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Suppose I have an SDP

$$\min_{X \in \mathbb{S}^{n}_{+}} f(X)$$ $$\text{s.t.} \quad X_{i,j} = c_{i,j} \quad \forall (i,j) \in I,$$ where $I \subseteq [n] \times [n]$ and $f$ is convex on the set of positive semidefinite matrices. Are there any methods for solving SDPs which are able to simplify the problem by eliminating the constrained variables?

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