# Questions tagged [mixed-integer-programming]

For questions about mathematical optimization problems involving both continuous and binary or general integer variables.

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### In an integer program, how I can force a binary variable to equal 1 if some condition holds?

Suppose we have a binary or continuous variable $x$, a binary variable $y$, and a constant $b$, and we want to enforce a relationship like If $x \gtreqless b$, then $y = 1$. How can we write this ...
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### In an integer program, how can I “activate” a constraint only if a decision variable has a certain value?

Suppose we have the constraint $$a_1x_1 + \cdots + a_nx_n \gtreqless b,$$ where $a_i$ and $b$ are constants and $x_i$ are decision variables. Suppose also that we want the constraint to hold if $y=1$ (...
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### What are good reference books for introduction to operations research?

The reference books should cover the wide range of problem-solving techniques and methods.
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### Why is it important to choose big-M carefully and what are the consequences of doing it badly?

The question here discusses the two different use of "big-M method", where one of them is the big-M in logical constraints and linearization in (mixed-)integer programming problems (that's what I'm ...
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### When to use indicator constraints versus big-M approaches in solving (mixed-)integer programs

Various optimization modeling languages and solvers allow for both indicator constraints (see for example here, here and here) and traditional binary variable and big-M approaches can be used to model ...
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### Single reference for Mixed Integer Programming formulations to linearize, handle logical constraints and disjunctive constraints, do Big M, etc?

Is there a single crisp and accessible reference which covers how to generate Mixed Integer Programming formulations to linearize products, handle logical constraints and disjunctive constraints, do ...
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### Optimization Problem Libraries

Can someone please make a list of optimization problem libraries so that the community can add to and refine it? I know a few off the top of my head.
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### Mixed-integer optimization with bilinear constraint

So I have an optimization problem of the following form: \begin{aligned} \max_{x,y} \quad & \sum_i x_i \\ \text{s.t.} \quad & \sum_i x_iy_i \leq a \\ \quad & x_{\min} \leq x \leq x_{\max} ...
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### How can I strengthen a family of constraints in the presence of a clique constraint?

Suppose $x_i$ are binary variables, $y_j$ are arbitrary variables, $a_j$ and $b$ are constants, and I have the following linear constraints: \begin{align} x_i + \sum_j a_j y_j &\le b &&\...
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### How does a warm start work in LP/MIP?

Can someone explain how warm starts/ MIP starts work? How do solvers like CPLEX/GUROBI use warm start with the Simplex algorithm? I am interested in understanding how the entire warm start pipeline ...
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### State-of-the-art algorithms for solving linear programs

Průša and Werner (2019) show that the general linear programming problem reduces in nearly linear time to the LP relaxations of many classical NP-hard problems (assuming sparse encoding of instances)....
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### How to model a mixed-integer linear programming formulation in Python using Gurobi?

I can remember that I spent some time in understanding how to formulate my first model. So I aimed at presenting a complete model here, wishing to save some time for students or researchers needing it....
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### How to linearize min function as a constraint?

I'm trying to solve an optimization problem including following constraint, and I need to linearize it in a maximization nonlinear programming model. Please help me to reformulate it with mixed ...
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### How to construct my mixed integer programming problem with constraint of minimum consecutive ones

My target is to formulate a binary sequence with fixed size $N$ = 10, such as $[1, 0, 0, 0 ,1, 1, 0, 1, 0, 0]$. However, I want to constrain this sequence so that when 1 appears, it has to appear at ...
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### Modeling floor function exactly

Suppose we want to enforce a constraint $$y=\lfloor{x}\rfloor$$ where $x$ is some continuous variable. One option is to use $$x-1\leq{y}\leq{x},\quad y\in\mathbb{Z},$$ which fails on the edge case ...
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### Feeding known lower bounds to solvers

Given an optimization problem that aims at minimizing some objective function, a lower bound that is valid for all feasible solutions, and your solver of choice: For what theoretical and/or practical ...
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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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### Working with absolute values in constraint in a LP or MILP

Having all the approaches explained in the blog called "OR in an OB World" (this address) in my mind, I would like to ask the following question: What is the best practice to make a constraint linear ...
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### Modelling precedence relations

I have two tasks $i$ and $k$ with durations $d_i$ and $d_k$, where $d_i$ and $d_k$ are nonnegative variables. I would like to model that $i$ may precede $k$ or $k$ may precede $i$ and that they may ...
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### How can I find the shortest path visiting all nodes in a connected graph as MILP?

I am looking for an idea on how to formulate the following problem as a MILP. Given a connected graph, find the shortest path route starting from a node (not given) and visit all other nodes. All ...
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### How to visit a subset of network nodes in a single trip?

I have a connected network where I want to visit a set of destinations which may require visiting intermediate nodes as well because there may be no direct edge between source and destination nodes. I ...
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### How to compare two different formulations of a problem?

I somewhat know how to compare two MILP formulations of a problem that both use the same set of decision variables (as in the classical MTZ vs DFJ formulations of the TSP). I was wondering how two ...
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### What are the tradeoffs between "exact" and Reinforcement Learning methods for solving optimization problems

Exact methods, e.g., models that utilize an MIP approach with a specified objective and constraints, have advantages like the following: Using off the shelf solvers Optimality gap provability ...
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### How to evaluate the performance of open source solver?

I am looking for a reliable open source solver to solve LP and MILP (with a few thousand variables). How can I evaluate the performance of a given solver for a particular use case?
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### Application of complex numbers in Linear Programming?

The theory surrounding Linear Programming is based on variables, bounds and coefficients that take on values in $\mathbb R$, the set of real numbers. I have long wondered whether there might be ...
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### Estimation of the size of Branch-and-Bound trees using ML

A short background: A paper [1] published in 2006 intends to show that the time needed to solve mixed-integer programming problems by branch and bound can be roughly predicted early in the solution ...
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### Using CPLEX "solution pool" to count feasible points

Some problems call for a count of the number of integer "lattice" points contained in a feasible region (rather than for locating the minimum or maximum objective function value in that region). See ...
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### 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) ...
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### The effect of choosing big M properly

I have a set of linearized constraints that are modelled using big-Ms. Now, it is, of course, common knowledge to make the value of M and small as possible in order to provide tighter LP relaxations ...
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### How could we simplify solving the large scale MIPs without using any advanced methods like decompositions?

Many practical optimization models (specially MIPs) are NP-Hard and solving them need much time even with the modern solvers like CPLEX or GUROBI. One of the best way (but not easy) is using ...
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### What do solvers like Gurobi and CPLEX do when they run into hard instances of MIP?

MIP is NP-Hard, so it is possible that an instance is very difficult and has multiple local minima that the search can get stuck in. With a Metaheuristic Algorithm, the stochastic and approximate ...
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### Is there a heuristic approach to the MILP problem?

I have the following optimization problem which is a MILP. I can solve it with a MILP solver. \begin{align}\min_t&\quad t\\\text{s.t.}&\quad d_{c}-t\le \sum_{n=1}^{N} B_{n,c}x_{n}\le d_{c}+t,...
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### How to use warm start to solve MIPs efficiently?

I'm working on the scheduling model which takes a long time to solve to optimality (even for a small instance), therefore I would like to use a warm start (MIP start) to solve the problem. I'm using ...
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### How to remove or replace sub tour elimination constraints in the VRP variant models?

In many of vehicle routing problems variant (VRP), which can be formulated using MIPs, to avoid creating sub tour, we need to use sub tour elimination constraints (SEC). One of the known SEC is (I ...
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### Algorithms vs LP or MIP

Is there a way of writing an algorithm with if-, while-statements to find an optimal solution without using linear-programming (LP)/MIP? If so, what would the benefits be against the LP/MIP? Is it ...
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### Google OR Tools: Iterative Assignment Problem

This question is a Google OR-Tools specific implementation of recommendation from a previous question. In short, the movie theater problem encompasses assigning viewers to seats such that the distance ...
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### Mixed Integer Programming: Iterative assignment problem

I have a real world problem, which is analogous to the below toy problem, which I call 'The Movie Theater Problem' (TMTP.) In TMTP, movie viewers are assigned seats which principally balances two ...
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### How to create A, b and c matrices from very large .lp file?

I am working on Scaling MIP. I use Gurobi within Clion. I need to extract the coefficient of Xs (A matrix), right-hand side (B matrix), and an objective function(c matrix) from the .lp file and ...
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### Mixed Integer programming - Problem modelling coincidence restriction in scheduling match problem

I am trying to model and solve a problem for maximize audience of matches that must be scheduled in different slots (I am using python PulP library). Below I explain the problem and the model process ...
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### "Best practices" for formulating MIPs

Often there are many alternatives ways for formulating a MIP. For example: The model contains inequality constraints that must hold with equality in an optimal solution. The model contains ...
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### The rationale to improve MTZ?

Currently I need to solve a quite specific problem involving symmetric TSP as a sub-problem (i.e., a Hamiltonian cycle is a necessary condition for optimizing some problem-specific variables that ...
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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 ...
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