Mathematical Programming for Constrained Minimal Problems. Part 3. Combined Gradient-Restoration Algorithm

摘要

The problem of minimizing a function f(x) subject to a constraint phi(x) = O is considered. Here, f is a scalar, x an n-vector, and phi a q-vector. A combined gradient-restoration algorithm is presented: this is an iterative algorithm characterized by a displacement delta x leading toward the minimum point while simultaneously leading toward constraint satisfaction. This displacement, generated by minimizing the first-order change of the function subject to the constraint employed in linearized form and a quadratic constraint on delta x, has the direction of the gradient of the augmented function F(x, lambda)= f(x) +(lambda sup T)phi(x). The descent properties of the algorithm are studied, and schemes to determine the optimum stepsize are discussed. Three numerical examples are presented. (Author)