Fix and Bound: An efficient approach for solving large-scale quadratic programming problems with box constraints

This repository contains the source code of the B&B algorithm described in the paper "Fix and bound: an efficient approach for solving large-scale quadratic programming problems with box constraints" for solving non-convex BoxQPs to global optimality:

$$ \begin{align*} \min_{} \quad & \frac{1}{2} {\bf x }^\top {\bf Q} {\bf x} + {\bf c}^\top {\bf x} \\ \textrm{s.t.} \quad & {\bf x}\in [0, 1]^n,\\ \end{align*} $$

where ${\bf Q}\in \mathbb{R}^{n\times n}$ is symmetric and not positive semidefinite and ${\bf c}\in \mathbb{R}^n$.

M. Locatelli, V. Piccialli, A. M. Sudoso (2024). Fix and bound: an efficient approach for solving large-scale quadratic programming problems with box constraints. Mathematical Programming Computation, https://doi.org/10.1007/s12532-024-00270-y.

Installation

The B&B algorithm is implemented in C++/MATLAB and requires the following solvers:

• SDPNAL+
• Gurobi
• SNOPT

SDPNAL+ is called by using the MATLAB Engine API for C++, whereas Gurobi and SNOPT are called from MATLAB functions.

Ubuntu and Debian instructions:

1. Install CMake and Armadillo:

sudo apt-get update
sudo apt-get install cmake libarmadillo-dev

4. Open the makefile boxqp_cpp/Makefile
   • Set the variable matlab_path to the MATLAB installation folder.
5. Compile the code:

make

This code has been tested on Ubuntu 20.04 LTS with MATLAB R2020b Update 7, SDPNAL+ 1.0, Gurobi 10.0.2 and SNOPT 7.7. To speed up the B&B search, the code features a configurable pool of POSIX threads.

Configuration

Various parameters used in the B&B algorithm can be modified via the configuration file boxqp_cpp/config.txt:

• BRANCH_AND_BOUND_TOL - Optimality gap tolerance
• BRANCH_AND_BOUND_PARALLEL - Thread pool size: single thread (1); multi-thread (> 1)
• BRANCH_AND_BOUND_MAX_NODES - Maximum number of nodes
• BRANCH_AND_BOUND_VISITING_STRATEGY - Best first (0); depth first (1); breadth first (2)
• BRANCH_AND_BOUND_FIXING - Do not fix variables (0); try to fix variables (1)
• BRANCH_AND_BOUND_FIXING_TOL - Tolerance for fixing variables (only if fixing is enabled)
• MATLAB_SESSION_THREADS_ROOT - Number of threads for the MATLAB session at the root
• MATLAB_SESSION_THREADS_CHILD - Number of threads for the MATLAB session of child nodes
• MATLAB_SOURCE_FOLDER - Full path of the MATLAB folder containing source files (boxqp_matlab)
• SDP_SOLVER_TOL - Accuracy tolerance of SDPNAL+ in the relative KKT residual
• SDP_SOLVER_VERBOSE - Do not display SDP log (0), display SDP log (1)
• GUROBI_FOLDER - Gurobi installation folder
• SNOPT_FOLDER - SNOPT installation folder
• SNOPT_LICENSE - SNOPT license file (.lic)
• SDPNAL_FOLDER - SDPNAL+ installation folder
• CP_MAX_ITER - Maximum number of cutting-plane iterations
• CP_TOL - Tolerance between two consecutive cutting-plane iterations
• CP_MAX_INEQ - Maximum number of triangle inequalities to separate
• CP_PERC_INEQ - Fraction of the most violated triangle inequalities to add
• CP_EPS_INEQ - Tolerance for checking the violation of triangle inequalities
• CP_EPS_ACTIVE - Tolerance for detecting active triangle inequalities

Usage

./bb <config_file> <data_file> <log_file> <sol_file>

• config_file - configuration file
• data_file - problem data ${\bf Q}$, ${\bf c}$
• log_file - log file
• sol_file - optimal solution ${\bf x^\star}$

File data_file contains the problem data in the following format:

n
c_1 c_2 ... c_n
Q_11 Q_12 ... Q_1n
Q_21 Q_22 ... Q_2n
...
Q_n1 Q_n2 ... Q_nn

Log

The log file reports the progress of the algorithm at the node level:

• N - node size
• BIN - number of variables in set $N$
• PARENT - parent node ID
• NODE - node ID
• LB_PAR - lower bound of the parent node
• LB - lower bound
• TIME (s) - running time in seconds
• CP_ITER - number of cutting-plane iterations
• CP_FLAG - termination flag of the cutting-plane procedure
  • -1 - maximum number of iterations
  • 0 - node must be pruned
  • 1 - less than $10n$ triangle inequalities
  • 2 - 3 - lower bound not significantly improved
• CP_INEQ - number of inequalities added in the last cutting-plane iteration
• SDP_FIX - number of SDPs solved for fixing variables
• TIME_FIX - time spent in seconds for fixing variables
• N_FIX - number of fixed variables
• UB - current upper bound
• GUB - global upper bound
• I_FIX - current branching decision
• NODE_GAP - gap at the node
• GAP - overall gap
• OPEN - number of open nodes

At the end of the file, it shows the statistics:

• TIME - overall computational time in seconds
• NODES - overall number of nodes
• ROOT_GAP - gap at the root node
• GAP - overall optimality gap
• OPT - optimal objective function value

Log file example:

DATA_PATH: /boxqp_instances/spar200-075-2.in
LOG_PATH: /log_spar200-075-2.txt

BRANCH_AND_BOUND_TOL: 0.0001
BRANCH_AND_BOUND_PARALLEL: 2
BRANCH_AND_BOUND_MAX_NODES: 1000
BRANCH_AND_BOUND_VISITING_STRATEGY: 0
BRANCH_AND_BOUND_FIXING: 1
BRANCH_AND_BOUND_FIXING_TOL: 0.01
MATLAB_SESSION_THREADS_ROOT: 4
MATLAB_SESSION_THREADS_CHILD: 2
MATLAB_SOURCE_FOLDER: /bb-boxqp-fixing/boxqp_matlab/
SDP_SOLVER_TOL: 0.0001
SDP_SOLVER_VERBOSE: 0
CP_MAX_ITER: 10
CP_TOL: 0.0001
CP_MAX_INEQ: 100000
CP_PERC_INEQ: 0.1
CP_EPS_INEQ: 0.0001
CP_EPS_ACTIVE: 0.00
GUROBI_FOLDER: /gurobi951/
SNOPT_FOLDER: /snopt7_matlab/
SNOPT_LICENSE: /snopt7.lic
SDPNAL_FOLDER: /SDPNAL+/

|    N|  BIN|  PARENT|    NODE|      LB_PAR|          LB|  TIME (s)| CP_ITER| CP_FLAG|   CP_INEQ| SDP_FIX| TIME_FIX|   N_FIX|          UB|         GUB| I_FIX|     NODE_GAP|          GAP|  OPEN|
|  200|  126|      -1|       0|        -inf|    -22266.4|       364|       3|       1|     76669|       1|  58.1208|      19|      -22163|     -22163*|    -1|   0.00466673|   0.00466673|     0|
|  180|  106|       0|       1|    -22266.4|    -22223.6|       110|       1|       1|     59120|       0|        0|       0|      -22062|      -22163|    38|   0.00273636|   0.00273636|     0|
|  180|  106|       0|       2|    -22266.4|    -22210.9|       240|       2|       1|     60048|       1|  40.9975|      10|      -22163|      -22163|    38|   0.00215997|   0.00273636|     1|
|  179|  105|       1|       3|    -22223.6|    -22183.4|       156|       1|       1|     59324|       1|  38.8991|       2|      -22062|      -22163|   199|  0.000921238|   0.00215997|     2|
|  179|  105|       1|       4|    -22223.6|    -22187.3|       161|       1|       1|     59630|       1|  41.3265|       3|    -22099.4|      -22163|   199|    0.0010967|   0.00215997|     3|
|  169|   95|       2|       5|    -22210.9|    -22172.6|       188|       1|       1|     58267|       1|  60.8439|      17|      -22163|      -22163|    81|  0.000432555|    0.0010967|     4|
|  175|  101|       4|       6|    -22187.3|    -22153.1|        62|       0|       0|     56624|       0|        0|       0|    -22099.4|      -22163|    97| -0.000447315|    0.0010967|     5|
|  169|   95|       2|       7|    -22210.9|    -22171.6|       156|       1|       1|     55533|       1|  37.3501|       7|      -22136|      -22163|    81|  0.000385916|  0.000921238|     4|
|  175|  101|       4|       8|    -22187.3|      -22163|        57|       0|       0|     56624|       0|        0|       0|    -22095.4|      -22163|    97|  9.47342e-08|  0.000921238|     5|
|  176|  102|       3|       9|    -22183.4|    -22156.2|        60|       0|       0|     57021|       0|        0|       0|    -22056.5|      -22163|   119|  -0.00030788|  0.000432555|     4|
|  176|  102|       3|      10|    -22183.4|    -22157.1|        67|       0|       0|     57021|       0|        0|       0|    -22029.7|      -22163|   119|   -0.0002681|  0.000432555|     3|
|  151|   77|       5|      11|    -22172.6|    -22163.1|        58|       0|       0|     45312|       0|        0|       0|      -22163|      -22163|    18|  3.10148e-06|  0.000385916|     2|
|  151|   77|       5|      12|    -22172.6|    -22139.5|        55|       0|       0|     45312|       0|        0|       0|      -22138|      -22163|    18|  -0.00106123|  0.000385916|     1|
|  161|   87|       7|      13|    -22171.6|    -22145.9|        63|       0|       0|     49688|       0|        0|       0|    -22124.8|      -22163|    18| -0.000771785| -0.000771785|     0|
|  161|   87|       7|      14|    -22171.6|    -22149.5|        52|       0|       0|     49688|       0|        0|       0|      -22136|      -22163|    18| -0.000609419| -0.000609419|     0|

TIME: 1145 sec
NODES: 15
ROOT_GAP: 0.00466673
GAP: 0
OPT: -22163