E. Polak, ON THE USE OF CONSISTENT APPROXIMATIONS IN THE SOLUTION OF SEMIINFINITE OPTIMIZATION AND OPTIMAL-CONTROL PROBLEMS, Mathematical programming, 62(2), 1993, pp. 385-414
Citations number
27
Categorie Soggetti
Operatione Research & Management Science",Mathematics,"Operatione Research & Management Science",Mathematics,"Computer Applications & Cybernetics
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.