qss 0.1.2

QSS: Quadratic-Separable Solver

QSS: Quadratic-Separable Solver

QSS solves problems of the form

minimize    f(x) + g(x)
subject to  Ax = b

Where f is a convex quadratic function given by f(x) = 0.5 x^T P x + q^T x + r and g is a function that is separable in each entry of x, i.e. g(x) = g_1(x_1) + ... + g_n(x_n).

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

Clone the repository and run

pip3 install ./

Usage

After installing qss, import it with

import qss

This will expose the QSS class which is used to instantiate a solver object. It takes the following arguments:

solver = qss.QSS(data,
                 eps_abs=1e-4,
                 eps_rel=1e-4,
                 alpha=1.4,
                 rho=0.1,
                 max_iter=np.inf,
                 precond=True,
                 reg=True,
                 use_iter_refinement=True,
                 polish=False,
                 verbose=False,
                 )

Use the solve() method when ready to solve:

results = solver.solve()

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 'r' should be numpy arrays, and 'r' should be a scalar.

  • 'g' is a list of separable function definitions. Each separable function is declared as a dictionary with the following keys:

    • 'func_name': 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.
• 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.
• polish: boolean specifying whether to attempt to polish the final solution. Still in development, best left as False for now.
• 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 >= 0)
• "is_neg": g(x) = I(x <= 0)
• "is_bound": g(x; lb, ub) = I(lb <= x <= 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 if x == 0, 1 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 if |x| <= M, 2M|x| - M^2 else}
  • M > 0 is a scalar.
  • Default: M = 1.
• "is_int": g(x) = I(x is an integer)
• "is_finite_set": g(x; S) = I(x is 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

minimize 0.5 * ||Gx - h||_2^2 subject to x >= 0

qss can be used to solve this problem as follows:

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["A"] = sp.sparse.csc_matrix((1, p)) # All zeros meaning no constraints
data["b"] = np.zeros(1)
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

And install qss in development mode:

pip3 install -e ./ -r requirements.txt

Finally, test to make sure the installation worked:

pytest tests/