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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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 ...
nflgreaternba's user avatar
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1 answer
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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 ...
nflgreaternba's user avatar
1 vote
1 answer
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Exploiting identical subproblems in column generation

As suggested, this is a new post. I have the following column generation model. With the following master problem $MP$: \begin{align} &\text{minimize} &\sum_t \sum_s \text{slack}_{ts} \\ &\...
nflgreaternba's user avatar
3 votes
1 answer
272 views

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 ...
nflgreaternba's user avatar
1 vote
1 answer
87 views

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}...
nflgreaternba's user avatar
2 votes
1 answer
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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 ...
Sina_Alef's user avatar
1 vote
1 answer
105 views

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,...
Sina_Alef's user avatar
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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, \...
CHE's user avatar
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1 answer
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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 ...
mdslt's user avatar
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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?
user12849's user avatar
1 vote
1 answer
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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 ...
mdslt's user avatar
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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 ...
IsalanOnkar's user avatar
4 votes
2 answers
231 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 (...
Rom's user avatar
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1 vote
1 answer
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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}^...
Lin Sen's user avatar
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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 ...
Nada.S's user avatar
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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 ...
Nada.S's user avatar
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1 answer
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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 \\ && ...
1 vote
2 answers
100 views

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 ...
XXia's user avatar
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4 votes
1 answer
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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 ...
Инженер человеческих душ's user avatar
1 vote
2 answers
181 views

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 ...
A.Omidi's user avatar
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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 ...
user11638's user avatar
2 votes
1 answer
144 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, ...
West Wind's user avatar
4 votes
1 answer
126 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+...
Romio Rodriguez's user avatar
4 votes
1 answer
361 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 ...
133crem's user avatar
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2 votes
1 answer
549 views

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 ...
Sune's user avatar
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5 votes
1 answer
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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 ...
mars's user avatar
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2 votes
1 answer
178 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, ...
Eduardo Silva's user avatar
7 votes
1 answer
425 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 ...
J. Dionisio's user avatar
8 votes
1 answer
266 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 ...
user123456's user avatar
4 votes
1 answer
224 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^...
mdslt's user avatar
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6 votes
0 answers
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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) ...
Enthusiast's user avatar
4 votes
3 answers
205 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 ...
Christian Komusiewicz's user avatar
10 votes
1 answer
404 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 ...
pele's user avatar
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6 votes
2 answers
459 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 ...
mars's user avatar
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6 votes
0 answers
104 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? ...
gmn's user avatar
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8 votes
1 answer
251 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 ...
gmn's user avatar
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9 votes
1 answer
282 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 ...
Dr. Soomin Woo PhD's user avatar
5 votes
0 answers
77 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}^{'} | \...
mdslt's user avatar
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6 votes
2 answers
532 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 ...
gmn's user avatar
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4 votes
1 answer
194 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 ...
lobstersalesman's user avatar
6 votes
1 answer
403 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}\\ ...
pozyavas's user avatar
6 votes
2 answers
269 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 ...
KGM's user avatar
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1 vote
0 answers
303 views

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..!
user3237357's user avatar
7 votes
1 answer
82 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 ...
Alberto Santini's user avatar
5 votes
0 answers
117 views

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 ...
A.Omidi's user avatar
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4 votes
1 answer
641 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 ...
independentvariable's user avatar
6 votes
2 answers
578 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'...
Bhr's user avatar
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5 votes
1 answer
882 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?
Sina Haji Kazemi's user avatar
4 votes
1 answer
261 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 ...
Bhr's user avatar
  • 455
6 votes
1 answer
285 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 ...
Bhr's user avatar
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