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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Implementing column generation in SAS
I am currently trying to implement this model in SAS. Unfortunately, I always get this error. How can I solve this and finally get the final roster $x$ and the total value of $slack_{ts}$ (preferably ...
CommunityBot
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Exploiting identical subproblems in column generation
As suggested, this is a new post. I have the following column generation model. The notation is as follows: $i\in I$ refers to the nurses, $t\in T$ to the days and $s\in S$ for the shifts. With the ...
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Branch-and-Price for VRPs: Enforcing branching on arcs
The following is based on my understanding, which may have flaws, so please correct me if I claim something wrong.
When solving VRPs with Branch-and-Price, one can enforce integrality by branching on ...
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How to implement self-adjusting smoothing in column generation?
I am currently trying to implement the "Smoothing with a Self-Adjusting Parameter" methodology (on the right in Table 1) from Pessoa et. al (2018) in my MILP. Unfortunately, I am not quite ...
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Gurobi - Problem with adding a non-linear constraint to a column object
I have implemented the following Column Generation model in Gurobi. Unfortunately, I currently have the problem that the constraint in the master problem is not linear and therefore I cannot add the ...
CommunityBot
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Dual stabilization doesnt work properly
I am currently trying to speed up my column generation approach to reduce the tailing-off effect. I have modified my master problem as follows. For this I have introduced $\phi_{ts}^+,\phi_{ts}^-,\...
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How can I add columns from the subproblem to the master problem in my column generation model?
I have one last question about my column generation model. I am currently trying to implement it in code, and I am successfully solving the subproblem. So I can solve the dual values of the master ...
CommunityBot
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Determine whether column generation heuristic is optimal?
I am currently working on a machine scheduling problem formulated as MILP and trying to solve it with Column Generation. Since I do not use Branch-n-Price, but only Column Generation as heuristic, it ...
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How to improve a column generation algorithm
I have been busy the last few weeks implementing this model with Column Generation in Gurobi and then solving it. The whole thing is now working quite well and I have also run a few calculation ...
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What is delayed column generation?
I came across the term "Delayed Column Generation" yesterday and am wondering how this differs from classic Column Generation, or whether the terms are used synonymously.
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Add multiple columns to the Master Problem in a column generation approach
I have read several times now that with a column generation approach in the subproblem you do not necessarily have to look for the column with the lowest negative reduced cost, but only one with ...
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How to modify master problem and individual sub problems in column generation?
This is a follow-up post regarding this one. I deleted this new post once before, as I was unhappy with the formulation. I have the following basic nurse scheduling MILP, which tries to cover the ...
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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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Help with formulating the objective function of my subproblem
i have a pretty basic question regarding column generation. I have the following scheduling problem i would like to solve with column generation:
\begin{align}
&\min\sum_{t}^{}\sum_{s}^{}slack_{ts}...
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How to evaluate the quality of a solution obtained using the price-and-branch method for an IP problem?
I have solved an integer programming problem using the price-and-branch (not branch-and-price) approach exactly as same as described in this question and obtained a feasible solution. As this approach ...
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Column generation for solving minimum cost multicommodity flow problem with additional edge disjoint constraints
I'm working on an integer programming problem that's fundamentally a minimum cost multicommodity flow (MCMF) problem with additional constraints. I'm exploring the use of column generation to solve it,...
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Question on Column generation
I am implementing a Column Generation algorithm to solve a set partitioning problem. The master problem takes the form :
$\min \sum_{i \in I} c_i \lambda_i$
s.t
$\sum_{i \in I} a_{ji} \lambda_i = 1, \...
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Materials or sample code for column generation book (GERAD 25th Anniversary Series)
As far as I know, one of the interesting resources to learn the basic concept of the decomposition methods is column generation book (GERAD 25th Anniversary Series), which has been mentioned in some ...
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Termination criterion for the Phase 1 algorithm in column generation
To initialize column generation, I obtain a feasible set of columns via the following Phase 1 algorithm. The restricted master problem for this algorithm is formulated as follows.
\begin{equation}
min ...
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In column generation is it possible that a new column from sub problem doesn't enter master's base? [duplicate]
In some iterations, when a new column is added to the master, it does not enter the base and a stop condition occurs, or a new column with a better objective function is found. is it correct?
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Partially modify LHS of a constraint set?
I have implemented a column generation algorithm in Gurobi in Python. The most computationally expensive aspect of this algorithm is the reconstruction of the master problem when a column is added (or ...
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Is there existing code for the set partitioning formulation of routing problems?
The set partitioning formulation of routing problems is pretty well-accepted as perhaps the best exact method to solve routing problems. However, these methods are difficult to implement as many ...
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Column generation: set partitioning vs set covering
I am working with a column generation algorithm and have noticed that convergence is much faster when my master is a set covering problem ($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 ...
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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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