# Bounded Condition Number

I am wondering if there are ways to bound the condition number of a matrix that is being optimized?
For instance, for a simple 2 by 2 matrix A that I am trying to maximize the product of eigenvalues subject to a maximum ratio of the two eigenvalues given by tau. In practice there are other constraint LMIs in this particular problem, but I think this captures the basic issue, namely that the condition # is quasi-convex.

``````  minimize(det_inv(A))
subject to
A==semidefinite(2);
lambda_max(A)>=tau*lambda_min(A);
``````

yields an error: Invalid constraint: {convex} >= {concave}
even when tau =1 and the second constraint is always
satisfied by the solution even if that equation is commented out (e.g. the largest eigenvalue is >= the second eigenvalue).

Is there a smart way to recast this problem such that the constraint equation that limits the value of the condition number is convex?

“Linear Matrix Inequalities in System and Control Theory,” by Boyd, ElGhaoui, Feron, Balakrishnan

Section 3.1 Minimizing Condition Number by Scaling
Section 3.2 Minimizing Condition Number of a Positive-Definite Matrix

Also, there is some material in “Condition number of a positive definite matrix” in Section 6.2.2 "Eigenvalue optimization: of the “MOSEK Modeling Cookbook” Release 3.1 https://docs.mosek.com/modeling-cookbook/sdo.html

Also search for “condition number” (several scattered pages) in “Convex Optimization”, Boyd and Vandenberghe http://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf

All of this is easy to code in CVX.

Thanks Mark! Will read through those.

For anyone trying to do this, the constraint equation