in mathematical programming problem, Math. Oper. Res. 4 (1979), pp. 458 – 463.CrossRefGoogle Scholar
  15. [15]
    V.F. Demyanov and V.N Malozemov, On the theory of nonlinear minimax problems, Russ. Math. Surv.. 26 (1971), 57 - 115.CrossRefGoogle Scholar
  16. [16]
    R.T. Rockafellar, Directional differentiability of the optimal value in a nonlinear programming problem, Math. Prog. Studies 21 (1984), pp. 213 – 226.CrossRefGoogle Scholar
  17. [17]
    J.P. Aubin, Lipschitz behavior of solutions to convex minimization problems, Math. Oper. Res. 9 (1984), pp. 87 - 111.CrossRefGoogle Scholar
  18. [18]
    R.T. Rockafellar, Lipschitzian properties of multifunctions, J. Nonlin. Anal., to appear in 1985.Google Scholar
  19. [19]
    R.T. Rockafellar, Convex algebra and duality in dynamic models of production, in Mathematical Models of Economic (J. Los, ed.), North-Holland, 1973, pp.351–378.Google Scholar
  20. [20]
    R.T. Rockafellar, Monotone Processes of Convex and Concave Type, Memoir no. 77, Amer. Math. Soc., Providence RI, 1967.Google Scholar
  21. [21]
    R.T. Rockafellar, Maximal monotone relations and the second derivatives of nonsmooth functions, Ann. Inst. H. Poincaré, Analyse Non Linéaire 2 (1985), pp. 167 - 184.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1985

Authors and Affiliations

  • R. T. Rockafellar
    • 1
  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA

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