What is the tangent cone of the set of low-rank positive semidefinite matrices?

Question

Consider a subset of the set of all $n$-by-$n$ symmetric real matrices, as defined below

$$S := \{ M \in \mathbb{R}^{n \times n} \mid \mbox{rank}(M) \leq r, M \succeq 0 \}$$

Given a matrix $M \in S$, what is the Bouligand and Clarke tangent cone of $S$ at $M$? Equivalently speaking, what is $T_S(M)$ for a $M \in S$?

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Below are two papers that answers two related questions:

• The tangent cone of the set of low-rank matrices

• The tangent cone of the set of positive semidefinite matrices