# Optimality for Convex Functions of Semidefinite Matrices

**Theorem 1** Let \(f(X)\) be a convex, twice continuously differentiable function of \(X\in \mathbb{R}^{n\times n}\). Consider the optimization
\(\underset{X \succeq 0}{\operatorname{minimize}} f(X).\)
We consider a rank-constrained factorization of the form
\(\operatorname{minimize}_{U \in \mathbb{R}^{n \times k}} g(U)=f\left(U U^{T}\right).\)
If \(U\) is a second-order stationary point of (2), and \(\mathrm{rank}(U)<k\), then \(UU^T\) is a global minimum for (1).

**Proof.** The KKT conditions (which are in this case both necessary and sufficient) for (1) are
\(\nabla f(X)\succeq 0, \quad \nabla f(X)X=0.\)