MATLAB scripts for alternating direction method of multipliers

MATLAB scripts for alternating direction method of multipliers

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein

This page gives MATLAB implementations of the examples in our
paper on distributed optimization with the alternating direction method of multipliers.

These scripts are serial implementations of ADMM for various problems. In cases where the scripts solve distributed consensus problems (e.g., distributed

-regularized
logistic regression), the code runs serially instead of in parallel. These scripts give an idea of the structure and flavor of ADMM; an implementation in C/C++ that follows the structure laid out in our scripts and exploits parallelism can be competitive with
state-of-the-art solvers for these problems. These are presented in MATLAB for readability. Unless otherwise noted on this page, most scripts stand alone and work in Matlab without further modification. Some scripts may require CVX.

As output, each example script displays the primal residual

, the primal feasibility
tolerance

, the dual residual

,
and the dual feasibility tolerance

. See section 3.3 of the paper
for more details on these quantities. Also included are plots of the objective value and the primal and dual residual by iteration. Note that the objective value at any particular iteration can go below the true solution value

because
the iterates need not be feasible (e.g., if the constraint is

, we can have

for
some

).

You can use the grabcode function
built in to MATLAB to obtain these codes. Please report any issues to the authors.

Examples

Basis pursuit

ADMM function

Example

Sparse inverse covariance selection

ADMM function

Example

Huber fitting

ADMM function

Example

Intersection of polyhedra

ADMM function

Example

Lasso

ADMM function

Example

Lasso with LSQR, an iterative method for solving sparse systems


ADMM function

Example

Least absolute deviations

ADMM function

Example

Linear programming

ADMM function

Example


-regularized logistic regression

ADMM function

Example

Distributed

-regularized logistic regression

ADMM function - also requires l2_log, l2_log_grad, record_bfgs_iters,
and LBFGS-B for Matlab.

Example

Regressor selection (nonconvex problem)

ADMM function

Example

Quadratic programming

ADMM function

Example

Sum-of-norms regularization (group lasso)

ADMM function

Example

Sum-of-norms regularization (group lasso) with feature splitting

ADMM function

Example

Support vector machine

ADMM function

Example

Total variation minimization

ADMM function

Example