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Does anybody know a method to solve the following problem numerically (or analytically, if there is a general method, but I doubt it).

I am given a matrix $A \in \mathbb{C}^{n \times n}$ and I want to minimize

$||A- \sum_{i=1}^{n}\lambda_i C_i||$ in the Frobenius-norm for given $C_i$ and unknown $\lambda_i \ge 0$ with $\sum_{i=1}^{n} \lambda_i =1$ and $\langle C_i,C_j \rangle = \delta_{i,j}.$

By compactness of the possible set of $\lambda_i$ the problem has always an actual minimum and also the convexity (since we are taking convex combinations of the $\lambda_i$) could be useful if we want to find a numerical algorithm.

But I am not very familiar with these kind of optimisation questions, so I would love to hear about possible algorithms to tackle this problem.

Sergio Parreiras
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user167575
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    The key to seeing this really is a duplicate of the linked problem above is noting that 1) $|A|_F = | \mathop{\textrm{vec}}(A) |_2$, and 2) squaring the objective does not change the solution (though it makes it easier to solve). – Michael Grant Oct 06 '15 at 13:59

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If you square the norm, and use the definition of the Frobenius norm, it will expand to $\text{trace}~A^TA - 2\sum_i \lambda_i \text{trace} A^TC_i + \lambda^T\lambda$. In other words, a convex quadratic, which together with your constraints means that you have a convex quadratic program, which is an easily solved standard problem class.

Johan Löfberg
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