BARRIER FUNCTIONS AND THEIR MODIFICATIONS

INTRODUCTION

In the mid-1950s and the early 1960s, Frisch (1955) and Carroll (1961) proposed the use of Barrier Functions (BFs) for constrained optimization. Since then, the BFs have been extensively studied, with particularly major work in the area due to Fiacco and McCormick (1968) who developed the Sequential Unconstrained Minimization Technique (SUMT). Currently, methods based on barrier functions make up a considerable part of modern optimization theory.

BARRIER FUNCTIONS

Consider the constrained optimization problem

(1)

where Ω = {x: g _i(x) ≥ 0, i = 1, ..., m}, f : ℜⁿ → ℜ is convex, all g _i : ℜⁿ → ℜ are concave, m > n, and X ^∗ is the set of values minimizing f (x)on Ω. Frisch's logarithmic barrier function F: int Ω × ℜ₊₊ → ℜ is defined by formula

(2)

and Carroll's hyperbolic barrier function C: int Ω × ℜ₊₊ → ℜ is defined as

We assume that X ^∗ is bounded and ln t = −∞ for t ≤ 0; then for any μ > 0, there exists a minimum of F(x, μ)in ℜⁿ, which we write as

(3)...