Questions tagged [column-generation]
Questions on algorithms that solve mathematical models by generating columns (variables) in the pricing step during solution on demand.
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In column generation is it possible that a new column from sub problem doesn't enter master's base?
In some iterations, when a new column is added to the master, it does not enter the base and a stop condition occurs, or a new column with a better objective function is found. is it correct?
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Partially modify LHS of a constraint set?
I have implemented a column generation algorithm in Gurobi in Python. The most computationally expensive aspect of this algorithm is the reconstruction of the master problem when a column is added (or ...
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Is there existing code for the set partitioning formulation of routing problems?
The set partitioning formulation of routing problems is pretty well-accepted as perhaps the best exact method to solve routing problems. However, these methods are difficult to implement as many ...
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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 (...
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Question about Pricing Problem in Column Generation
In column generation, we need to solve the following pricing problem :
$$\min c_j-\bf{c}^T_B\bf{B}^{-1}\bf{N}_j$$
In the book, I saw authors say that according to duality theory, $\bf{y}^T = \bf{c}^...
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How should one proceed with column generation when the subproblem generates only columns with positive reduced costs?
I try to solve a MILP with Column generation. The Master Problem is a minimization problem with " $\le$ " constraint which lead to non-positive dual values. The problem is that the ...
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Why the column generation algorithm remains stuck at the same solution?
I was trying to solve a MILP by a column generation algorithm, and I noticed that the algorithm was stuck at the same solution, I tried to diversify the pool of initial columns, but the problem I ...
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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 ...
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Can one strengthen the Lagrangian dual bound in column generation when there are multiple subproblems?
Consider a Linear Programme (LP)
\begin{align}
\min && \sum_{i \in I} c_i x_i \\
\text{s.t.} && \sum_{i \in I} a_{ij} x_i &\geq b_j & \quad & \forall j \in J \\
&& ...
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Combination of lagrangian relaxation and column generation
I am solving an integrated scheduling problem and have dealt with coupling constraints using Lagrangian relaxation to decompose the problem into two separate problems. However, it is still difficult ...
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Quality of the defined initial solution in column generation
In the context of column generation, specifically, price and branch or delayed column generation like Gilmore-Gomory (GG-CG) procedure, it needs to define an initial feasible solution as a trigger of ...
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Dual prices remain stable but negative in column generation
I arrive at an objective function value that I know to be optimal. However, dual prices remain negative (which leads me to believe another pattern can be added, but when I do, the dual prices don't ...
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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, ...
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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 ...
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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 ...
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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+...
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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 ...
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Adding variable to constraint in pyomo
I am trying to implement a delayed column generation approach in pyomo.
However, I cannot find a way to add a variable to a constraint. I can add variables to the objective function, simply by adding ...
5
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1
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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 ...
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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, ...
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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 ...
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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 ...
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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^...
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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) ...
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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 ...
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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 ...
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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 ...
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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 ...
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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? ...
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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}^{'} | \...
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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 ...
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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 ...
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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 ...
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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}\\
...
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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 ...
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Set Covering and Column Generation
I need some set covering and column generation examples in OR-Tools python.
If anyone can provide or direct me to some resources that would be great..!
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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 ...
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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 ...
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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 ...
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Materials or sample code for column generation book (GERAD 25th Anniversary Series)
As far as I know, one of the interesting resource to learn the basic concept of the decomposition methods is column generation book (GERAD 25th Anniversary Series), which has been mentioned in some ...
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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 ...
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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'...
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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?
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 = \...
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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 ...
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