An algorithm on minimizing generalized eigenvalues with linear matrix inequality constraints

There are a large number of problems in engineering, especially in systems and control, that can be formulated as convex or quasiconvex optimization problems which involve linear matrix inequalities. Except in a few cases, closed-form or analytic solutions do not seem to exist, and therefore the problems can only be solved by iterative methods. In this paper, we propose an interior point method on minimizing the largest eigenvalue of a symmetric definite pencil subject to linear matrix inequality constraints. We also provide a convergence analysis for the proposed method.