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I am wondering if the following problem is well studied. Given two matrices $A, B \in R^{n\times d}$, solve $$\max_{x:\|x\|\leq 1}\min_{y:\|y\|\leq 1} \|Ax - By\|^2,$$ where $x, y \in R^d$. Assume $n\geq d$, and $A$, $B$ are full column-rank.

I have derived something, but I wonder if it's already been studied somewhere?

Jiaji Huang
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  • what do you get if you solve the inner minimization problem? – LinAlg Dec 14 '20 at 19:08
  • The inner one is a norm constrained least squares, relatively well studied. See for example, this post https://math.stackexchange.com/questions/2399321/least-squares-with-euclidean-l-2-norm-constraint – Jiaji Huang Dec 14 '20 at 19:10
  • What I am getting at what the maximization problem is like. The full problem might have been studied as an adjustable robust optimization problem. – LinAlg Dec 14 '20 at 20:56
  • Could you give any reference? – Jiaji Huang Dec 14 '20 at 21:32
  • There are plenty of papers and even a book on adjustable robust optimization, but I am not sure if they consider this specific function. In your case, $w_f$ is the uncertain parameter and $w_g$ is the adjustable variable. – LinAlg Dec 14 '20 at 21:54

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I have a partial solution and am glad to see more discussions.

First, let's solve the inner minimization problem for any given $x$, $$y^\ast(x)\triangleq \arg\min_{\|y\|\leq 1} \|Ax-By\|^2 = (B^\top B + \lambda I)^{-1}B^\top Ax,$$ where $\lambda=0$ if $(B^\top B)^{-1}B^\top Ax \leq 1$. Otherwise $\lambda$ is such that $(B^\top B + \lambda I)^{-1}B^\top Ax=1$. In general, we can write $\lambda=\lambda(x)$ to emphasize that the $\lambda$ value depends on $x$. This is not hard. A detailed derivation can be found at Least Squares with Euclidean ($ {L}_{2} $) Norm Constraint

Now substitute $y^\ast(x)$ to the outer maximization problem, solve $$\max_{\|x\|\leq 1} \|Ax-B(B^\top B + \lambda I)^{-1}B^\top Ax\|^2 \equiv \max\|[I-B(B^\top B + \lambda I)^{-1}B^\top] Ax\|^2.$$ If $\lambda$ were a constant, the maximizer is simply the leading right singular vector of $[I-B(B^\top B + \lambda I)^{-1}B^\top] A$.

However in reality, $\lambda$ also depends on $x$. So I'm not sure what we can do with this. Could anyone give more ideas?

Jiaji Huang
  • 151