Algorithms for the solution of optimization problems with two numerical precision parameters

We generalize the theory of consistent approximations and algorithm implementations presented in [E Polak, 1993] so as to enable us to solve infinite-dimensional optimization problems whose discretization involves two precision parameters. A typical example of such a problem is an optimal control problem with initial and final value constraints. Our main result is a two-discretization parameter algorithm model, with an associated convergence theorem. We illustrate its applicability by implementing it using an approximate steepest descent method and applying it to a simple two point boundary value optimal control problem. Our numerical results (not only the ones in this paper) show that algorithms based on our new model perform quite well and are fairly insensitive to the selection of user-set parameters. Also, they appear to be superior to some alternative, ad hoc schemes.