Questions tagged [column-generation]
Questions on algorithms that solve mathematical models by generating columns (variables) in the pricing step during solution on demand.
66
questions
26
votes
5
answers
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views
Solving pricing problem heuristically in column generation algorithm for VRP
In the set covering/column generation approach for the VRP (Balinski and Quandt (1964), or e.g. this tutorial), the basic idea is:
Generate some routes.
Solve the set covering problem using those ...
- 13.1k
23
votes
9
answers
699
views
Reference for column generation applications
When talking about column generation algorithms, the main example is the cutting stock problem. I'm aware that variations of vehicle routing problem (VRP) can be solved using a column generation ...
- 5,864
20
votes
3
answers
432
views
Variable bounds in column generation
Consider the set covering problem
\[
\begin{align}
\min&\ \sum_{j=1}^nc_jx_j\\
s.t.:&\ \sum_{j=1}^na_{ij}x_j\geq 1,\quad \forall i=1,\dots,m\\
&\ 0\leq x_j \leq 1
\end{align}
\]
...
- 1,133
19
votes
2
answers
399
views
Automating the column generation decomposition process
When trying a decomposition technique such as column generation, most of the times my approach is to look at the problem and then:
Decide what a column should represent
Write the Master Problem
Write ...
- 293
19
votes
3
answers
1k
views
Column generation stabilisation
Has anyone performed a benchmark of the various stabilisation techniques in column generation? Which ones perform better for set pack/partition/cover problems like VRP? And are there theoretical ...
- 1,235
18
votes
1
answer
260
views
Optimality in a simultaneous column and row generation procedure
What is the optimality argument in a simultaneous column and row generation procedure? By column and row generation procedure I mean a procedure in which every time a column in generated, several ...
- 1,345
15
votes
2
answers
931
views
Column generation for TSP
For teaching purposes, I would like to solve the Travelling Salesman Problem with a column generation approach.
In the academic literature, an approach is proposed (for example here), where columns ...
- 12.9k
14
votes
4
answers
585
views
Online Education for OR and Developing Decision Support Systems
I am looking for educational programmes which can be conducted online, such as
full MSc., degree certificates,
postgraduate courses/modules,
MOOCs.
Topics I am looking for are on advanced ...
- 317
14
votes
2
answers
769
views
Obtaining optimality gaps when using hybrid exact-heuristic approaches to vehicle routing problems
I'm starting to read about column generation-based approaches to vehicle routing problems (VRP). Let's say that I want to solve very large instances of an intricate VRP, I'm not looking to always ...
- 383
14
votes
1
answer
597
views
Tools to implement branch-and-price algorithms
As far as I know, implementing a branch-and-price algorithm is a task far from trivial. However, there are tools such as SCIP or the BCP framework of COIN-OR that help implement such algorithms.
I ...
- 383
13
votes
1
answer
795
views
Branch and Price algorithm is exact?
I know that the Column Generation algorithm delivers an exact solution when you are solving a linear programming optimization problem. I want to know that, does this column generation approach deliver ...
- 233
12
votes
1
answer
524
views
When using column generation, can a sub-problem be ignored once and for all once it fails to generate a new column?
Suppose you are solving a minimization problem with a column generation procedure, and that you have $2$ sub-problems. At some point, one of the $2$ sub-problems is no longer able to generate a column ...
- 12.9k
12
votes
1
answer
138
views
Stabilization techniques in column generation when subproblems are not solved optimally
Which stabilization techniques in column generation, like penalty function approaches and dual price smoothing, can be used when subproblems are not solved optimally? (I am particularly interested in ...
10
votes
1
answer
366
views
Integer column generation without branch & price
Consider the following situation. I have an integer program which I want to solve using column generation. After a suitable decomposition, the master problem has decision variables that select the ...
- 123
10
votes
1
answer
312
views
DOCplex adding columns
When using DOCplex to implement column generation, is it possible to add columns as an object, as opposed to creating a variable and then modifying the coefficients in the constraints?
Edit: In ...
- 1,345
9
votes
1
answer
267
views
Odd behavior in the branch bound and price algorithm with column generation
I have two questions on branch-bound-and-price.
Why does it sometimes fail? For instance, the paper by Dabia et al shows results where the proposed algorithm fails for some cases. The paper does not ...
9
votes
0
answers
93
views
Adjusting duals heuristically
I am solving a scheduling problem-to find shifts and task schedules- using column generation. In essence, it is a set covering problem with additional constraints. The problem seems to be that the ...
8
votes
1
answer
234
views
Why is the tailing off effect only a problem in column generation?
Why is the tailing off effect only a problem in column generation? If all columns were pregenerated, and one used the simplex method, wouldn't one see the tailing off effect?
Is it simply not an issue ...
- 622
8
votes
1
answer
265
views
Column generation: decreasing value of restricted master problem
I am using column generation to solve a minimization problem.
At a given iteration, my subproblem finds a column with reduced cost $-1$, and in the following restricted master problem, this new column ...
- 81
7
votes
1
answer
407
views
Can't solve a specific instance of Gilmore & Gomory Cutting Stock with Delayed Column Generation
I am now somewhat comfortable with delayed column generation, but one specific example has been bugging me, as it's quite simple (on the second iteration we stop) and I can't reach the optimal ...
- 516
7
votes
1
answer
80
views
Branch and price with two sets of exponential variables
There are classes of problems which allow for "natural" extended formulations using multiple exponential-size sets of variables.
For example, 2-echelon routing problems can be modelled ...
- 2,073
6
votes
2
answers
560
views
Branch and Price Algorithm
Can branch and price be a good solution approach for a routing problem with min-max objective function? For example, minimizing the max length of any vehicle route in a VRP.
In the literature, I haven'...
- 455
6
votes
2
answers
406
views
Metaheuristics vs Column generation for VRP
I am solving a vrp and its variants (pick up drop off, milk-run problem) using genetic algorithms. The issue that I am getting with the genetic algorithm is the randomness associated with the ...
- 619
6
votes
2
answers
264
views
Heuristic solution to the graph partitioning problem
I am working on a graph partitioning problem. A static column generation based solution was proposed in How to partition a graph with optimal number of groups?
But I need some MILP solver to solve ...
- 2,211
6
votes
1
answer
488
views
Column Generation algorithm for vehicle routing problem
I want to solve a VRP with a column generation algorithm. The objective of the problem is makespan minimization. In more detail, I want to minimize the arrival time of the last vehicle in the depot. I ...
- 455
6
votes
2
answers
489
views
What kind of data structure should be used to store labels when implementing a labeling algorithm?
The shortest path problem with resource constraints is a common subproblem when doing column generation. It is often solved with a labeling algorithm. The procedure is very well explained here and ...
- 622
6
votes
1
answer
262
views
Column generation approach for CVRP
I want to use a column generation based heuristic to solve a capacitated Vehicle Routing Problem. I know the basics of the algorithm but I don't have much experience in coding. is there any code about ...
- 455
6
votes
1
answer
201
views
When using column generation, can I delete a node with negative reduced cost from my subproblem?
I am solving a minimization problem with a column generation procedure. The master problem is of the form
$$
\min \sum_{i\in \Omega}c_i \lambda_i
$$
subject to
$$
\sum_{i\in \Omega \mid v \in i } \...
- 12.9k
6
votes
1
answer
395
views
Dual of a model to obtain reduced costs
I have the following model which I am going to solve with column generation.
\begin{align}
\max & \sum_{b \in B} \sum_{s \in S} \sum_{r \in \Omega_s}\beta_{bs}p_r y_{br}\label{objective-set1}\\
...
- 61
6
votes
0
answers
207
views
Benders decomposition for a dense MILP
I am trying to solve a large MILP, but it seems like dense problems can be very difficult for moderns solvers. I tried to solve the problem described below considering only constraints (1) and (2) ...
- 109
6
votes
0
answers
103
views
Dual instability, degeneracy, tailing off effect - Which are the causes and which are the effects?
Dual instability, degeneracy, and the tailing off effect are often mentioned together in papers. However, I cannot seem to find a clear explanation on which of these cause the other and vice versa? ...
- 622
5
votes
1
answer
259
views
A Question on A tutorial on column generation and branch-and-price for vehicle routing problems by Dominique Feillet
I am reading A tutorial on column generation and branch-and-price for vehicle routing problems by Dominique Feillet to learn the column generation approach, but I have a problem. in section 3.3 ...
- 455
5
votes
1
answer
300
views
Reduced cost in column generation
I am following this tutorial to learn column generation.
`
https://co-at-work.zib.de/slides/Donnerstag_24.9/cgbpdw-coatwork-annotated.pdf
I have a couple of questions I thought someone experienced in ...
- 619
5
votes
1
answer
830
views
Does pyomo support column generation?
I want to use column generation technique to solve cutting stock problem.
Is there any other python modeling library that supports column generation?
5
votes
0
answers
75
views
Linear programming approach to dynamic programming - an initial pair of state-decisions
I aim to solve the following Bellman equation:
\begin{equation}
v(\vec{s}) = \min_{\vec{x} \in \Xi_{\vec{s}}} \big\{c(\vec{s}, \vec{x}) + \lambda \times \sum_{\vec{s}^{'}\in S} p(\vec{s}^{'} | \...
- 573
4
votes
3
answers
203
views
Is this ILP formulation for Group Closeness Centrality a column generation approach?
I want to solve the Group Closeness Centrality problem where the input is a graph $G=(V,E)$ and integer $k$ and we want to find a vertex set $S$ of size $k$ minimizing the total distance of the ...
4
votes
1
answer
587
views
Column generation for a linear optimization problem
I have an LP that has exponentially many constraints, and just linearly many variables. The dual of the problem, therefore, has exponentially many variables, while just linearly many constraints. My ...
- 3,980
4
votes
1
answer
198
views
Column generation when intractable variables appear in the objective function
Is it possible to implement a column generation for a problem that the variables in the "complicating" constraint appear in the objective function?
Suppose the MIP is:
\begin{align}
z = \...
- 2,086
4
votes
1
answer
255
views
Column Generation algorithm
I want to solve a VRP with a column generation algorithm. The objective of the problem is makespan minimization. but there is a point in calculating the arrival time of the vehicle in each node. the ...
- 455
4
votes
1
answer
183
views
How do you know if Column Generation is any faster than standard MIP formulations?
I'm currently in the process of learning about column generation, so I apologize in advance if I show a gross lack of understanding about it.
Anyways, what I've gathered so far is that Column ...
4
votes
1
answer
220
views
Accelerating an integer programming model
I am working on a scheduling problem, where I am solving it through column generation. The pricing problem of this algorithm is an integer programming model as follows:
\begin{equation}
F_1 \Big\{V^...
- 573
4
votes
2
answers
188
views
Column generation: set partitioning vs set covering
I am working with a column generation algorithm and have noticed that convergence is much faster when my master is a set covering problem ($Ax\ge 1$) compared to when it is a set partitioning problem (...
- 41
4
votes
1
answer
68
views
Can dual values smoothing lead to generating duplicate columns?
When using smoothing for dual values stabilization in column generation, the duals used in the subproblem lie on the segment joining the stability center (inside the dual feasible region) and the ...
4
votes
1
answer
301
views
How do we branch a Cutting Stock problem using Branch and Price?
I'm sure this is a straightforward question, but since I started learning integer programming recently this isn't clear to me.
Consider solving a 1 dimensional cutting stock problem using delayed ...
- 41
4
votes
0
answers
85
views
Problem related to machine's performance while solving an LP-relaxation of a set-covering problem using the Column Generation
PS: I am familiar with performance variability encountered while solving large-scale MIPs. My question may not be explicitly related to it but maybe to the inefficiency of python packages.
I have ...
- 668
4
votes
1
answer
119
views
How to find the vectors to be added as the columns in the master problem of Dantzig-Wolfe Decomposition?
I have a Dantzig-Wolfe decomposition question with the following questions
\begin{align}
&Maximize: 2x_1 +3x_2+4x_3+2x_4 \\
s.t. \quad & x_1 +x_2+2x_3+x_4 \le 15\\
& x_1 +x_2+2x_3+...
3
votes
0
answers
359
views
On solving the Restricted Master Problem in Column Generation technique
I am working on developing a column generation (CG) based optimization framework for a large-scale airline crew pairing problem (a set-covering problem). First, I generate an initial feasible solution ...
- 668
2
votes
1
answer
103
views
General question regarding column generation
Assume one have an integer problem.
Is the master problem in the column generation algorithm the original integer problem, or its linear relaxation?
Thank you in advance.
2
votes
1
answer
133
views
Must the Newly Generated Column be used in RMP in the Column Generation Method?
When I used Column Generation, I found a column with reduced cost < 0, and added it to the Restricted Master Problem.
Must the newly added column be used in the solution of RMP at once? If not, ...
- 21
2
votes
1
answer
161
views
Dantzig-Wolfe decomposition and optimal integer solutions
It is well known that applying the Dantzig-Wolfe Decomposition in the one period cutting stock problem with trim loss minimization leads to an optimal real solution that shares the round up property, ...
- 153