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

Abstract

We study the minimization of a function f : ℝⁿ → 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.