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Consider the optimization over two vectors $x$ and $y$

$$\max_{x,y} \min\left(\frac{x^TAx}{y^TAy},\frac{y^TBy}{x^TBx}\right)$$ for two positive definite matrices $A$ and $B$. This problem can be simplified by the variable change $$c=B^{1/2}x, \quad\quad d=A^{1/2}y,$$ in which case I get the much simpler problem $$\max_{\|c\|^2,\|d\|^2} \min\left(\frac{\|c\|^2 \lambda_1}{\|d\|^2},\frac{\|d\|^2 \lambda_2}{\|c\|^2}\right),$$ where $\lambda_1$ and $\lambda_2$ are the largest eigenvalues of $$B^{-1/2}AB^{-1/2} \quad \mathrm{and} \quad A^{-1/2}BA^{-1/2}.$$

Now to my question. If I have three (or more) terms, how to proceed? That is (for three terms) $$\max_{x,y,z} \min\left(\frac{x^TAx}{y^TAy+z^TAz},\frac{y^TBy}{x^TBx+z^TBz},\frac{z^TCz}{x^TCx+y^TCy}\right).$$

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    $\begingroup$ I don't understand the notation. What do you mean by taking the maximum of a pair of functions of $x,y$ over $x,y$? How do you order the pairs? $\endgroup$
    – Ben McKay
    Commented May 25, 2016 at 9:22
  • $\begingroup$ I try to maximize the minimum. And I do that over the variables x and y. $\endgroup$
    – Tom Steiglitz
    Commented May 26, 2016 at 6:25

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