A QUADRATICALLY CONVERGENT LOCAL ALGORITHM ON MINIMIZING THE LARGEST EIGENVALUE OF A SYMMETRICAL MATRIX

Optimization involving eigenvalues arise in many engineering problems. We propose a new algorithm on minimizing the largest eigenvalue over an affine family of symmetric matrices. Under certain assumptions it is shown that, if started close enough to the minimizer x*, the proposed algorithm converges to x* quadratically. The proposed algorithm can be readily extended to minimizing the largest eigenvalue over an affine family of Hermitian matrices. Also, it has been extended to minimizing sums of the largest eigenvalues of a symmetric or Hermitian matrix.