A finite steps algorithm for solving convex feasibility problems

This paper develops an approach 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. Our approach is closely related to the classical alternating projection method for solving convex feasibility problems. As an application, we propose an algorithm for linear programming with linear matrix inequality (LMI) constraints. We show how to compute the solution of LMI optimization using a finite steps algorithm with an easy implementation. This solution can be 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.