On Substochastic Inverse Eigenvalue Problems with the Corresponding Eigenvector Constraints

TL;DR

This paper addresses the inverse eigenvalue problem for substochastic matrices with eigenvector constraints, formulating it as a nonconvex optimization problem and proposing an alternating minimization algorithm with proven convergence.

Contribution

It introduces a novel approach to solve the substochastic inverse eigenvalue problem with eigenvector constraints via nonconvex optimization and develops an efficient alternating minimization algorithm.

Findings

The proposed algorithm converges reliably.

Numerical experiments demonstrate the method's efficiency.

The approach effectively constructs substochastic matrices with specified spectra.

Abstract

We consider the inverse eigenvalue problem of constructing a substochastic matrix from the given spectrum parameters with the corresponding eigenvector constraints. This substochastic inverse eigenvalue problem (SstIEP) with the specific eigenvector constraints is formulated into a nonconvex optimization problem (NcOP). The solvability for SstIEP with the specific eigenvector constraints is equivalent to identify the attainability of a zero optimal value for the formulated NcOP. When the optimal objective value is zero, the corresponding optimal solution to the formulated NcOP is just the substochastic matrix desired to be constructed. We develop the alternating minimization algorithm to solve the formulated NcOP, and its convergence is established by developing a novel method to obtain the boundedness of the optimal solution. Some numerical experiments are conducted to demonstrate the efficiency of the proposed method.