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 ~ t e 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 m a n y simple constraints such as those often appearing in optimal control.

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We consider distributed algorithms for solving dynamic programming problems whereby several processors participate simultaneously in the computation while maintaining coordination by information exchange via communication links. A model of asynchronous distributed computation is developed which requires very weak assumptions on the ordering of computations, the timing of information exchange, the amount of local information needed at each computation node, and the initial conditions for the algorithm. The class of problems considered is very broad and includes shortest path problems, and finite and infinite horizon stochastic optimal control problems. When specialized to a shortest path problem the algorithm reduces to the algorithm originally implemented for routing of messages in the ARPANET.

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