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Discontinuous piecewise linear optimization

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Abstract

A theoretical framework and a practical algorithm are presented to solve discontinuous piecewise linear optimization problems dealing with functions for which theridges are known. A penalty approach allows one to consider such problems subject to a wide range of constraints involving piecewise linear functions. Although the theory is expounded in detail in the special case of discontinuous piecewiselinear functions, it is straightforwardly extendable, using standard nonlinear programming techniques, tononlinear (discontinuous piecewise differentiable) functions.

The descent algorithm which is elaborated uses active-set and projected gradient approaches. It is a generalization of the ideas used by Conn to deal with nonsmoothness in thel 1 exact penalty function, and it is based on the notion ofdecomposition of a function into a smooth and a nonsmooth part. The constrained case is reduced to the unconstrained minimization of a (piecewise linear)l 1 exact penalty function. We also discuss how the algorithm is modified when it encounters degenerate points. Preliminary numerical results are presented: the algorithm is applied to discontinuous optimization problems from models in industrial engineering. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

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Author notes

  1. Marcel Mongeau

    Present address: Laboratoire Approximation & Optimisation, Université Paul Sabatier, 31062, Toulouse Cedex 04, France

Authors and Affiliations

  1. T.J. Watson Research Center, P.O. Box 218, 10598, Yorktown Heights, NY, USA

    Andrew R. Conn

  2. Centre de Recherches Mathématiques, Université de Montréal, H3C 3J7, Canada

    Marcel Mongeau

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  1. Andrew R. Conn
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Supported by the Natural Sciences and Engineering Council of Canada and the Centre de Recherches Mathématiques, Université de Montréal.

This research was supported in part by the Advanced Research Projects Agency of the Department of Defense and was monitored by the Air Force Office of Scientific Research under Contract No. F49620-91-C-0079. The United States Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation hereon.

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Conn, A.R., Mongeau, M. Discontinuous piecewise linear optimization. Mathematical Programming 80, 315–380 (1998). https://doi.org/10.1007/BF01581171

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