On the Goldstein-Levitin-Polyak gradient projection method

Abstract-This paper considers some aspects of a gradient projection method proposed by Goldstein [l], Levitin and Polyak [3], and more recently, in a less general context, by McCormick [lo]. We propose and analyze some convergent step-size rules to be used in conjunction with the method. These rules are similar in spirit to the efficient 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 consbaints in a f ~te number of iterations. As a resnlt 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. As an example we propose some quadratically convergent combinations of the method with Newton’s method. Such combined methods appear to be very efficient for large-de problems with many simple constraints such as those often appearing in optimal control. method of steepest descent for unconstrained minimization (Q = H). It should be noted, of course, that in order for the method (1) to be effective, the set Q must be such that the projection operation Pe (.) can be easily carried out. Levitin and Polyak point out several such cases, the most representative of which is when H = R “ and Q implies upper and (or) lower bounds on all the variables of the problem: I