It is well know (see Boyd & Vandenberghe's Convex Optimization, Sec. 8.4.2) that the maximum volume ellipsoid inscribed in a given convex polytope in $\cal{H}$-form can be computed by solving the Semi-Definite-Program

$\mbox{max} \qquad \log \det B$

$\mbox{s.t.} \qquad \vert\vert B a_i\vert\vert_2 + a_i^{\rm T} d \leq b_i, \quad B \succcurlyeq 0$

I am looking for a way to add to this problem contraints on the ratios of the ellipsoid semi-axes. In other words, if $\lambda_i, i=1,\ldots,n$ are the eigenvalues of $B B^{\rm T}$,

$r_i = \frac{\lambda_i}{\lambda_1}, \quad i=2,\ldots,n$

It seems that if we add this new constrains the original problem is not even convex anymore. Any ideas?


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