An Exact Potential Method for Constrained Maxima

Abstract

The main result of the paper consists of the theorem that under certain, natural assumptions the local conditional maximum $x_0 $ of the function f on the set \[ A = \left\{ {x \notin R^n |\phi _i (x) \geqq 0,\psi _j (x) = 0,i = 1, \cdots ,k,j = 1, \cdots ,l} \right\} \] is identical with the unconditional maximum of the potential function \[ p(x,\mu ) = \mu f(x) + \sum _{i = 1}^k {{\operatorname{neg}}(\phi _i (x))} - \sum _{j = 1}^l {| {\psi (x)} |} ,\qquad x \in R^n ,\quad \mu \geqq 0, \] for $\mu $ sufficiently small. There is also provided a draft of a modified gradient procedure for maximizing the potential $p(x,\mu )$ since it is generally nonsmooth even for differentiable f, $\phi _i $ and $\psi _j $.