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Mathematical Programming

, Volume 41, Issue 1–3, pp 327–339 | Cite as

The upper and lower second order directional derivatives of a sup-type function

  • Hidefumi Kawasaki
Article

Abstract

The purpose of this paper is to give a formula for expressing the second order directional derivatives of the sup-type functionS(x) = sup{f(x, t); t ∈ T} in terms of the first and second derivatives off(x, t), whereT is a compact set in a metric space and we assume thatf, ∂f/∂x and2f/∂x2 are continuous on ℝ n × T. We will give a geometrical meaning of the formula. We will moreover give a sufficient condition forS(x) to be directionally twice differentiable.

Key words

Envelope sup-type function second order directional derivative nondifferentiable function 

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References

  1. [1]
    A. Auslender, “On the differential properties of the support function of theε-subdifferential of a convex function,”Mathematical Programming 24 (1982) 257–268.Google Scholar
  2. [2]
    A. Ben-Tal and J. Zowe, “A unified theory of first and second order conditions for extremum problems in topological vector spaces,”Mathematical Programming Study 19 (1982) 39–76.Google Scholar
  3. [3]
    A. Ben-Tal and J. Zowe, “Directional derivatives in nonsmooth optimization,”Journal of Optimization Theory and Applications 47 (1985) 483–490.Google Scholar
  4. [4]
    V.F. Dem'yanov and A.B. Pevnyi, “Expansion with respect to a parameter of the extremal values of game problems,”U.S.S.R. Computational Mathematics and Mathematical Physics 14 (1975) 33–45.Google Scholar
  5. [5]
    J.B. Hiriart-Urruty, “Approximating a second-order directional derivative for nonsmooth convex functions,”SIAM Journal on Control and Optimization 20 (1982) 783–807.Google Scholar
  6. [6]
    J.B. Hiriart-Urruty, “Limiting behavior of the approximate first order and second order directional derivatives for a convex function,”Nonlinear Analysis: Theory, Methods, and Applications 6 (1982) 1309–1326.Google Scholar
  7. [7]
    J.B. Hiriart-Urruty, “Approximate first-order and second-order directional derivatives of a marginal function in convex optimization,”Journal of Optimization Theory and Applications 48 (1986) 127–140.Google Scholar
  8. [8]
    H. Kawasaki, “An envelope-like effect of infinitely many inequality constraints on second-order necessary conditions for minimization problems,”Mathematical Programming. Google Scholar
  9. [9]
    H. Kawasaki, “Second order necessary optimality conditions for minimizing a sup-type function,” submitted toMathematical Programming. Google Scholar
  10. [10]
    A. Shapiro, “Second-order derivatives of extremal-value functions and optimality conditions for semi-infinite programs,”Mathematics of Operations Research 10 (1985) 207–219.Google Scholar

Copyright information

© The Mathematical Programming Society, Inc. 1988

Authors and Affiliations

  • Hidefumi Kawasaki
    • 1
  1. 1.Department of MathematicsKyushu University 33FukuokaJapan

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