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The BT-Algorithm for Minimizing a Nonsmooth Functional Subject to Linear Constraints

Part of the Ettore Majorana International Science Series book series (EMISS, volume 43)

Abstract

We study the minimization of a function f : ℝ n → ℝ subject to linear constraints
$$\min \,f(x)\,subject\,to\,Ax \leqslant a,$$
(1.1)
where, in contrast to the standard situation, we do not require f to have continuous derivatives (so-called nonsmooth f). More precisely, we are content if the gradient of f exists almost everywhere and if, at each x where the gradient is not defined, the subdifferenttal
$$\partial f(x)\,:\, = \,conv\,\{ g \in \,\mathbb{R}\,:\,g\,\lim \,\nabla f({x_i}),\,{x_i}\, \to \,x,\,\nabla f({x_i})\,exists,\,\nabla f({x_i})\,converges\} $$
(1.2)
is a nonempty set. This is true e.g. for locally Lipschitz f and thus in particular for convex f. To simplify the presentation we restrict our development to the case of a convex f since it is in this framework that things are most easy to explain; further we skip the linear constraints in (1.1). The general case (1.1) with weakly semi-smooth f (see [16]) is presently under consideration and seems to require only technical changes.

Keywords

Line Search Travelling Salesman Problem Trust Region Cluster Point Quadratic Programming Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1989

Authors and Affiliations

  • J. Zowe
    • 1
  1. 1.Math. Univ. of BayreuthBayreuthFederal Republic of Germany

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