A Globally Convergent Method for (Constrained) Nonlinear Continuous L_l Approximation Problems

Abstract

Let f,g : X × Rⁿ → R be continuously differentiable functions of their parameters, and consider the problem

$$\begin{gathered} find a \in {R^n} to \min imise \left\| {f(x,a)} \right\| \hfill \\ subject to g(x,a) \leqslant 0, x \in X \hfill \\ \end{gathered} $$

where X ⊂ R^N is a Cartesian product of closed intervals, and the norm is the L_l norm defined on X . A globally convergent method is given for this problem, based on the use of an exact penalty function, and capable of converging at a second order rate if exact second derivative information is also available. For the particular case of the unconstrained problem, and when X is an interval of the real line, an algorithm is presented, and the effectiveness of this is illustrated by the numerical solution of a number of nonlinear problems.