Convexity of Sum of $k$-smallest Eigenvalue

2019-05-05

An answer to this question on the Scientific Computing Stack Exchange.

Question

If I have a real positive definite matrix $A\in\mathbb{R}^{n\times n}$, and denote its eigenvalues as $\lambda_1\leq \lambda_2 \leq ... \leq \lambda_n $.

Define the function as $f(A)=\sum_{i=1}^{k} \lambda_i$ for a constant $k<n$. What do we know about the convexity of $f(A)$? Is it convex or concave?

Answer

cvxpy treats that function as being concave (link).