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 $$\nabla f(X)\succeq 0, \quad \nabla f(X)X=0.$$