An Interior-Point Method for Minimizing the Maximum Eigenvalue of a Linear Combination of Matrices

Abstract

We consider the problem of finding $${\lambda ^{opt}}: = \mathop {\inf }\limits_{x \in {R^m}} \{ {\lambda _{\max }}({A^{(0)}} + \sum\limits_{i = 1}^m {{x_i}{A^{(i)}}} )\} $$ (1) and an optimal solution xopt if it exists. Here, the A (i) are given symmetric matrices in IR n×n for 0 ≤ i ≤ m, and αmax(A) is the largest eigenvalue of a matrix A. Problem (1) can be rewritten as a convex difFerentiable problem with a positive definite constraint. Positive definiteness can easily be verified numerically (via Cholesky decomposition). Moreover, the cone of positive definite matrices A is characterized by the domain of the smooth convex barrier function $$ - \log \;{\det _ + }(A) $$ .