Maximum value of a linear function over an euclidean norm

Question

I am trying to solve:

$\underset{S}{max}$ $y^TQ^{-1}S$

$subject$ $to$ $||S||_2 \leq p$

Where $y$ is a given vector, $Q$ is an $n\times n$ positive definite matrix and $S$ is the optimization variable. The KKT conditions written below are a bit inconsistent for me:

$||S||_2 \leq p$ (constraint),

$\lambda \geq 0$ (dual variable),

$\lambda(||S||_s-p)=0$ (compl. slackness),

$y^TQ^{-1}+ \frac{\lambda S^T}{2||S||_2} = 0$ (first order condition of Lagrangian).

I can't go further. What is the real solution for this simple convex model?

Answer 1

To maximize the inner product, you should have $S$ parallel to $Q^{-T}y$. Hence $S= tQ^{-T}y$ for some positive scalar $t$. To maximize the expression, you pick $t$ such that the constraint is active, and you will obtain $S = p \frac{Q^{-T}y}{||Q^{-T}y||}$