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I am trying to solve the following optimization problem.

$$ \max_{A} \min_{i,j,i\neq j} ||A(x_i-x_j)||_2^2 \\ \text{s.t } ||A||_F^2 <=1 $$

$A$ is a matrix and $x_i$ are given vectors. The min of convex functions is not a convex/concave function. Is there any tricks to convert it to a convex problem or an approximation of a convex problem?

dsp_guy2020
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  • Is it possible to impose the constraint $|A(x_i-x_j)| \geq d$ and then adjust $d$ such that $d = \min\limits_{i,j}|A(x_i-x_j)|$ for some $A$? – jokersobak Feb 24 '23 at 02:02
  • $||A(x_i-x_j)|| \geq d$ is not a convex constraint. – dsp_guy2020 Feb 24 '23 at 13:33
  • It's important what are the directions of projection for linear operator A, because if this direction coincides with x_i - x_j then min = 0. So, it's worth to transform A to diagonal form by using principle directions. – jokersobak Feb 24 '23 at 21:56

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