Metadata-Version: 2.1
Name: qss
Version: 0.1.3
Summary: QSS: Quadratic-Separable Solver
Home-page: https://github.com/lukevolpatti/qss
Author: Luke Volpatti
License: Apache 2.0
Project-URL: Bug Tracker, https://github.com/lukevolpatti/qss/issues
Platform: UNKNOWN
Requires-Python: >=3.6
Description-Content-Type: text/markdown
Requires-Dist: cvxpy
Requires-Dist: numpy
Requires-Dist: qdldl
Requires-Dist: scipy

# QSS: Quadratic-Separable Solver
QSS solves problems of the form 
$$\begin{equation*} \begin{array}{ll} \text{minimize} & (1/2) x^T P x + q^T x + r + g(x) \\\\ \text{subject to} & Ax = b \end{array} \end{equation*}$$
where $x \in \bf{R}^n$ is the decision variable being optimized over. The
objective is defined by a positive definite matrix $P \in \bf{S}^n_+$, a vector
$q \in \bf{R}^n$, a scalar $r \in \bf{R}$, and a $g$ that is separable in the
entries of $x$, i.e., $g$ can be written as 
$$g(x) = \sum_{i=1}^n g_i(x_i).$$
The constraints are defined by a matrix $A \in \bf{R}^{m \times n}$ and a vector
$b \in \bf{R}^m$. 

To use QSS, the user must specify $P$, $q$, $r$, $A$, $b$, as well as the $g_i$ from a built-in collection of separable functions. 

## Installation
```
pip install qss
```

## Usage
After installing `qss`, import it with
```python
import qss
```
This will expose the QSS class which is used to instantiate a solver object:
```python
solver = qss.QSS(data)
```
Use the `solve()` method when ready to solve:
```python
results = solver.solve(eps_abs=1e-4,
                       eps_rel=1e-4,
                       alpha=1.4,
                       rho=0.1,
                       max_iter=np.inf,
                       precond=True,
                       warm_start=False,
                       reg=True,
                       use_iter_refinement=False,
                       verbose=False,
                       )
```

### Parameters
- `data`: dictionary with the following keys:
    - `'P'`, `'q'`, `'r'`, `'A'`, `'b'` specify the quadratic part of the objective and the linear constraint as in the problem formulation above. `'P'` and `'A'` should be `scipy.sparse` CSC matrices, `'q'` and `'b'` should be `numpy` arrays,  and `'r'` should be a scalar. `'A'` and `'b'` can be excluded from `data` or set to `None` if the linear equality constraints are not needed. 
    - `'g'` is a list of separable function definitions. Each separable function is declared as a dictionary with the following keys:
        - `'g'`: string that corresponds to a valid separable function name (see below for a list of supported functions).
        - `'args'`: `'weight'` (default 1), `'scale'` (default 1), `'shift'` (default 0) allow the `'g'` function to be applied in a weighted manner to a shifted and scaled input. Some functions take additional arguments, see below. 
        - `'range'`: tuple specifying the start index and end index that the function should be applied to.

        Note that the zero function will be applied to any indices that don't have another function specified for them.
- `eps_abs`: scalar specifying absolute tolerance.
- `eps_abs`: scalar specifying relative tolerance.
- `alpha`: scalar specifying overstep size.
- `rho`: scalar specifying ADMM step size.
- `max_iter`: maximum number of ADMM iterations to perform.
- `precond`: boolean specifying whether to perform matrix equilibration.
- `warm_start`: boolean specifying whether to warm start upon a repeat call of
  `solve()`.
- `reg`: boolean specifying whether to regularize KKT matrix. May fail on certain problem instances if set to `False`.
- `use_iter_refinement`: boolean, only matters if `reg` is `True`. Helps mitigate some of the accuracy loss due to regularization. 
- `verbose`: boolean specifying whether to print verbose output.

### Returns
A list containing the following:
- `objective`: the objective value attained by the solution found by `qss`. 
- `solution`: the solution vector.

### Separable functions
The following separable functions are supported: 
- `"zero"`: $g(x) = 0$
- `"abs"`: $g(x) = |x|$
- `"is_pos"`: $g(x) = I(x \geq 0)$
- `"is_neg"`: $g(x) = I(x \leq 0)$
- `"is_bound"`: $g(x; lb, ub) = I(lb \leq x \leq ub)$
    - Default: `lb` = 0, `ub` = 1.
- `"is_zero"`: $g(x) = I(x == 0)$
- `"pos"`: $g(x) = \max\\{x, 0\\}$
- `"neg"`: $g(x) = \max\\{-x, 0\\}$
- `"card"`: $g(x) = \\{0 \text{ if } x = 0, 1 \text{ else}\\}$
- `"quantile"`: $g(x; \tau) = 0.5 |x| + (\tau - 0.5) x$ 
    - `tau` in `(0, 1)` is a scalar.
    - Default: `tau = 0.5`.
- `"huber"`: $g(x; M) = \\{x^2 \text{ if } |x| \leq M, 2M|x| - M^2 \text{ else}\\}$
    - `M > 0` is a scalar.
    - Default: `M = 1`. 
- `"is_int"`: $g(x) = I(x \text{ is an integer})$
- `"is_finite_set"`: $g(x; S) = I(x \in S)$
    - `S` is a Python set of scalars.
- `"is_bool"`: $g(x) = I(x \in \\{0,1\\})$

The `t` (weight), `a` (scale), `b` (shift) parameters are used to shift and scale the above as follows: `t * g(ax - b)`.

### Example
Nonnegative least squares is a problem of the form
$$\begin{equation*} \begin{array}{ll} \text{minimize} & (1/2) \Vert Gx - h \Vert_2^2 \\\\ \text{subject to} & x \geq 0. \end{array} \end{equation*} $$
`qss` can be used to solve this problem as follows:
```python
import numpy as np
import scipy as sp
import qss

p = 100
n = 500
G = sp.sparse.random(n, p, density=0.2, format="csc")
h = np.random.rand(n)

data = {}
data["P"] = G.T @ G
data["q"] = -h.T @ G
data["r"] = 0.5 * h.T @ h
data["g"] = [{"g": "is_pos", "range": (0, p)}] # Enforce x >= 0

solver = qss.QSS(data)
objective, x = solver.solve()
print(objective)
```

## Development
To create a virtual environment, run
```
python3 -m venv env
```
Activate it with 
```
source env/bin/activate
```
Clone the `qss` repository, `cd` into it, and install `qss` in development mode:
```
pip install -e ./ -r requirements.txt
```
Finally, test to make sure the installation worked:
```
pytest tests/
```


