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


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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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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