pyDualBB

This python module provides python bindings for a so-called dual Branch-and-Bound (dualBB) algorithm in order to solve convex mixed-integer quadratically constrained quadratic programs of the form:

min  f(x) = 0.5 * x' * Q0 * x + c0' * x + r0
 x
s.t. g_i(x) = 0.5 * x' * Qi * x + ci' * x + r0 <= 0 for all i = 1, .., p
      h1(x) = A1*x - b1 <= 0
      h2(x) = A2*x - b2 == 0
      x_i integer for all i in I

for a small number p, i.e. a small number of quadratic constraints.

Here,

• all the matrices Q0, ... Qp are assumed to be symmetric positive definite, i.e. f and g_i are strictly convex.
• the matrices A1 and A2 have dimensions (m1, n) and (m2,n) with m2 <= n and A2 has full rank.

The algorithm is a multi-threaded and high-performance version of the original implementation. See here for a detailed description of the algorithm.

It is written in C++ and builds on CPLEX' generic callbacks. The packages requires an installation of Cmake (>= 3.16), a decent C++ compiler supporting C++17 and an installation of Cplex 20.10.

Install

Clone this repo and run

python3 setup.py install

inside the repo folder.

Example

import numpy as np
from pyDualBB import dualBB

# Set the cplex options for dualBB
# See the Cplex documentation for the specific values
cplex_options_dualbb = {
    "mip_strategy_nodeselect": 2,       # best estimate search
    "mip_strategy_search": 0,           # let cplex choose (default)
    "mip_strategy_heuristiceffort": 1,  # default
    "timelimit": 1800                   # 30 min
}

# Path to the problem as .lp / .mps or .mps.gz file
problem_file = "problem.mps.gz"
# Path to the .json file containing all matrices and vectors
json_file = "problem.json"

# Fixed lagrangian multipliers
alphas = np.array([0.01, 0.025, 0.05, 0.1, 0.2, 0.95])

# set the number of threads used for dualbb (all other available threads will be used by Cplex)
dualbb_threads = 4

status, total_runtime, incumbents, incRuntimes = dualBB(problem_file, json_file, alphas, options=cplex_options_dualbb, dual_branching_threads=dualbb_threads)

# Always cut-off the branch with higher dual bound (only for demonstration purposes!)
status, total_runtime, incumbents, incRuntimes = dualBB(problem_file, json_file, alphas, options=cplex_options_dualbb, only_single_branch=True, dual_branching_threads=dualbb_threads)

To do:

• Provide a matrix interface for dualBB instead of using .mps and .json files
• Parse all model matrices directly from the .mps/.lp/.mps.gz file