# GCG

Branch-and-Price & Column Generation for Everyone

Why use GCG?

On this page, we give a brief overview of what GCG is capable of (and what it is not).

# Why should you use GCG?

If you have solved mixed integer linear programs already, you will know that it can take forever to solve them, even if they are well thought out. Whether you did think your model through or you didn't, it will always be worth a try to solve them with GCG, because we deploy a variety of tools to possibly increase solving speed in orders of magnitudes.

## Introduction to GCG

GCG will decompose your problem into problems that can be solved far more easily (decomposition). Then, we apply a Dantzig-Wolfe reformulation to strengthen your formulation. This reformulation will convexify the "easy" constraints, meaning that the polyhedron (i.e. the space where your solution can lie in) is shrinked down as much as possible. Finally, we solve your problem using Branch-and-Price. In each node of the Branch-and-Bound tree we solve the pricing problems, getting variables that are promising.

### Features

As an extendible and easy-to-use toolkit, GCG...

• solves your problem using complex solving algorithmics without requiring any input or knowledge from your side
• has an explore function to investigate identified decompositions
• is a framework for implementing column generation algorithms For further use cases of GCG, please consult the User's Guide.

As a solver, GCG...

• detects hidden or apparent structures in the constraint matrix in order to apply DWR. Among others, it detects
• single-bordered structures
• staircase structures
• set partitioning master structures
• clustered and graph connectivity based structures
• uses a Dantzig-Wolfe reformulation (DWR) to solve arbitrary MIPs.
• automatically aggregates subproblems if possible
• offers an automated Benders' decomposition algorithm
• has branching rules for automatic branching on any problem, e.g. branching on original, Ryan-Foster branching or generic branching
• applies a wide variety of cuts to the problem, e.g. combinatorially or from basis)
• rates columns in its (parallel) exact or heuristic pricing
• has a large number of primal heuristics both in the original and the reformulated space
• applies dual variable stabilization by dual value smoothing

For more details on what and how GCG does things to solve your problem quickly, please consult the Developer's Guide.

## When is it sensible to use GCG?

In order for a decomposition to make sense, your problem has to exhibit some kind of structure. In many cases, it will have a certain structure (maybe even one that is not known to you), but if it does not have one at all, it can be detrimental to performance to try to use a decomposition anyways. However, if it has a structure there is a good chance that GCG will be able to solve it much faster than other state-of-the-art open-source solvers, such as SCIP, and often at least as fast as commercial solvers such as Gurobi or CPLEX.

GCG SCIP
SCIP Status : problem is solved [optimal solution found]
Solving Time (sec) : 28.48
Solving Nodes : 1
Primal Bound : -4.10000000000000e+01 (5 solutions)
Dual Bound : -4.10000000000000e+01
Gap : 0.00 %
SCIP Status : problem is solved [optimal solution found]
Solving Time (sec) : 337.72
Solving Nodes : 1088063 (total of 1088983 nodes in 2 runs)
Primal Bound : -4.10000000000000e+01 (389 solutions)
Dual Bound : -4.10000000000000e+01
Gap : 0.00 %