Consider the map $ A : K \subset \mathbb{R}^m \rightarrow \mathbb{R}^m$, with $K$ compact and convex, defined by
$$ A(c) = \text{arg}\min_{y \succeq 0} y^TDy + c^T y $$
Where $D$ is a positive semidefinite (not positive definite) matrix, and c is a vector. Also, suppose $D$ and $c$ are always such that there is a unique minimum so $A$ is a well-defined function. It can be shown that $A$ is continuous in this case.
Question: Is the map $A$ Lipschitz continuous?
I'm not too familiar with optimization techniques in general, and my naïve attempts to answer this question have so far failed. E.g. without the non-negatovity constraint, the answer is true, but I cannot figure out how to relate that to the constrained case. Any ideas would be greatly appreciated! Thanks!