ON THE GOLDSTEIN-LEVITIN-POLYAK GRADIENT PROJECTION METHOD.

This paper considers some aspects of a gradient projection method proposed by A. Goldstein, E. Levitin and B. Polyak and more recently, in a less general context, by G. McCormick. Some convergent stepsize rules to be used in conjunction with the method are proposed and analyzed. These rules are similar in spirit with the efficient L. Armijo rule for the method of steepest descent and under mild assumptions they have the desirable property that they identify the set of active inequality constraints in a finite number of iterations. As a result the method may be converted towards the end of the process to a conjugate direction, Quasi-Newton or Newton's method and achieve the attendant superlinear convergence rate. A quadratically convergent combination of the method with Newton's method is proposed as an example. Such combined methods appear to be very efficient for large scale problems with many simple constraints such as those often appearing in optimal control.