TWO-METRIC PROJECTION METHODS FOR CONSTRAINED OPTIMIZATION.

This paper is concerned with the problem min left brace f(x) vertical x an element of X right brace , where X is a convex subset of a linear space H, and f is a smooth real-valued function on H. The class of methods x//k// plus //1 equals P(x//k minus alpha //kg//k), is poposed, where P denotes projection on X with respect to a Hilbert space norm, g//k denotes the Frechet derivative of f at x//k with respect to another Hilbert norm, on H, and alpha //k is a positive scalar stepsize. It is then possible to match the first norm with the structure of X so that the projection operation is simplified while at the same time reserving the option to choose the second norm on the basis of approximations to the Hessian of f so as to attain a typically superlinear rate of convergence. The resulting methods are particularly attractive for large-scale problems with specially structured constraint sets such as optimal control and nonlinear multi-commodity network flow problems. The latter class of problems is discussed in some detail.