Nonsmooth algorithms for minimizing the largest eigenvalue with applications to inner numerical radius

Abstract

Nonsmoothness at optimal points is a common phenomenon in many eigenvalue optimization problems. We consider two recent algorithms to minimize the largest eigenvalue of a Hermitian matrix dependent on one parameter, both proven to be globally convergent unaffected by nonsmoothness. One of these algorithms models the eigenvalue function with a piece-wise quadratic function and is effective in dealing with nonconvex problems. The other algorithm projects the Hermitian matrix into subspaces formed of eigenvectors and is effective in dealing with large-scale problems. We generalize the latter slightly to cope with nonsmoothness. For both algorithms we analyze the rate of convergence in the nonsmooth setting, when the largest eigenvalue is multiple at the minimizer and zero is strictly in the interior of the generalized Clarke derivative, and prove that both algorithms converge rapidly. The algorithms are applied to, and the deduced results are illustrated on the computation of the inner numerical radius, the modulus of the point on the boundary of the field of values closest to the origin, which carries significance for instance for the numerical solution of a symmetric definite generalized eigenvalue problem and the iterative solution of a saddle point linear system.

Implicit Determinant Method for Solving an Hermitian Eigenvalue Optimization Problem

“…This example is used to demonstrate that the sequence generated by Algorithm 2.1 converges to a local extreme point locally quadratically. We consider the tridiagonal matrix B as in [15,17], where We set c be the eigenvector of A(ω 0 ) corresponding to λ 0 in Eq. ( 2 4.2 shows the computed results of Example 4.2.…”

“…For the same example, we compare the iterations and CPU time of our algorithms with the subspace method proposed by Kangal et al [14] and the support based algorithm proposed by Kangal and Mengi [15] in Table 4.5. The subspace method and the support based algorithm are methods of global convergence.…”

“…The subspace method and the support based algorithm are methods of global convergence. The codes of these two methods are available in [14] and [15] respectivly. We set the initial point ω 0 = 2 in Algorithm 3.2.…”

“…Eigenvalue optimization problem (1.1) has many applications. For examples, the computation of the stable radius [24] of a stable matrix, the computation of the H ∞ norm [13] of a linear system, the computation of the Crawford number [15] and quadratic constrained quadratic programming [25] for a frequently encountered case [8], can be converted into eigenvalue optimization problem (1.1).…”