Questions tagged [benders-decomposition]

For questions related to Benders decomposition, a type of optimization algorithm in which certain variables are optimized in a "master problem," the values of those variables are fixed, the remaining variables are optimized in a "subproblem," cuts are generated to be added to the master problem, and the method repeats.

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Stochastic programming MIP solvers

I am aware that Benders Decomposition is readily available in CPLEX and in SCIP; but are there any (free) solvers that provide off the shelf stochastic programming MIP algorithms or a nice to work ...
Albert Schrotenboer's user avatar
24 votes
4 answers
731 views

How to determine if a given problem seems to be a good fit to be solved using combinatorial Benders decomposition

Combinatorial Benders decomposition is a mathematical programming technique consisting into dividing a problem into a master problem and a sub problem. The master problem is solved to optimality (or ...
Renaud M.'s user avatar
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18 votes
4 answers
2k views

Relationship between Benders decomposition and Dantzig-Wolfe decomposition

It’s often said that “Benders decomposition is Dantzig-Wolfe applied to the dual”. How can this statement be made precise? I know that in Dantzig-Wolfe, cuts are added in one-to-one correspondence ...
David M.'s user avatar
  • 2,077
18 votes
2 answers
659 views

What are the modern optimization methods for large systems?

I came across the preface of Optimization Theory for Large Systems (you can read it in Amazon). The author claims in the table (page v) that some of the methods such as Dantzig Wolfe decomposition, ...
Antarctica's user avatar
  • 2,917
15 votes
2 answers
4k views

Is Apple's M1 suitable for Operations Research?

I am curious about the performance of Apple's M1 chip solving optimizations models, MIP, LP, and in solutions approach as benders or columns generations. I read that is a spectacular cpu to perform ...
orpanter's user avatar
  • 517
14 votes
3 answers
1k views

How to handle an IP sub-problem with an objective function in Benders Decomposition

I have a question on Benders Decomposition (BD). Suppose I have an MILP model which can be decomposed into a master problem (MP) including integer and continuous variables and a subproblem (SP) ...
whitepanda's user avatar
12 votes
1 answer
144 views

Benders subproblem feasible region dependent upon solution master problem

Suppose I want to solve a naturally MINLP problem of the following form: $$ \min_{x,y} \{c'x + y \mid Ax \leq b, Dx + Ey \leq f, G(x)y\leq g, x \in \mathbb{Z}, y \in \mathbb{R}^+\} $$ Here $G(x)$ ...
Albert Schrotenboer's user avatar
12 votes
1 answer
175 views

Improving cuts from sub-problem with problem-specific hierarchical information

I'm solving an assignment-alike problem with a Logic-based Benders decomposition-alike (LBBD) method. The master problem provides an assignment, which is checked in the sub-problem. Define the set of ...
Jasper's user avatar
  • 221
11 votes
2 answers
626 views

CPLEX Auto-Benders: How do I get the number of optimality and feasibility cuts?

I am using CPLEX's 12.9 auto-Benders decomposition feature (from the CPLEX Java API). Following the documentation I let CPLEX decide the decomposition strategy as ...
k88074's user avatar
  • 1,661
9 votes
3 answers
1k views

Is Dantzig-Wolfe and Benders' Decomposition still applied in Operations Research?

A year ago, I had taken my master's degree class Optimisation Theory, and we were learning Dantzig-Wolfe Decomposition and Benders' Decomposition. I found it very challenging to use these algorithms ...
dozgunay's user avatar
  • 139
9 votes
0 answers
200 views

Ill-conditioned LP in Benders decomposition

I have implemented a Benders decomposition for a constrained network flow but the LP solver (Gurobi) warns me of the ill-conditioning of the subproblem dual LP. As you can see below, the coefficients ...
Mauricio Zambon's user avatar
7 votes
3 answers
1k views

Textbook recommendation for linear programming decomposition fundamentals

I am looking for a textbook on linear programming decomposition fundamentals. The book should be clear and easy-to-follow for self study and should include examples to illustrate the concepts.
DSPinfinity's user avatar
7 votes
2 answers
346 views

Do Benders cuts exclude current solutions?

I am wondering if optimality cuts in Benders algorithm exclude the possibility to have the same solutions and as a result, have the same optimality cut? I don't know why it is not possible to have the ...
Amin's user avatar
  • 2,150
7 votes
2 answers
906 views

How to find extreme rays

I am applying Benders decomposition and the dual is unbounded. I need to find the extreme rays to proceed, but I am not sure how to do that. Following is an example problem, can someone explain how ...
John Bolton's user avatar
7 votes
1 answer
740 views

Classical Benders decomposition algorithm implementation details

Given the following problem: \begin{align} P: \min_{x,y}&\quad c^\top x+f^\top y\\ \text{s.t.}&\quad Ax+By=b\\ &\quad y\in Y\\ &\quad x\geq 0 \end{align} Problem $P$ is equivalent to: \...
Joris Kinable's user avatar
7 votes
2 answers
361 views

Logic-based Benders decomposition with integer master variables

I am looking for a paper/study in which a Logic-based Benders decomposition (LLBD) framework is used when the master problem is associated with integer variables (not specifically binary). In general ...
whitepanda's user avatar
6 votes
2 answers
365 views

Lagrangian Relaxation for Two-Stage Stochastic Program

I have a two-stage stochastic program as follows: \begin{align}\max&\quad f^\top y+\sum_{s}p_sc_s^\top x_s\\\text{s.t.}&\quad Ay=b\\&\quad W_sX_s+Ty \le h_s \quad \forall s \in S \\&\...
Amin's user avatar
  • 2,150
6 votes
2 answers
427 views

How to handle bigM in sub-problem of benders decomposition?

Suppose you want to solve a MIP with Benders decomposition and the binary variables ($y_i$) are fixed in the master problem but these variables are used in the sub-problem with bigM like $x_{ij} \le M....
Amin's user avatar
  • 2,150
6 votes
0 answers
210 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) ...
Enthusiast's user avatar
5 votes
1 answer
407 views

Optimality in L Shaped or Bender Decomposition

I was working on solving a two-stage stochastic problem using L Shaped method (Benders Decomposition). I have discussed the model here: Stochastic Facility Location Model. Do the single-cut/ multi-cut ...
mars's user avatar
  • 629
5 votes
3 answers
409 views

Combinatorial Feasibility Cuts for Benders Decomposition

Are there any advantages of adding constraints in the Benders master problem that ensure the feasibility of the subproblems? This would not add any feasibility cuts. Or is it beneficial to have a ...
Amogh Bhosekar's user avatar
5 votes
1 answer
367 views

Dantzig-Wolfe vs Benders Decomposition on the dual problem - Computational differences

My question is a follow-up to this one: Relationship between Benders’ decomposition and Dantzig-Wolfe decomposition. Here what is being discussed is the relationship between the two methods, and it is ...
J. Dionisio's user avatar
4 votes
2 answers
336 views

Benders subproblem with product of continuous and discrete variables

I am trying to solve the following problem. The decisions in the problem are $x, y, v, $ and $W$, where $x, y$ are binary and $v, W$ are continuous variables. \begin{equation}\label{eq:3} \begin{...
Pramesh Kumar's user avatar
4 votes
1 answer
389 views

Is Benders decomposition and the L-shaped method the same algorithm?

I've been studying the Benders decomposition method to solve stochastic integer problems. I've also stumbled across papers using a so called L-shaped-algorithm which also divides into master problem ...
Arctic_Skill's user avatar
4 votes
2 answers
177 views

How to fix unbalanced multi-commodity network flow with equal supply and demand?

I have a fairly large network with eleven commodities and arc capacities that are commodity-dependent (i.e. an arc may have a higher capacity for one commodity than another). I'm solving a protection-...
Emma Kuttler's user avatar
4 votes
1 answer
112 views

Benders Decomposition Implementation - Should I keep the old feasibility cut in master problem?

I am currently working on combinatorial subproblems. First, I need to solve the master problem, find all possible subproblems, solve each combination individually, and ensure all the subproblems are ...
Zain's user avatar
  • 41
4 votes
1 answer
156 views

What's the best way to speed up Benders Decomposition for a stochastic vehicle routing problem?

Currently I am working on an implementation of Benders Decomposition that solves a stochastic vehicle routing problem with synchronisation constraints. Sadly, at the moment it is not performing fast ...
Arctic_Skill's user avatar
3 votes
3 answers
772 views

Unbounded master problem in Benders decomposition

After a few iterations, my master problem with optimality cuts is still unbounded. I wonder if it's possible in theory? If it's possible, how to deal with the unbounded master problem?
htam_ujn's user avatar
3 votes
1 answer
537 views

Benders decomposition feasibility/ optimality cuts

I am trying to understand Benders Decomposition method. I am reading this book Decomposition techniques in mathematical programming by A Conejo, E Castillo, R Minguez. The book provides an example of ...
Jonn's user avatar
  • 333
3 votes
1 answer
448 views

Benders Decomposition for deterministic MILP

When I think of Benders Decomposition, I typically think of two-stage stochastic programs. However, I was wondering if there is any application to decomposing a large scale deterministic MILP with ...
EricO's user avatar
  • 33
3 votes
1 answer
262 views

Logical / combinatorial Benders Decomposition vs Cutting plane method

Is there a difference between logical and combinatorial Benders Decomposition and the cutting plane method? My understanding is that for all of these techniques, there is a MIP and based on solutions ...
PSLP's user avatar
  • 2,401
3 votes
2 answers
719 views

Decomposition methods for two-stage stochastic program with integer variables

In a stochastic programming problem, I have binary variables in the second stage. As an example, consider that the optimization problem is given by: \begin{align} &\text{minimize} &\gamma\\ &...
Sam's user avatar
  • 161
3 votes
1 answer
149 views

Benders Decomposition for Fixed Charge Transportation Problem

I am trying to write down the steps in Benders decomposition for the Fixed Charge Transportation Problem and was hoping someone could confirm/deny whether my understanding of it is correct. The ...
BftA's user avatar
  • 31
3 votes
1 answer
349 views

Implementation of Local Branching

I've been recently reading some papers where the authors use local branching specifically in Benders Decomposition (see for reference). Although I understand up to some extend how the algorithm works, ...
whitepanda's user avatar
3 votes
1 answer
96 views

How to modify Benders decomposition to handle overlapping or shared variables among the subproblems

I have a problem that can be separated into a master problem and two subproblems, SP-A and SP-B. SP-B share some variables with SP-A, and the shared/overlapping variables from SP-A cannot be fixed for ...
user4444's user avatar
3 votes
1 answer
522 views

Accessible introduction to L-shaped methods/Benders decomposition

I am looking for papers or other resources that provide an accessible introduction to L-shaped methods/Benders decomposition for solving stochastic linear programming-ideally something focused more on ...
Moving in rhythm with OR's user avatar
3 votes
1 answer
157 views

About Mathematical Programs

I am dealing with the following problem as follows. $$\min \sum_{i,j}( x_{i}+y_{j}+q_{ij}+w_{i})$$ $$\text{s.t.} x_{i}+y_{j}+q_{ij}+w_{i} \geq b_{ij}, \forall i,j$$ Is it possible to handle this ...
user avatar
3 votes
1 answer
132 views

Master Problem Infeasible after Iterations of Unbounded Sub Problem (Generic Benders Decomposition in non Stochastic MILP)

I am dealing with a large problem of MILP problem and interested to apply Benders Decomposition. I have already check the original problem and it was feasible to run in large number of computation ...
Lizard White's user avatar
3 votes
1 answer
1k views

How to compare the optimality gaps between a commercial solver and Benders

Suppose I have a mixed-integer-linear programming model where the objective is maximization. I solve my model in two different ways. First, I call CPLEX (a commercial solver), and then implement ...
whitepanda's user avatar
3 votes
1 answer
320 views

Stochastic Facility Location Model

I am solving a stochastic facility location model using Benders decomposition (L-shaped algorithm). In each scenario, I want to allocate demands from origin to a fixed number of closest open ...
mars's user avatar
  • 629
3 votes
1 answer
915 views

Implementing benders decomposition using Lazy and User cuts callback of Cplex

I am trying to implement benders decomposition for a simple fixed charge transportation problem for the purpose of learning. I implemented the classic Benders decomposition successfully by adding ...
Sam's user avatar
  • 87
3 votes
1 answer
353 views

Benders implementation on Cplex is very slow

I'm working on a location problem and I have an issue with the Benders decomposition. I'm using Cplex with Python. I coded a single cut and a multi-cut to compare. The single-cut implementation takes ...
Almufa's user avatar
  • 141
2 votes
2 answers
174 views

Two Genetic Algorithms to solve two subproblems is a bad decision or I'm doing something wrong?

I'm developing a heuristic based on U-NSGA-III and GA for continuous variables with a crossover operator from this article: https://www.researchgate.net/publication/331451524_CAM-...
Josa Ferreira's user avatar
2 votes
2 answers
83 views

In Benders Decomposition, do you add an optimality cut together with a feasibility cut?

in the process of implementing my first BD algorithm, I am unsure how to proceed in the case that the subproblem is unbounded. Obviously, it means you get an extreme ray that you can add to the RMP as ...
Arctic_Skill's user avatar
2 votes
1 answer
84 views

How do you derive the Benders feasibility cuts?

starting off with a MIP that I want to solve using Benders. so in Benders Decomposition, you add feasibility cuts in the following form: $v^j (b - Ax) \geq 0$ with $j \in J$ being the set of extreme ...
Arctic_Skill's user avatar
2 votes
1 answer
191 views

Difference between Optimality cuts and Feasibility cuts for L shaped method in stochastic programming?

What is the difference between Optimality cuts and Feasibility cuts for L shaped method in stochastic programming? Like for the following problem they used Optimality cuts, $$ \begin{aligned} & z=\...
falamiw's user avatar
  • 157
2 votes
1 answer
131 views

multi stage stochastic programming algorithm

I have a multi-stage stochastic programming model. I have 3 groups of variables: the first group takes values at the beginning of the planning horizon before the first realization and does not change ...
mahgol's user avatar
  • 21
2 votes
1 answer
283 views

Benders Decomposition cuts for MILP problem with further separable subproblems

I am solving an OR scheduling problem where I assign the patient to (day,OR) tuple in Master Problem. Once the assignment is made, a subproblem can be solved for each (day,OR) tuple independently ...
Amogh Bhosekar's user avatar
2 votes
1 answer
254 views

Benders decompositions: Number of iterations does not remain the same

I am solving an LP (i.e 118-bus system economic dispatch for 130% loading) using Benders decomposition. The problem takes 26 iterations to converge. This means that the process adds 25 cuts to the ...
Fouad Hasan's user avatar
2 votes
1 answer
168 views

Numerical problem regarding to classical benders cut of large scale problem

I am trying to implement benders decomposition for a simple two stage unit commitment problem. I implemented the classic Benders decomposition to add feasible cut and optimal cut to relax master ...
Lee Adolin's user avatar