ON THE USE OF CONSISTENT APPROXIMATIONS IN THE SOLUTION OF SEMIINFINITE OPTIMIZATION AND OPTIMAL-CONTROL PROBLEMS

We consider a pair consisting of an optimization problem and its optim ality function (P, theta), and define consistency of approximating pro blem-optimality function pairs, (P(N), theta(N)) to (P, theta), in ter ms of the epigraphical convergence of the P(N) to P, and the hypograph ical convergence of the optimality functions theta(N) to theta. We the n show that standard discretization techniques decompose semi-infinite optimization and optimal control problems into families of finite dim ensional problems, which, together with associated optimality function s, are consistent discretizations to the original problems. We then pr esent two types of techniques for using consistent approximations in o btaining an approximate solution of the original problems. The first i s a ''filter'' type technique, similar to that used in conjunction wit h penalty functions, the second one is an adaptive discretization tech nique that, can be viewed as an implementation of a conceptual algorit hm for solving the original problems.