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A Calculus of EPI-Derivatives Applicable to Optimization

Published online by Cambridge University Press:  20 November 2018

R. A. Poliquin  and
R. T. Rockafellar
R. A. Poliquin
Affiliation:
Department of Mathematics, University of Alberta, Edmonton, Alberta, T6G 2G1
R. T. Rockafellar
Affiliation:
Department of Mathematics, University of Washington, Seattle, Washington 98195, U.S.A.
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Abstract

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When an optimization problem is represented by its essential objective function, which incorporates constraints through infinite penalties, first- and secondorder conditions for optimality can be stated in terms of the first- and second-order epi-derivatives of that function. Such derivatives also are the key to the formulation of subproblems determining the response of a problem's solution when the data values on which the problem depends are perturbed. It is vital for such reasons to have available a calculus of epi-derivatives. This paper builds on a central case already understood, where the essential objective function is the composite of a convex function and a smooth mapping with certain qualifications, in order to develop differentiation rules covering operations such as addition of functions and a more general form of composition. Classes of "amenable" functions are introduced to mark out territory in which this sharper form of nonsmooth analysis can be carried out.

Keywords

90C4849A5258C2090C30epi-derivativesgeneralized second derivativesamenable sets and functionsnonsmooth analysiscomposite optimizationparametric optimizationsensitivity analysis
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 45 , Issue 4 , 01 August 1993 , pp. 879 - 896
Copyright
Copyright © Canadian Mathematical Society 1993

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