pyMIQP

This is a Mixed Integer Quadratic Programming solver for python exploiting sparsity.

Internally it uses CoinOR's general MINLP-solver Bonmin (which uses other CoinOR projects like Cbc and Ipopt) and prepares the necessary internals tuned for instances of Quadratic Programming like structure- and function-definitions including Jacobian / Hessian of the Lagrangian information).

In general, Bonmin tackles MINLP (Mixed Integer NonLinear Programming) problems which is more general than MIQP (Mixed Integer Quadratic Programming) problems, but the performance, when specialized commercial solvers (Gurobi, CPLEX, Mosek; some potentially limited to CMIQP -> convex) are unavailable, is good (only open-source + free solver considered in this Benchmark)!

For convex problems, Bonmin guarantees a global-optimum, while it only guarantees a local-optimum for nonconvex problems.

Status

Prototype

It's working for small constrained examples. But heavy-testing has not been done yet and it's to be expected, that there are bugs.

I'm rediscovering C++ right now and the C++-part in this project is sub-optimal at least!

Problem-definition

min           c.T x + 0.5 * x.T Q x
subject to:   glb <= A x <= gub
               lb <=   x <=  ub

              x_i in Z for all i in I and,
              x_i in R for all i not in I.

with:
      c:           n   real vector
      Q: symmetric n*n real matrix
         convex if Q is positive-semidefinite
      A:           m*n real matrix
      glb:         m   real vector
      gub:         m   real vector
       lb:         n   real vector
       ub:         n   real vector

From this follows, that only linear-constraints are supported (limited by this wrapper; not due to Bonmin)!

Implementation Overview

This wrapper uses pybind11 to combine C++ and Python.

The setuptools-based install is based on pybind/python_example.

C++Side

Eigen is used for most (sparse) calculations.

Python-side

The implementation assumes the usage of sparse-matrices (for Q + A), provided by scipy.sparse (see tests) and numpy-arrays (for vectors) provided by numpy.

Usage

See tests folder. E.g. test_0.py:

import numpy as np
import scipy.sparse as sp
from pyMIQP import MIQP

"""
Example from OPTI:
    https://www.inverseproblem.co.nz/OPTI/index.php/Probs/MIQP
    Example 1: Small MIQP
"""

Q = sp.csc_matrix(np.array([[1,-1],[-1,2]]))
c = np.array([-2, -6])

A = sp.csc_matrix(np.array([[1,1],[-1,2], [2,1]]))
glb = np.array([-np.inf, -np.inf, -np.inf])
gub = np.array([2,2,3])

xlb = np.array([0,0])
xub = np.array([np.inf, np.inf])

# 0, 1, 2 = cont, bin, int
var_types = np.array([2, 0])

miqp = MIQP(hessian_approximation=False) # redundant: default -> exact-hessian
miqp.set_c(c)
miqp.set_Q(Q)
miqp.set_A(A)
miqp.set_glb(glb)
miqp.set_gub(gub)
miqp.set_xlb(xlb)
miqp.set_xub(xub)
miqp.set_var_types(var_types)
miqp.set_initial_point(np.array([0, 0])) # redundant: default init-vec -> zeros
miqp.solve(algorithm="B-Hyb")            # redundant: default algorithm -> B-Hyb

print('sol-status: ', miqp.get_sol_status())
print('sol-obj: ', miqp.get_sol_obj())
print('sol-x: ', miqp.get_sol_x())
print('sol-time (ms): ', miqp.get_sol_time())

Install

Only the combination of Linux (Ubuntu-like) and python3 was tested!

Make sure to respect all those licenses (a lot of software involved).

Install Bonmin (system-wide)

• Prepare your system to be able to build software (sudo apt install build-essential and co.)
  • The following might suffice (no guarantees): sudo apt-get install gfortran gcc g++ autotools-dev automake autoconf
• Grab a release (e.g. from github)
  • Release because: we need the ThirdParty folder (not shipped with all kinds of source-grabbing approaches)!
  • We are approaching a free open-source build here (Cbc shipped; Ipopt shipped; ```Mumps````used as Linear-solver; will be grabbed)
    • Other configs were not tested!
  • cd to ThirdParty/Blas
  • ./get.Blas
  • cd to ThirdParty/Lapack
  • ./get.Lapack
  • cd to ThirdParty/Mumps
  • ./get.Mumps
  • cd to core-dir
  • ./configure --prefix=/usr --enable-cbc-parallel --enable-gnu-packages
    • (global-install is used so that bonmin's lib is automatically found)
  • make -j 4
  • (make test)
  • sudo make install

Grab Eigen

• Eigen is a header-only library and it's expected that the folder can be found at: src/Eigen
  • src/Eigen/src/Eigen is available

Install project

• sudo python3 setup.py install

ToDo

• C++ style-formatting: very ugly mixed-style code for now
• C++ code quality improvements: partially non-nice C++ Code
• Support more of Bonmin's options
• Support options to control verbosity (global verbose-switch only for now)
• Error-handling
• Tests
• (maybe) consider supporting MIQCQP (quadratic constraints) or MISOCP (second-order cone constraints)
• Provide Hessian-calculations to not use Ipopt's Hessian-approximation
• Write an interface for cvxpy's newly introduced QP-pipeline
  • Done

Alternatives

The only alternatives known to the author:

• Pyomo
  • Very complete and mature software with sparse documentation
  • Not limited to QPs
  • As far as the author knows: Bonmin-usage completely based on AMPL Solver Library (file-based?)
  • Usage resembles mathematical-modelling languages
• CasADi
  • Very complete software
  • Not limited to QPs