CVXQUAD

CVXQUAD is a collection of functions to be used with the MATLAB-based convex optimization tool CVX. It implements a new approximation strategy to treat the exponential cone as well as various functions based on matrix logarithm using symmetric cone solvers. 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.

Installation

Unpack the zip file https://github.com/hfawzi/cvxquad/archive/master.zip and add the folder to your MATLAB path.

Replacing successive approximation

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).

Example

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

Functions and sets

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

Citing

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}}
}