Questions tagged [mixed-integer-programming]

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

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32 votes
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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 ...
LarrySnyder610's user avatar
12 votes
2 answers
3k views

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 $y=1$ (...
LarrySnyder610's user avatar
10 votes
1 answer
3k views

What are good reference books for introduction to operations research?

The reference books should cover the wide range of problem-solving techniques and methods.
Rajasekhar Kadambur's user avatar
24 votes
2 answers
6k views

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 ...
EhsanK's user avatar
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19 votes
6 answers
8k views

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 ...
AndyT's user avatar
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13 votes
4 answers
2k views

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 ...
Mark L. Stone's user avatar
50 votes
8 answers
3k views

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.
Mark L. Stone's user avatar
7 votes
2 answers
2k views

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} ...
Johnny's user avatar
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7 votes
1 answer
429 views

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 &&\...
RobPratt's user avatar
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22 votes
2 answers
6k views

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 ...
Palaniappan Chellappan's user avatar
15 votes
2 answers
2k views

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)....
rasul's user avatar
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14 votes
4 answers
2k views

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....
Mostafa's user avatar
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7 votes
1 answer
9k views

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 ...
Vida's user avatar
  • 87
32 votes
8 answers
2k views

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 ...
David M.'s user avatar
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30 votes
3 answers
2k views

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 ...
fbahr's user avatar
  • 1,016
23 votes
9 answers
741 views

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 ...
EhsanK's user avatar
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18 votes
1 answer
2k views

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 ...
Oguz Toragay's user avatar
  • 8,642
7 votes
1 answer
627 views

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 ...
shaojie liu's user avatar
7 votes
3 answers
319 views

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 ...
Clement's user avatar
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25 votes
6 answers
2k views

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 ...
rasul's user avatar
  • 2,140
24 votes
4 answers
1k views

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 ...
fhk's user avatar
  • 1,069
19 votes
4 answers
4k views

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?
Rajasekhar Kadambur's user avatar
18 votes
3 answers
1k views

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 ...
Mark H's user avatar
  • 550
14 votes
3 answers
1k views

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 ...
hardmath's user avatar
  • 247
14 votes
1 answer
746 views

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 ...
Oguz Toragay's user avatar
  • 8,642
13 votes
4 answers
668 views

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 ...
Albert Schrotenboer's user avatar
13 votes
3 answers
1k views

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 ...
A.Omidi's user avatar
  • 8,642
12 votes
4 answers
4k views

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 ...
Skander H.'s user avatar
  • 2,139
10 votes
1 answer
2k views

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 ...
A.Omidi's user avatar
  • 8,642
10 votes
2 answers
973 views

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 ...
A.Omidi's user avatar
  • 8,642
10 votes
3 answers
1k views

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,...
KGM's user avatar
  • 2,211
8 votes
5 answers
3k views

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 ...
Georgios's user avatar
  • 1,193
6 votes
1 answer
221 views

TSP subtour elimination by assigning distance traveled

Given a set $S$ which we need to travel following TSP rules. I was wondering if this sub tour elimination method is good enough or not? Let $b_{i,j}$ denote edge from $i$ to $j$ is taken or not and $...
ooo's user avatar
  • 1,589
5 votes
1 answer
226 views

Does exchanging integer variables by binary variables strengthen a MIP?

Suppose that I have an MIP with a whole lot of continuous variables and some integer variables. In my case, this takes a very long to solve (in fact I wasn't able at all to solve it to optimality). So ...
Noah D.'s user avatar
  • 93
5 votes
2 answers
760 views

How to linearize specific range constraints?

I would like to know about the linearization of the $(If, Then)$ constraints as follows: $$\begin{array}{l} \text { If: } \\ 15 \leqslant x \leqslant 25 \\ \text { then: } \quad y=\color{blue}{a} x+\...
A.Omidi's user avatar
  • 8,642
4 votes
0 answers
115 views

Conditional constraint formulation [duplicate]

How can I create constraints to make sure $x=1$ if $k\geq 0$ and $x=0$ if $k<0$, where $x\in \{0,1\}$ and $k\in \mathbb{R}$? Here is my attempt: \begin{equation}\label{cons:1} \begin{aligned} ...
tcokyasar's user avatar
  • 1,249
2 votes
1 answer
186 views

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 ...
jbuddy_13's user avatar
  • 519
2 votes
1 answer
123 views

Linearize product of $x\cdot y \text{ with } x,y \in \{-1,0,1\}$ for MILP

I have a problem where I have many products between variables drawn out of $\{-1,0,1\}$. Could you suggest a linearization in terms of variables in $\{-1,0,1\}$ or $B_1 - B_2$ where $B_i \in \{0,1\}$ ...
worldsmithhelper's user avatar
2 votes
2 answers
163 views

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 ...
jbuddy_13's user avatar
  • 519
1 vote
2 answers
409 views

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 ...
asdf's user avatar
  • 105
-2 votes
1 answer
73 views

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 ...
brtin_'s user avatar
  • 19
31 votes
4 answers
3k views

"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 ...
Rolf van Lieshout's user avatar
26 votes
1 answer
590 views

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 ...
Omicron_Persei_11's user avatar
26 votes
4 answers
1k views

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
21 votes
5 answers
659 views

Tightness of an LP relaxation without using objective function

How can we measure the tightness of a linear programming relaxation for a mixed integer linear program without using the objective value? I would like to get a measure in terms of the feasible set and ...
Mertcan Yetkin's user avatar
18 votes
4 answers
1k 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
17 votes
1 answer
4k views

What is the "big-M" method? And are there two of them?

I’ve seen the "big-$M$ method" referred to in different ways. What is the "big-$M$ method" and why does it seem to mean two different things?
LarrySnyder610's user avatar
16 votes
2 answers
2k views

Why do we need to measure the difficulty of mixed-integer programming problems?

I'm doing a project about the estimation of the difficulty of mixed-integer programming problems. The MIP instances are from MIPLIB 2017. And there are three categories of MIPs provided by MIPLIB 2017,...
Karen Jin's user avatar
  • 161
16 votes
2 answers
657 views

Branching rules in commercial MIP solvers

I am working on a branch-and-cut algorithm, and I have spent quite some effort into improving the branching decisions that are made by commercial solvers, such as CPLEX and Gurobi. However, it was ...
Albert Schrotenboer's user avatar
15 votes
1 answer
266 views

Has the expressibility of 'non-integrality testing' as extension to MILP been studied before?

It turns out that extending MILP with any of the constraints $y=\lfloor x\rfloor$, $y=\lceil x\rceil$, $0 < x$, or $x\notin \mathbb{Z}$ is 'equally hard'. (see my answer here, and below) ...
Discrete lizard's user avatar