I have a problem of the form $$\sup_{x\in\Bbb{C}^n}\left\{\frac{\|Ax\|_\infty}{\|Bx\|_\infty}\right\}$$ where $A$, $B$ are matrices with different number of rows and $x$ is an $n$ dimensional vector. Is there a way to find a tight bound to the expression or to convert this into a linear programming problem?
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No. Since $x$ is allowed to be any complex vector, there is no natural simplex to use for casting this as a linear programming problem. You can observe (assuming the infinity norms are the compatible norms they usually denote) that $\frac{||A(cx)||_\infty}{||B(cx)||_\infty}=\frac{||c||\cdot||Ax||_\infty}{||c||\cdot||Bx||_\infty} = \frac{||Ax||_\infty}{||Bx||_\infty}$, so you may choose to work on the unit sphere, but this is not a simplex. |
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