TWO-METRIC PROJECTION METHODS FOR CONSTRAINED OPTIMIZATION*

This paper is concerned with the problem min $\{ f(x)\mid x \in X\} $ where X is a convex subset of a linear space H, and f is a smooth real-valued function on H. We propose the class of methods $x_{k + 1} = P(x_k - \alpha _k g_k )$, where P denotes projection on X with respect to a Hilbert space norm $\| \cdot \|$ , $g_k $ denotes the Frechet derivative of f at $x_k $ with respect to another Hilbert space norm $\| \cdot \|_k $ on H, and $\alpha _k $ is a positive scalar stepsize. We thus remove an important restriction in the original proposal of Goldstein [1] and Levitin and Poljak [2], where the norms $\| \cdot \|$ and $\| \cdot \|_k $ must be the same. It is therefore possible to match the norm $\| \cdot \|$ with the structure of X so that the projection operation is simplified while at the same time reserving the option to choose $\| \cdot \|_k $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 attrac...

[1]  M. Projected Newton Methods and Optimization of Multicommodity Flows , 2022 .

[2]  A. Goldstein Convex programming in Hilbert space , 1964 .

[3]  J. Moreau Convexity and duality , 1966 .

[4]  D. Bertsekas On the Goldstein-Levitin-Polyak gradient projection method , 1974, CDC 1974.

[5]  Robert G. Gallager,et al.  A Minimum Delay Routing Algorithm Using Distributed Computation , 1977, IEEE Trans. Commun..

[6]  Stella Dafermos,et al.  Traffic Equilibrium and Variational Inequalities , 1980 .

[7]  Robert G. Gallager,et al.  Flow Control and Routing Algorithms for Data Networks , 1980 .

[8]  D. Bertsekas,et al.  Optimal short-term scheduling of large-scale power systems , 1981, 1981 20th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes.

[9]  D. Bertsekas Projected Newton methods for optimization problems with simple constraints , 1981, 1981 20th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes.

[10]  T. Magnanti,et al.  Equilibria on a Congested Transportation Network , 1981 .

[11]  Philip E. Gill,et al.  Practical optimization , 1981 .

[12]  Eliezer M. Gafni,et al.  The integration of routing and flow-control for voice and data in a computer communication network , 1982 .

[13]  Dimitri P. Bertsekas,et al.  Constrained Optimization and Lagrange Multiplier Methods , 1982 .

[14]  D. Bertsekas,et al.  Projection methods for variational inequalities with application to the traffic assignment problem , 1982 .

[15]  Dimitri P. Bertsekas,et al.  Second Derivative Algorithms for Minimum Delay Distributed Routing in Networks , 1984, IEEE Trans. Commun..