An Ideal Penalty Function for Constrained Optimization

This chapter presents an ideal penalty function for constrained optimization. The penalty function is used in the usual way, that is, for any given value of the parameters , S , a vector , ( S ) is obtained that minimizes ϕ( , , S ) without constraints. There is an outer iteration in which and S are changed so as to cause the solutions , ( , S ) → *. A well-known penalty function is one with = 0, in which case this convergence is ensured by letting σ i → ∞, i = 1, 2, …, m . The chapter presents some optimality results for Lagrange multipliers, which show that the optimum choice of the (or ) parameters for the Powell/Hestenes/Rockafellar penalty function is determined by a maximization problem in terms of these parameters.