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  • A Calculus of EPI-Derivatives Applicable to Optimization

  • R. A. Poliquin, R. T. Rockafellar
  • Journal:
    Canadian Journal of Mathematics / Volume 45 / Issue 4 / 01 August 1993
    Published online by Cambridge University Press:
    20 November 2018, pp. 879-896
    Print publication:
    01 August 1993
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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.