README.md

GeneralizedEigenvalueMinimization.jl

An (experimental) Julia package for solving the following optimization problem

$$minimize λ over x∈Rⁿ, λ∈R subject to λB(x)-A(x)≻0 B(x)≻0 C(x)≻0 where A(), B() and C() are affine functions of x, and the interpretation of the inequalities is that the matrices on the left are positive definite.$$

The package implements (or plans to implement) a few solution methods:

1. Bracketing over λ: for a fixed λ the problem reduces to eigenvalue minimization problem, for which efficient algorithms exist.

2. The method of centers for minimizing generalized eigenvalues described in Boyd, Stephen, and Laurent El Ghaoui. “Method of Centers for Minimizing Generalized Eigenvalues.” Linear Algebra and Its Applications 188–189 (July 1, 1993): 63–111. https://doi.org/10.1016/0024-3795(93)90465-Z. Also available online at https://web.stanford.edu/~boyd/papers/gevc.html. The method is also implemented in Robust Control Toolbox for Matlab as gevp function (internally relying on LMI Toolbox).