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For example consider the problem

$\min_X f(X)$

s.t. $\lambda_i(X+A)=\lambda_i(B)$ for $i \in {1,...,N}$

where $A$ and $B$ are full rank N by N matrix, $\lambda_i(X)$ is the i-th eigenvalue of $X$

Is it the correct way to represent the eigenvalue constraints?

If yes, how can it be handled?

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    You can find the eigenvalues of $B$ and substitute $X=Y-A$ into $f$ to reduce this to the problem of minimizing $f$ with respect to $Y$ for given eigenvalues of $Y$. In case you know $Y$ to be symmetric, you can explicitly write $Y$ in terms of its eigenvectors and eigenvalues and then minimize with respect to the orthogonal matrix of eigenvectors.2012-12-17

1 Answers 1

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I too am interested in knowing how to solve this. Specifically, I want to solve $\min ||y - Ax||^2 \\ st \quad -\delta <= \lambda_\max (A) <= \delta$

One approach that comes to mind is using projected gradient descent -- since the function $ \lambda_\max (A) = \max_x \frac{x'Ax}{x'x} $ is convex.

Not sure how one would handle arbitary eigenvalues.