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Project Description
# CySCS
[![Build Status](https://travis-ci.org/ajfriend/cyscs.svg?branch=master)](https://travis-ci.org/ajfriend/cyscs)

`cyscs` is a Python interface, written in Cython, for [SCS](https://github.com/cvxgrp/scs) a numerical optimization package written in C for solving convex cone problems. The main advantage of this interface over the existing
Python interface is the `Workspace` object, which allows for re-use
of matrix factorizations, which can reduce solve time when applied
a sequence of related problems.

SCS solves convex cone programs via operator splitting.
It can solve: linear programs (LPs), second-order cone programs (SOCPs),
semidefinite programs (SDPs), exponential cone programs (ECPs), and
power cone programs (PCPs), or problems with any combination of these
cones.

Most users will not interact with `cyscs` directly. The most common use-case is as
a back-end to a convex optimization modeling framework like [CVXPY](http://www.cvxpy.org).

Advanced users can consult the interface notes below, the [tutorial IPython notebook](tutorial.ipynb) or the [parallel tutorial IPython notebook](parallel_tutorial.ipynb). For more complete definitions of the input data format, convex cones, and output variables, please see the [`SCS README`](https://github.com/cvxgrp/scs/blob/master/README.md) or the [SCS Paper](http://web.stanford.edu/~boyd/papers/scs.html)

## Installation
### Pip
- XXX: not yet uploaded to PyPI, so `pip` won't work
- `pip install cyscs`

### Building from source with Cython
Users can also install by
- cloning this GitHub repo with `git clone --recursive https://github.com/ajfriend/cyscs.git`
- installing dependencies `pip install numpy scipy cython`
- (optionally, for tests) `pip install pytest`
- running `python setup.py install --cython` inside the `cyscs` directory
- (optionally) run tests with `make test`

## Basic Usage
The basic usage is almost identical to the existing SCS Python interface:
```python
import cyscs as scs
sol = scs.solve(data, cone, warm_start=None, **settings)
```

Note that `cyscs` can be used as a drop-in replacement to the current
Python interface `scs` using the import statement `import cyscs as scs`.
(Some advanced features like warm-starting are *not* identical
across both interfaces. Converting between them manually is easy, however.)

We describe the arguments to `cyscs.solve()` briefly below. For more detail, please see the [`SCS README`](https://github.com/cvxgrp/scs/blob/master/README.md)

- `data` is a Python `dict` with keys:
- `'A'`: `scipy.sparse.csc` matrix, i.e., a matrix in Compressed Sparse Column format with `m` rows and `n` columns
- `'b'`: 1D `numpy` array of length `m`
- `'c'`: 1D `numpy` array of lenth `n`
- `cone` is a Python `dict` with potential keys:
- `'f'`: `int` of linear equality constraints
- `'l'`: `int` of linear inequality constraints
- `'q'`: `list` of `int`s giving second-order cone sizes
- `'s'`: `list` of `int`s giving semidefinite cone sizes
- `'ep'`: `int` of primal exponential cones
- `'ed'`: `int` of dual exponential cones
- `'p'`: `list` of `float`s of primal/dual power cone parameters
- `warm_start` is an **optional** `dict` of `numpy` arrays (with keys `'x'`, `'y'`, and `'s'`) used to warm-start the solver; if these values are close to the final solution, warm-starting can reduce the number of SCS iterations
- `cyscs.solve()` also accepts optional keyword arguments for solver settings:
- `use_indirect`
- `verbose`
- `normalize`
- `max_iters`
- `scale`
- `eps`
- `cg_rate`
- `alpha`
- `rho_x`
- settings are passed as keyword arguments:
- `cyscs.solve(data, cone, max_iters=100)`
- `cyscs.solve(data, cone, alpha=1.4, eps=1e-5, verbose=True)`
- `cyscs.solve(data, cone, warm_start, use_indirect=True)`
- default settings can be seen by calling `scs.default_settings()`
- `sol` is a `dict` with keys:
- `'x'`: `numpy` array
- `'y'`: `numpy` array
- `'s'`: `numpy` array
- `'info'`: `dict` containing solver status information

### Warm-starting
The solver can be warm-started, that is, started from a point close to the final solution in the hope of reducing the solve-time. You must supply `numpy` arrays for for **all** of the warm-started variables `x`, `y`, and `s`. Pass them as dictionary to the `warm_start` parameter in `cyscs.solve()`:

```python
ws = {'x': x, 'y': y, 's': s}
sol = scs.solve(data, cone, warm_start=ws)
```

Output from previous solves can be used to warm-start future solves:
```python
sol = scs.solve(data, cone, eps=1e-3)
sol = scs.solve(data, cone, warm_start=sol, eps=1e-4)
```


### Data Formats
Below are the integer and floating-point format expectations for input data.
If the formats are not exactly correct, `cyscs` will attempt to convert the data for you.

`cyscs.solve()` expects `b`, `c`, `x`, `y`, and `s` to be one-dimensional `numpy` arrays with `dtype` `'float64'`.

`solve()` also expects `A` to be a `scipy.sparse.csc` matrix such that the values of the matrix have `dtype` `'float64'`, and the attributes `A.indices` and `A.indptr` are `numpy` arrays with `dtype` `'int64'`:

```python
>>> A.dtype
dtype('float64')
>>> A.indices.dtype, A.indptr.dtype
(dtype('int64'), dtype('int64'))
```

Note that, by default, `scipy.sparse.csc` matrices have `indptr` and `indices` arrays with `dtype` `int32`. If the matrices are not converted ahead of time, `cyscs` will do the conversion internally, without modifying the original `A` matrix. However, it may be more efficient to construct an `A` with the correct `dtype`s initially, rather than convert.

### Data Immutability
`cyscs.solve()` will not modify the input data in `data`, `cone`, or `warm_start`. Copies of the data will be made for internal use, and new `numpy` arrays will be created to be returned in `sol`.


## Factorization Caching with `cyscs.Workspace`
When using the **direct** solver (`use_indirect=False`), a single matrix factorization is performed and used many times in SCS's iterative procedure.
This factorization depends on the input matrix `A` but **not** on the vectors
`b` or `c`. When solving many problems where `A` is fixed, but `b` and `c` change, the `cyscs.Workspace()` object allows us to cache the initial factorization and reuse it across many solves, without having to re-compute it. This can save time when solving many related problems.

The `Workspace` is instantiated with the same `data` and `cone` dictionaries
as `cyscs.solve()`, along with optional settings:
```python
work = scs.Workspace(data, cone, **settings)
```

Once the `Workspace` object is created, we can call its solve method
```python
sol = work.solve(new_bc=None, warm_start=None, **settings)
```

which will re-use the matrix factorization that was computed when the `Workspace` was initialized. Note that all of the parameters to `work.solve()` are optional. `new_bc` is a dictionary which can optionally provide
updated `b` or `c` vectors (any other keys, including `A`, are ignored).

The return value, `sol`, is a `dict` with keys `x`, `y`, `s`, and `info`, just as in `cyscs.solve()`.

### `Workspace` state
`work.solve()` will operate on the data contained in the `work` object:

- `work.data`
- `work.settings`

The user can modify the state of the `Workspace` object between calls to `work.solve`.

#### `work.data`
Note that `work.data` is a `dict` with keys `b` and `c`, but **not** `A`. This is because `A` is copied and stored internally (along with its factorization) upon initialization, and cannot be modified.

Due to the data copy, the user is now free to delete or modify the `A` matrix that they passed into the `cyscs.Workspace` constructor, as this will have no effect on the `Workspace` object.

#### `work.settings`
Only some of the values in `work.settings` can be modified between calls to `work.solve()`.

The following settings are **fixed** at `Workspace` initialization time:

- `use_indirect`
- `rho_x`
- `normalize`
- `scale`

A copy of the `dict` of **fixed settings** is given by `work.fixed`. If any of the `work.settings` differ from `work.fixed` when `work.solve()` is called, an `Exception` will be raised. Calling `work.fixed` returns a **copy** of the underlying `dict`, which cannot be modified. XXX: make a test for this

The following settings **can** be modified between calls to `work.solve()`:

- `verbose`
- `max_iters`
- `eps`
- `cg_rate`
- `alpha`

#### `work.info`

When calling `sol = work.solve()`, solver status information is available
through the `sol['info']` dictionary. This same information is also available through the attribute `work.info`.

This attribute is useful, for instance, if you'd like to know the solver setup time after calling `Workspace()` but before calling `work.solve()`, which you can access with `work.info['setupTime']`.

#### Immutable `work` state
Upon initialization, `A` is copied, stored, and factored internally.
Any changes made to the `scipy` sparse input matrix `A` after the fact
will not be reflected in the `work` object.
Similarly, `work.cone` is fixed at initialization and cannot be modified.
To avoid confusion, we do not expose `A` or `cone` to the user
through the `work` object.

### `work.solve()` arguments

#### `new_bc` and `settings`
Passing a `new_bc` dictionary or additional settings to `work.solve()` provides one last chance to modify the problem data before calling the solver. Any changes are written to the `work` object and persist to future calls to `work.solve()`. In fact,
```python
new_b = dict(b=b)
work.solve(new_bc = new_b, eps=1e-5, alpha=1.1)
```

is exactly equivalent to

```python
work.data['b'] = b
work.settings['eps'] = 1e-5
work.settings['alpha'] = 1.1
work.solve()
```

If `new_bc` is passed to `work.solve()`, only the keys `b` and `c` will be used to update `work.data`. If an `A` key exists, it will be ignored.

More commonly, a user might simply update the original `data` dictionary and pass it to `work.solve()`:

```python
work = scs.Workspace(data, cone)
...
data['b'] = b # update the b vector
sol = work.solve(new_bc=data)
```

#### `warm_start`
You can also provide a dictionary of warm-start vectors to the `warm_start` parameter, which may help reduce the solve time.

In the example below, we first solve a problem to a tolerance of `1e-3`, and use that solution as a warm-start for solving the problem to a higher tolerance of `1e-4`. The second call to `work.solve()` will generally take fewer iterations than if we hadn't provided a `warm_start`, and also
benefits from not having to re-compute the matrix factorization.

```python
work = scs.Workspace(data, cone)
sol = work.solve(eps=1e-3)
sol = work.solve(warm_start=sol, eps=1e-4)
```

## Example Library
`cyscs` comes with a small examples library,
`cyscs.examples`, which
demonstrates the proper problem input format,
and can be used to easily test the solver.

For example,
```python
import cyscs as scs
data, cone = scs.examples.l1(m=100, seed=0)
sol = scs.solve(data, cone)
```

solves a simple least L1-norm problem.

## Python GIL

`cyscs.solve()`, `Workspace` initialization, and `Worksapce.solve()` all release the Python GIL when running the underlying C solver code. This allows for multithreaded parallelism, so that multiple SCS problems can be solved at once
on multicore machines.

Problems can be solved using, for example, the `concurrent.futures.ThreadPoolExecutor` or `concurrent.futures.ProcessPoolExecutor` interfaces.
Since SCS releases the GIL, we can benefit from using the `ThreadPoolExecutor` since it does not require launching separate python interpreters or the serialization of data for communication between processes. `ProcessPoolExecutor` requires both of these.

For examples, see the [parallel tutorial IPython notebook](parallel_tutorial.ipynb).
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