On the Convergence of Some Feasible Direction Algorithms for Nonlinear Programming

Abstract

Abstract : Recently Goldstein has considered the problem of determining conditions on the selection of directions in feasible direction algorithms for unconstrained maximization problems which assure the gradient vanishes in the limit. This paper extends his work (when specialized to Euclidian spaces) in various ways, notably by considering constrained maximization problems and by allowing a wider choice in the selection of the step size. For example we allow the ''optimal'' step size among others. We give conditions on the directions which assure convergence to a stationary point in feasible direction algorithms for maximizing a real valued continuous function on a closed set. We also apply this result to establish convergence to a stationary point of several known methods (e.g., Cauchy's method of steepest ascents, the Newton-Raphson method, and ufrank and Wolfe's method), a simple variant of one of Zoutendijk's methods that eliminates the need for his ''anti-zigzagging'' procedures, and some new second order methods that require quadratic programs to be solved at each stage. (Author)