### Abstract

The purpose of this paper is to give a formula for expressing the second order directional derivatives of the sup-type function S(x) = sup{f(x, t); t ∈ T} in terms of the first and second derivatives of f(x, t), where T is a compact set in a metric space and we assume that f, ∂f/∂x and ∂^{2}f/∂x^{2} are continuous on ℝ^{n}× T. We will give a geometrical meaning of the formula. We will moreover give a sufficient condition for S(x) to be directionally twice differentiable.

Original language | English |
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Pages (from-to) | 327-339 |

Number of pages | 13 |

Journal | Mathematical Programming |

Volume | 41 |

Issue number | 1-3 |

DOIs | |

Publication status | Published - May 1 1988 |

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### All Science Journal Classification (ASJC) codes

- Software
- Mathematics(all)

### Cite this

**The upper and lower second order directional derivatives of a sup-type function.** / Kawasaki, Hidefumi.

Research output: Contribution to journal › Article

*Mathematical Programming*, vol. 41, no. 1-3, pp. 327-339. https://doi.org/10.1007/BF01580771

}

TY - JOUR

T1 - The upper and lower second order directional derivatives of a sup-type function

AU - Kawasaki, Hidefumi

PY - 1988/5/1

Y1 - 1988/5/1

N2 - The purpose of this paper is to give a formula for expressing the second order directional derivatives of the sup-type function S(x) = sup{f(x, t); t ∈ T} in terms of the first and second derivatives of f(x, t), where T is a compact set in a metric space and we assume that f, ∂f/∂x and ∂2f/∂x2 are continuous on ℝn× T. We will give a geometrical meaning of the formula. We will moreover give a sufficient condition for S(x) to be directionally twice differentiable.

AB - The purpose of this paper is to give a formula for expressing the second order directional derivatives of the sup-type function S(x) = sup{f(x, t); t ∈ T} in terms of the first and second derivatives of f(x, t), where T is a compact set in a metric space and we assume that f, ∂f/∂x and ∂2f/∂x2 are continuous on ℝn× T. We will give a geometrical meaning of the formula. We will moreover give a sufficient condition for S(x) to be directionally twice differentiable.

UR - http://www.scopus.com/inward/record.url?scp=34250086751&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34250086751&partnerID=8YFLogxK

U2 - 10.1007/BF01580771

DO - 10.1007/BF01580771

M3 - Article

AN - SCOPUS:34250086751

VL - 41

SP - 327

EP - 339

JO - Mathematical Programming

JF - Mathematical Programming

SN - 0025-5610

IS - 1-3

ER -