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?
Asked
Active
Viewed 188 times
1 Answers
0
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.
Eric Towers
- 67,037
-
Does it make any difference if i limit the optimization space to real number vectors? – ARM Feb 07 '14 at 22:59
-
@ARM: No. You just get the real unit sphere instead of the complex unit sphere. – Eric Towers Feb 07 '14 at 23:01