TY - JOUR

T1 - TWO-METRIC PROJECTION METHODS FOR CONSTRAINED OPTIMIZATION.

AU - Gafni, Eli M.

AU - Bertsekas, Dimitri P.

N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 1984

Y1 - 1984

N2 - 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.

AB - 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.

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U2 - 10.1137/0322061

DO - 10.1137/0322061

M3 - Article

AN - SCOPUS:0021522309

VL - 22

SP - 936

EP - 964

JO - SIAM Journal on Control and Optimization

JF - SIAM Journal on Control and Optimization

SN - 0363-0129

IS - 6

ER -