Questions tagged [mixed-integer-programming]

A mixed-integer programming (MIP) problem is a linear program where some of the decision variables are constrained to take integer values.

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Relaxation of the binary optimization problem doesn't make sense in terms of the original problem - how do I repose the model to make sense?

Suppose I have a set of binary variables $x\in \{0,1\}^N$, where in the context of my optimization problem these binary parameters mean to include the parameter in a model or not include it, where ...
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Minmax combinatorial optimization with target set size constraint

I have a combinatorial optimization problem whose objective is in the "min-max" form. Suppose that there is a row stochastic matrix $P=[p_1^T,\cdots,p_n^T]^T\in\mathbb{R}^{n\times n}$ whose ...
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Reformulate indicator function using mixed integer program

Consider the following constraint: $$ \sum_i I(a_i x \leq b) \leq m. $$ Can we reformulate this constraint using big-M constraint? A similar question can be found in Involving indicator function as a ...
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Changing a model in to a mixed integer linear programing model

Trying to learn about integer programming in quarantine and I've come across a problem that stumped me. I searched the net but couldn't see anything similar and would appreciate another set of eyes on ...
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What objective function I should consider?

Given an undirected graph $G=(V,E), the Vertex Coloring Problem (VCP) requires to assign a color to each vertex in such a way that colors on adjacent vertices are different and the number of colors ...
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Where can I find good practices on building models for optimization?

I am sort of new to mathematical optimization and have to build some fairly complicated models for my thesis. I was wondering where I could find literature to help me develop more efficient versions ...
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Formal robustness verification of neural networks: MILP vs SMT

I'm not sure if this is the right place to ask this questions. I'm working my way into the field of formal verification of neural networks. The goal is to analytically evaluate the robustness of ...
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How to write the KKT condition?

$Min f=\sum\limits_{t\in T\text{}}{\left[ \lambda _{grid,t}^{WM,P}p_{grid,t}^{WM}+k\left( 1-{{v}_{it}} \right) \right.}$ ${{v}_{i,t}}-{{v}_{j,t}}\le {{\tilde{z}}_{ij}}s_{ij,t}^{*}+\tilde{z}_{ij}^{*}{{...
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How to enforce graph continuity constraint?

I am working on a single source multiple destinations problem formulation using MILP. The network I have designed is as shown in the figure below Graph Network. The objective function is to minimize ...
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Assignment problem with batching costs

I am studying an assignment problem with batching costs, and I would like to know if there is a standard name or algorithm for this problem. I know this problem can be formulated as mixed-integer ...
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What will be an efficient joint clustering solution to this problem?

There is a system with $S$ transmitters/service points, $U$ subscribers/users. The link quality/gain between any transmitter indexed by $s$ and a user indexed with $u$ is denoted by $h_{u,s}$. Any ...
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Proving that an inequality is a split inequality (in cutting plane, Integer Programming)

As I know, to prove if a given inequality is a split inequality or not, we need to find $\pi$ and $\pi_0$ such that $πx ≤ π_0$ and $πx ≥ π_0 +1$ and the given inequality must be valid for both $Π_1$ ...
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Making a variable in a linear program be equal to a ramp function

In a linear program, I have a variable $y$ that must obey $y=\max(x-a,0)$ where $x$ is a linear combination of the other variables and $a$ is a constant. $x$ and $y$ and $a$ are always non-negative. ...
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Conditional constraint activated by binary variable [closed]

For each time step $t$, $T_1(t),...,T_n(t)$ are continuous variables, $z(t)$ are binary variables. $T_c(t)$ is known. I am trying to express the following constraint in a Linear Programm. $$ T_i(t+1) =...
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Using strict inequalities for constraints in linear programming

I started to work a bit with linear programming methods and would like to know why we can't use strict inequalities in our constraints, i.e., why is the equivalence in constraints excluded? What can I ...
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Why does the simplex algorithm not accept negative decision variables?

I would like to know why the Simplex algorithm does not accept negative decision variables? I read this article on Wikipedia but couldn't find a satisfying answer.
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Mixed Integer Programming - Model Formulation for A Resource Allocation Problem

Crossposted at Stack Overflow and Operations Research SE There are a number of orders, which needs to be shipped. For each order, there may be 1 to 3 route options. The problem here is to find out ...
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linear program formulation discontinuous function

I have the following discontinuous function: $f(x) = 10 $ if $x = 0$ or $f(x) = 5 + 8x $ if $x > 0$ where $ x \in [0,30]$ I'd like to formulate as a bilinear program using binary variables but I am ...
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How can I compute the number of solutions to an under-constrained equation over some integers?

I have system of under-constrained equations over a fixed set of integers, defined as follows, how can I find the number of solutions to this problem? A computational reference will do (in-fact ...
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Modeling that there is no feasible solution to a linear system in mixed integer programming

My question is about how to construct a mixed integer programming to model that there is no feasible solution to a given linear system. Specifically, given $x\in \mathbb{R}^{n}$ and $z\in \{0,1\}^{d}$,...
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MIP modelling of piecewise linear function

Suppose we have the following piecewise linear function $f(x)$: $3x + 8, x\in [0,5]$ $33 - 2x, x\in [5,10]$ $3 + x, x\in [10,20]$ How to model the relations between $f(x)$ and $x$ using ...
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Integer programming formulation for the entire region of a triangle except the region of a rectangle inside it

Suppose we have a triangle-shaped region such as: $x-y\ge0$ $x+y\le8$ $x,y\ge0$ and integers How to write an IP formulation to describe all real points inside this triangle except the points inside ...
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How to write a constraint for time bound binary integers?

In these constraints $u_{g,t}^G$ is on at t=1, then $u_{g,t}^{G,start}$ should become 1 after $t_g^{G,start}$ time-steps then after it stays 1 for $t_g^{G,cr}$ time-steps then $u_{g,t}^{G,on}$ ...
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How to solve the mixed integer continuous programming problem?

We have a mixed integer continuous programming problem that described in the follow, $$\max\limits_{\left\{\kappa_{k},\mathbf{w}_{k}\right\}} \left\|\mathbf{\kappa}\right\|_{0}$$ $$\mathrm{s.t.}~\...
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MILP formulation of constraint

I'm very new to MILP, and I'm testing python-mip. I'm trying to model a simple power distribution system with some constraints, but I'm stuck.... I want to find a valid power balance between a device, ...
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'Weird' Indexing In The MILP Literature

Fairly new to MILP. I posted this question as an issue on the Gurobi examples repo too, but it might be more appropriate to ask it here. NOTE: Since I "need at least 10 reputation to post images&...
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Solving a system of piecewise linear equations with multiple variables in minimum function

General problem I am facing a problem from the transportation field. For a given network, the maximum flow $q$ on a link is either the sum of the flows from upstream links, entering at the upstream ...
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Column Generation?

I implemented a method to solve a large integer program but am not sure how its called. The idea is the same as column generation, just less explicit. Take the following problem as a simple example (...
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Valid inequality with chvatal gomory for problem with binary variables

I am not able to find a valid inequality with chvatal gomory for my problem that increases the value of the relaxed problem. My problem: Determine the optimal allocation of n workers to m machines in ...
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Integer solutions to a large sparse equation

How might one find integer solutions to: $$ \textrm{diag}(x)Ax = Bx $$ $A$ and $B$ are integer-valued, square, non-invertible, sparse, and large. The values of $x$ should be $0$ or $1$ (ideally; ...
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Mathematical formulation of an optimization problem

I have the following optimization problem at hand. There are N integer numbers, $a_{i},a_{i+1},\dots,a_{n}$ where $a_{i} > 0$. We need to partition these numbers ...
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Any nice/good way to allow an “or” option in a linear program?

I am looking for a way to express an "or" option in a system of linear inequalities for a linear program I am working on. I will explain what I mean precisely: Lets say I have a set of ...
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If Else Statement in Linear Program, making it Mixed Integer Program [closed]

How to write the following if-else condition in Linear Programming or in Mixed Integer Program? If a <= b then c = a else ( a > b) then c = b a, b, c are all variables. Is it possible to rewrite ...
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How to write hard ILP problem?

I want to write/create hard ILP problems. When I say hard I mean the question should be solvable in at least 20 minutes with non-commercial ILP solvers. Can someone show me a direction or know how to ...
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linearizing non-strict inequality If.. else conditional constraint that includes decision variables

I would like to linearize a conditional constraint as follow: 0 <= x1 <= 1. If 0 <= x1 < 1, then x2 = 1, else x2 = 0. I find it very difficult because it is not a typical If...then ...
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Linear/Integer Programming: Scheduling task with regular intervals

Given is the following problem: There are $N$ items that have to be processed in given intervals over a year. For example, item $i_1$ is processed about every 14 days, item $i_2$ is processed monthly. ...
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Chance-Constrained Optimization

Let $x \in \mathbb{R}^n$ and $\xi \in \Xi \subset \mathbb{R}^n$ be a random vector with discrete probability distribution $p(\cdot)$, assuming a finite number of values $\{\xi_i\}_{i = 1}^S$. Let $A \...
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Assigning tasks to minimize cost

I have a set of jobs that must be done on a given day in sequence. $J_1, J_2, J_3$, with deadlines $D_1, D_2, D_3$. There are three workers $W_1, W_2, W_3$ that can execute the tasks each one with ...
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Multilevel integer programming

Recently, I have been dealing with an integer program that involves finding a $4$-level integral solution $$x \in \{-2, -1, 1, 2\}$$ or even $x \in \{\pm n, \pm n-1, \dots, \pm 1\}$ which is ...
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Linearizing the product of a binary and a continuous variable

I have an MIP optimization problem that has a constraint $p\geq xy$, where $x$ is a binary variable, $p$ and $y$ are non-negative continuous variables. I tried the Big-M method. However, the upper ...
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Linearize objective function for intlinprog

This is the predecessor to the following question: I have some constraints based on which I have written a Matlab program that provides me some feasible binary bits for Finite state machines(FSM). ...
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Formulating Constraints into Mixed-Integer Linear Programming

Is there a way to formulate the following Linear Program in a mixed-integer LP that I could solve with most linear programs in R/Python that support Mixed Integer Linear Programs (MILP)? So my ...
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1answer
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possible bin packing question

what is known: you are given 3 packs of boards of different lengths. ie pack A is 3.9m, B is 3m and C 2.4m and an arbitrary distance to cover, d. how do I pose and then solve this question? 3.9A + 3B ...
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Modeling following constraints in MILP

I want to know how I should formulate the following constraints in my MIP problem? $$x= x_1 z_1+ \dots +x_n z_n \text{ and } y_1 \le y \le y_n \text{ and } z_1+\dots+z_n=1$$ OR $$y= y_1 w_1+\...
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Why For any integer point in the feasible region the right side of this equation is less than 1 and the left side is an integer?

I wanted to find Why Gomory's cut works. wikipedia Gomory's cut explains: An integer programming problem be formulated (in Standard Form) as: \begin{aligned}{\text{Maximize }}&c^{T}x\\{\text{...
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Linear programming formulation confusion

I have a silly confusion. For this constraint here, $a_{i1,j1} + a_{i2,j2} ≤ 1$ if $0 < |i1−i2|+|j1−j2| < d$. I understand this constraint but I want to ensure that this encompasses all $i$ ...
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Modeling problem (of Operational Research)

Consider a logistics system consisting of $n$ production sites and $m$ warehouses. For a given product, the monthly production capacity of the production sites is $p_i$ units, with $i = 1,\dots, n$. ...
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Are all mixed integer nonlinear problems non-convex? If not, is my problem non-convex? And what solvers can we use?

My apologies for asking (basic) questions on mixed-integer nonlinear problems. Question 1: Are all mixed-integer nonlinear problems non-convex? If not, why do people say it is very hard to solve it? ...
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How to relax logical constraints

Consider two $m\times 1$ vectors, $x\equiv (x_1,x_2,...,x_m),\tilde{x}\equiv (\tilde{x}_1,\tilde{x}_2,...,\tilde{x}_m)$. Let $x\leq \tilde{x}$ if and only if $x_i\leq \tilde{x}_i$ for each $i=1,...,m$....
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How can I solve this nonlinear-projection optimization problem?

How can I solve this nonlinear projection problem? Can you give me any idea? Constraints are as shown in the image. The norm can be anyone. Variables : $z_1,\;z_2,\;\underline{x}\in\mathbb{R}^d,\;\...

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