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A Python package for type-II Anderson accelerated Douglas-Rachford splitting (A2DR).

# a2dr

`a2dr` is a Python package for solving large-scale non-smooth convex optimization problems with general linear constraints, with separable objective functions accessible through their proximal operators. It exploits the separability of the objective functions and the sparsity in the linear constraints, and utilizes the power of Anderson acceleration to achieve fast and robust convergence and scalability to multiple processors. The current version is `0.2.3.post2`.

It is an implementation of type-II Anderson accelerated Douglas-Rachford splitting, based on our paper A. Fu, J. Zhang, and S. Boyd (2019).

### Installation

To install using pip (recommended), run:

```pip install a2dr
```

To install `a2dr` from source, first make sure that you have setuptools installed. Then follow the steps below:

1. Clone the `a2dr` git repository.
2. Navigate to the top-level of the cloned directory and run:
```python setup.py install
```

To test the installation with nose, first make sure that you have nose installed. Then run:

```nosetests a2dr
```

The requirements are:

Please file an issue on Github if you want Python 2 support.

### Problem

`a2dr` solves problems of the following form:

``````minimize         f_1(x_1) + ... + f_N(x_N)
subject to       A_1x_1 + ... + A_Nx_N = b.
``````

where f_i (i=1,...,N) are convex, closed and proper, and are only accessible through their proximal operators. Notice that f_i can take infinite values, and in particular constraints can be included in the objectives with the help of indicator functions.

#### Prox-affine forms

The above formulation is also referred to as prox-affine forms in the literature (see e.g., Epsilon). When it is seen as a standard form for generic convex optimization problems, the major advantage of prox-affine forms compared to the more widely used conic forms include:

• Privacy: suitable for peer-to-peer optimization with privacy requirements.
• In practice, the data and source code that define the proximal oracle can be securely encrypted (e.g., via compilation) so that privacy is preserved. For example, in Python, we can convert the `.py` file containing the proximal operator function into an encrypted `.so` file via the Cython extension.
• Compactness: straightforward canonicalization/transformation and lower dimensional representations.
• In general, the prox-affine form often requires less variables than the conic form (see e.g., the portfolio optimization example here). The compactness advantage is also partly exemplified by the comparison between `a2dr` and `SCS` in the sparse covariance estimation example in our paper.

For a bit more detailed introduction to prox-affine forms and the comparisons with conic forms, see our companion slides.

### Usage

After installing `a2dr`, you can import `a2dr` using

```import a2dr
```

This module exposes a function a2dr (the solver), which can be used via `a2dr.a2dr`, or directly imported using

```from a2dr import a2dr
```

The function a2dr is called with the command

```a2dr_result = a2dr(p_list,
A_list=[],
b=np.array([]),
v_init=None,
n_list=None,
max_iter=1000,
t_init=1/10,
eps_abs=1e-6,
eps_rel=1e-8,
precond=True,
ada_reg=True,
anderson=True,
m_accel=10,
lam_accel=1e-8,
aa_method='lstsq',
D_safe=1e6,
eps_safe=1e-6,
M_safe=int(max_iter/100),
verbose=True)
```

#### Parameters:

The arguments `p_list`, `A_list` and `b` correspond to the problem data.

• `p_list` is the list of proximal operators of f_i. Each element of `p_list` is a Python function, which takes as input a vector v and parameter t > 0 and outputs the proximal operator of f_i evaluated at (v,t).
• `A_list` is the list of A_i. The lists `p_list` and `A_list` must be given in the same order i = 1,...,N.
• `b` is the vector b. Notice that `A_list` and `b` are optional, and when omitted, the solver recognizes the problem as one without linear constraints. Also notice that in such cases, `A_list` and `b` have to be omitted together, and either `v_init` or `n_list` has to be provided to declare the dimension of each x_i.

For information on the other optional hyper-parameters, please refer to our companion paper (Algorithm 2) and the source code comments of the function a2dr in solver.py.

#### Returns:

The returned object `a2dr_result` is a dictionary containing the keys `'x_vals'`, `'primal'`, `'dual'`, `'num_iters'` and `'solve_time'`:

• The output `x_vals` is a list of x_1,...,x_N from the iteration with the smallest residuals.
• `primal` and `dual` are arrays containing the primal and dual residual norms for the entire iteration process, respectively.
• The value `num_iters` is the total number of iterations, and `solve_time` is the algorithm runtime.

When the linear equality constraint is infeasible, the solver will print the corresponding flag and return `None` for all the outputs.

#### Other tools

The module `a2dr` also comes with several additional tools that facilitates the transformation of the problems into the required input format described above as well as tests and visualization. In particular, it come with a package for proximal operators, which can be imported via

```import a2dr.proximal
```

It also comes with some tests and visualization tools, which can be imported via

```import a2dr.tests
```

#### Example

We showcase the usage of the solver function a2dr as well as the the tool packages `a2dr.proximal` and `a2dr.tests` with the following example. More examples can be found in the examples/ directory.

```# Non-negative least squares (see our companion paper for more details)
import numpy as np
import numpy.linalg
from scipy import sparse
from a2dr import a2dr
from a2dr.proximal import *
from a2dr.tests.base_test import BaseTest

# Problem data.
np.random.seed(1)
m, n = 150, 300
density = 0.001
X = sparse.random(m, n, density=density, data_rvs=np.random.randn)
y = np.random.randn(m)

# Convert problem to standard form.
prox_list = [lambda v, t: prox_sum_squares_affine(v, t, F=X, g=y),
prox_nonneg_constr]
A_list = [sparse.eye(n), -sparse.eye(n)]
b = np.zeros(n)

# Solve with DRS.
drs_result = a2dr(prox_list, A_list, b, anderson=False)
# Solve with A2DR.
a2dr_result = a2dr(prox_list, A_list, b, anderson=True)
bt = BaseTest()
bt.compare_total(drs_result, a2dr_result)
```

### Citing

If you wish to cite `a2dr`, please use the following:

``````@article{a2dr,
author       = {Fu, A. and Zhang, J. and Boyd, S.},
title        = {Anderson Accelerated {D}ouglas-{R}achford Splitting},
journal      = {http://stanford.edu/~boyd/papers/a2dr.html},
year         = {2019},
}

@misc{a2dr_code,
author       = {Fu, A. and Zhang, J. and Boyd, S.},
title        = {{a2dr}: Anderson Accelerated {D}ouglas-{R}achford Splitting, version 0.2.3.post2},
howpublished = {\url{https://github.com/cvxgrp/a2dr}},
year         = {2020}
}
``````

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