Constrained Optimization Using a Nondifferentiable Penalty Function

Abstract

Consider the problem of finding the local constrained minimum $x_0 $ of the function f on the set \[ F = \left\{ {x \in R^n |\phi _i (x) \geqq 0,\quad \psi _{j + k} (x) = 0;\quad i = 1,2, \cdots ,k;\quad j = 1,2, \cdots ,l} \right\}.\] One method of solution is to minimize the associated penalty function \[ p_0 (x) = \mu f(x) - \sum _{i = 1}^k {\min } \left( {0,\phi _i (x)} \right) + \sum _{j = 1}^l {\left| {\psi _{j + k} (x)} \right|} \quad {\text{for }} x \in R^n ,\quad \mu \geqq 0. \] Let $x(\mu )$ be the minimum of this penalty function. It is known that, provided $\mu $ is sufficiently small, $x(\mu ) = x_0 $. However, until recently, a serious drawback to this particular penalty function was that its first order derivatives are not everywhere defined. Thus, well-known gradient type methods usually applied to unconstrained optimization problems were necessarily excluded. This paper presents a method that enables a modified form of the gradient type approaches to be applied to a perturbation of the penalty function $p_0 $ above. It is also shown how one can thereby solve the original constrained problem. The main theorem consists of necessary and sufficient conditions for x to be a minimum of the perturbed problem. Along with the theoretical results some numerical experience is included.