## On Eigenvalue Optimization of a Symmetric Matrix

**Abstract:**

Optimization involving eigenvalue arises in many engineering problems. In this talk, we describe an algorithm on minimizing the largest eigenvalue over an affine family of symmetric matrices. It is shown that, if started close enough, the algorithm converges to the solution quadratically. To this end, based on some results on analyticity of functions involving eigenvalues and the knowledge of eigenvalue multiplicity at the solution, we first show that the underlying problem has a smooth reformulation near the solution.

Special consideration is then given to the new formulation and it leads to the algorithm. Finally, we describe a scheme on the eigenvalue multiplicity estimate near the solution. It is shown that there exists an open neighborhood around the solution so that the scheme applied to any point in that neighborhood will always give the correct eigenvalue multiplicity at the solution.