A Penalty Function Method Converging Directly to a Constrained Optimum

Abstract

Consider the following nonlinear optimization problem: find a local minimum $x_0 $ of the function f on the set $F = \{ {x \in R^n |\phi _i (x) \geqq 0,i = 1, \cdots ,M} \}$. A penalty function p is defined as follows: \[ p(x) = \mu f(x) - \sum _{i = 1}^M {\min \left( {0,\phi _i (x)} \right)} ,\qquad x \in R^n .\] Under some natural assumptions p has an unconstrained minimum that coincides with $x_0 $. This function is, however, not everywhere differentiable, even for differentiable f and $\phi _i $’s, so the standard unconstrained minimization methods are not directly applicable. The major result of this paper is a method that constructs a single sequence which minimizes p and converges directly to $x_0 $. The method is a continuation of earlier results, which were, however, only approximate in nature. The paper also provides a proof of convergence of the method for the convex problem along with some encouraging numerical results, for both convex and nonconvex problems.