Speaker
Description
In this talk, I will address eigenvalue-optimization-based matrix nearness problems such as the stability radius of a matrix or a time invariant system, $\mathcal{H}^\infty$ norm computation, the structured distance to singularity.
These are formulated here as 2-variable optimization problems of functionals depending either on a single or on several target eigenvalues of the matrix.
It is well-known that the classical Newton-bisection method (proposed, e.g., in [1]) may fail to converge.
For this, in 2016, Mitchell and Overton [3] proposed a very effective convergent alternating iteration method, the Hybrid Expansion Contraction (HEC), which was then extended to passivity optimization in [4].
The method we propose here uses a two-level nested iteration, where, at the inner level, a constrained gradient system is integrated to optimize the functional over the set of matrices of specified norm and, at the outer level, to compute the optimal norm.
However, an important modification of the method exploits the free gradient system, allowing for guaranteed convergence (see [2]). A discussion and a comparison wrt the HEC methodology will conclude the talk.
References.
[1] N. Guglielmi, M. Gürbüzbalaban, and M.L. Overton.
Fast approximation of the $\mathcal{H}^{\infty}$ norm via optimization over spectral value sets.
SIAM J. Matrix Anal. Appl., 34(2): 709--737, 2013.
[2] N. Guglielmi and C. Lubich.
Matrix nearness problems and eigenvalue optimization.
https://arxiv.org/abs/2503.14750, 2025.
[3] T. Mitchell and M.~L. Overton.
Hybrid expansion–contraction: a robust scalable method for approximating the $\mathcal{H}^\infty$ norm.
IMA J. Numer. Anal., 36(3): 985--1014, 2016.
[4] T. Mitchell and P. Van Dooren.
Root-max problems, hybrid expansion-contraction, and quadratically convergent optimization of passive systems.
SIAM J. Matrix Anal. Appl., 44(2): 753--780, 2023.