CVXQUAD is a collection of functions to be used with the MATLAB-based convex optimization tool CVX. It augments CVX with various convex/concave functions based on matrix logarithm such as the von Neumann entropy, or the quantum relative entropy (see below for list of functions). This package is based on the paper:
Semidefinite approximations of matrix logarithm
Hamza Fawzi, James Saunderson and Pablo A. Parrilo
available at https://arxiv.org/abs/1705.00812.
July 14, 2021: Added support for approximations that are formal upper or lower bounds on log -- the Pade approximations from the paper above are neither upper nor lower bounds. The new approximations are based on Gauss-Radau quadrature, see doc/log_approx_bounds.pdf for more details. Functions quantum_entr, quantum_rel_entr, trace_logm and op_rel_entr_epi_cone now have an additional parameter, 'apx', which can be used to specify which approximation to be used (apx=-1,0,+1).
Unpack the zip file https://github.com/hfawzi/cvxquad/archive/master.zip and add the folder to your MATLAB path.
Update: CVX 2.2 can now handle the exponential cone natively when used with Mosek 9. See CVX's documentation. If you have access to Mosek, no need to use any approximations.
To replace the successive approximation functionality of CVX whenever the exponential cone is used (e.g., when using rel_entr or in GP mode), copy the file "exponential/exponential.m" to the folder "sets" in your CVX installation (you may want to keep a copy of the existing file in case you want to revert to the successive approximation method).
The following code uses the quantum_rel_entr function of CVXQUAD to compute the nearest correlation matrix to a given matrix M, in the quantum relative entropy sense.
n = 4;
M = randn(n,n);
M = M*M';
cvx_begin
variable X(n,n) symmetric
minimize quantum_rel_entr(M,X)
subject to
diag(X) == ones(n,1)
cvx_end
| Function | ||
|---|---|---|
| rel_entr_quad(x,y) | x.*log(x./y) | convex in (x,y) |
| quantum_entr(X) | -trace(X*logm(X)) | concave in X |
| quantum_rel_entr(X,Y) | trace(X*(logm(X)-logm(Y))) | convex in (X,Y) |
| trace_logm(X,C) | trace(C*logm(X)) | concave in X (C fixed positive semidefinite matrix) |
| trace_mpower(X,t,C) | trace(C*X^t) | concave in X for t in [0,1] convex in X for t in [-1,0] or [1,2] (C fixed positive semidefinite matrix) |
| lieb_ando(X,Y,K,t) | trace(K' * X^{1-t} * K * Y^t) | concave in (X,Y) for t in [0,1] convex in (X,Y) for t in [-1,0] or [1,2] (K is a fixed matrix) |
| Set | |
|---|---|
| op_rel_entr_epi_cone | Operator relative entropy cone |
| matrix_geo_mean_hypo_cone | Matrix geometric mean hypograph cone |
| matrix_geo_mean_epi_cone | Matrix geometric mean epigraph cone |
To cite the package in your work, you can use the following bibtex code:
@article{cvxquad,
title={Semidefinite approximations of the matrix logarithm},
author={Fawzi, Hamza and Saunderson, James and Parrilo, Pablo A.},
year={2018},
journal={Foundations of Computational Mathematics},
note={Package cvxquad at \url{https://github.com/hfawzi/cvxquad}}
}