A Penalty Function Method Converging Directly to a Constrained Optimum

Consider the following nonlinear optimization problem: find a local minimum x₀ of the function f on the set F = \ x ∈ Rⁿ |φ ᵢ (x) 0,i = 1, ⋯ ,M \. A penalty function p is defined as follows: \[ p(x) = μ f(x) - ∑ i = 1^M {min ( {0,φ _i (x)} )} , x ∈ R^n .\] Under some natural assumptions p has an unconstrained minimum that coincides with x₀. This function is, however, not everywhere differentiable, even for differentiable f and φ 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 x₀. 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.