2 alpha fixup, >0 matrices are pd

Hello! (First question--please forgive me if its unclear.)

I am interested in efficient/approximate optimization techniques for minimizing a norm of a convex combination of symmetric, positive semi-definite matrices. In particular I am wondering what techniques exist to efficiently solve the following problem:

$\alpha^{*}=\arg\min\big\{\operatorname{trace}((\sum_i\alpha_i \mathcal{I}_i)^{-1}):\sum_i\alpha_i=1 mathcal{I}_i)^{-1}):\sum_i\alpha_i=1, \alpha_i\ge0 \big\}$

where $\mathcal{I}_i$ are positive semi-definite matrices and $\operatorname{trace}$ is the sum of the eigenvalues. Edit: Furthermore, assume $\mathcal{I}_i\succ0$ for at least one $i$.

Approximations are acceptable--I am interested in any "standard" approaches, should they exist. Should some other scalar function be more convenient this would be interesting to me too (ie determinant, nuclear norm, spectral radius, etc). The matrices are often sparse and could be extremely large.

The simplest idea I had was to lower bound the sum of matrices using Weyl's theorem, ie, $\lambda_i(A+B)\ge \lambda_j(A)+\lambda_{n+i-j}(B)$ for $j\ge i$ giving me an upper bound to the trace of the inverse of the sum. I am wondering if another bound would be tighter or better still some clever variational approach exists.

The motivation (shouldn't be relevant to the answer) for this problem comes from statistical machine learning. I would like to develop techniques for characterizing maximum composite likelihood estimators (a generalization of the maximum likelihood estimator). Classical statistical analysis yields a convex combination of the expected Hessian matrices and is as above. (The $\mathcal{I}$ denotes the Fisher information matrix.)

There is some similarity to the Eigenvalues of Matrix Sums question although I believe this question is sufficiently different to ask anew.

Thanks,

Josh

1

# Minimize trace of inverse of convex combination of matrices.

Hello! (First question--please forgive me if its unclear.)

I am interested in efficient/approximate optimization techniques for minimizing a norm of a convex combination of symmetric, positive semi-definite matrices. In particular I am wondering what techniques exist to efficiently solve the following problem:

$\alpha^{*}=\arg\min\big\{\operatorname{trace}((\sum_i\alpha_i \mathcal{I}_i)^{-1}):\sum_i\alpha_i=1 \big\}$

where $\mathcal{I}_i$ are positive semi-definite matrices and $\operatorname{trace}$ is the sum of the eigenvalues.

Approximations are acceptable--I am interested in any "standard" approaches, should they exist. Should some other scalar function be more convenient this would be interesting to me too (ie determinant, nuclear norm, spectral radius, etc). The matrices are often sparse and could be extremely large.

The simplest idea I had was to lower bound the sum of matrices using Weyl's theorem, ie, $\lambda_i(A+B)\ge \lambda_j(A)+\lambda_{n+i-j}(B)$ for $j\ge i$ giving me an upper bound to the trace of the inverse of the sum. I am wondering if another bound would be tighter or better still some clever variational approach exists.

The motivation (shouldn't be relevant to the answer) for this problem comes from statistical machine learning. I would like to develop techniques for characterizing maximum composite likelihood estimators (a generalization of the maximum likelihood estimator). Classical statistical analysis yields a convex combination of the expected Hessian matrices and is as above. (The $\mathcal{I}$ denotes the Fisher information matrix.)

There is some similarity to the Eigenvalues of Matrix Sums question although I believe this question is sufficiently different to ask anew.

Thanks,

Josh