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 and∂ 2 f/∂x 2 are continuous on ℝn × T. We will give a geometrical meaning of the formula. We will moreover give a sufficient condition forS(x) to be directionally twice differentiable.
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Kawasaki, H. The upper and lower second order directional derivatives of a sup-type function. Mathematical Programming 41, 327–339 (1988). https://doi.org/10.1007/BF01580771
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DOI: https://doi.org/10.1007/BF01580771