A finite steps algorithm for solving convex feasibility problems

Authors:
M. Ait Rami;U. Helmke;J. B. Moore
Affiliations:
Department of Mathematics, University of Würzburg, Würzburg, Germany 97074;Department of Mathematics, University of Würzburg, Würzburg, Germany 97074;Department of Information Engineering, Research School of Information Sciences and Engineering, Australian National University, Canberra, Australia 0200
Venue:
Journal of Global Optimization
Year:
2007

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Abstract

This paper develops a new variant of the classical alternating projection method for solving convex feasibility problems where the constraints are given by the intersection of two convex cones in a Hilbert space. An extension to the feasibility problem for the intersection of two convex sets is presented as well. It is shown that one can solve such problems in a finite number of steps and an explicit upper bound for the required number of steps is obtained. As an application, we propose a new finite steps algorithm for linear programming with linear matrix inequality constraints. This solution is computed by solving a sequence of a matrix eigenvalue decompositions. Moreover, the proposed procedure takes advantage of the structure of the problem. In particular, it is well adapted for problems with several small size constraints.