Clik here to view.
How to identify constraints that make problem not solvable in polynomial time?
I am reading this paper, available for free viewing, which contains an example of job shop scheduling, shown below.The details of the variable definitions, etc., can be found in the paper, but it's a...
View ArticleDeriving a valid inequality
Given a set of facilities $I$ and days $J$, each facility $i \in I$ has a capacity of $C_i$, and a set of days $J$ where in each day $j \in J$ there's a total demand of $q_j$ that can be satisfied by...
View ArticleStationarity conditions for IPs
Let's consider the following (MQ)IP:$\min x^T Q x$s.t. $g(x) \geqslant 0$$x_i \in \mathbb{Z}$$i \in I$By ignoring the integrality constraints we end up with the QP:$\min x^T Q x$s.t. $g(x) \geqslant...
View ArticleHow to linearize the following constraints
Given the following two expressions:$ x - \frac{1}{T}\sum_{i} y_{i}$$ x - \frac{1}{Q}\sum_{i} \beta_{i} y_{i}$where $x \in \mathbb{Z}_{+}$, $y \in \mathbb{R}_{+}$, and $T$, $Q$ and $\beta_{i}$ are...
View ArticleWhat is the best way to constrain a binary matrix so that at most one row has...
I have a binary variable $x_{i,j}$ for $i\in\{1,\ldots,m\}$ and $j\in\{1,\ldots,n\}$ and the constraint is to have at most one row that has ones. I wrote this as: $$x_{i,j}+x_{i',j'}\leqslant1,\forall...
View ArticleApplications of Knapsack and Cutting Stock in Pure Math
I'm giving a seminar to PhD students in pure math, and one of the things I'd like to do is show that more applied optimization can also make its way into pure Mathematics. As for classical problems,...
View ArticleOptimization problem with the Harmonic number
I have an optimization problem:\begin{align*}\text{ minimize } \sum_{i=1}^n H(x_i) \\\text{ subject to } Ax \geq b, x\geq 0, x\in \mathbb{Z}^n\end{align*}where $H(n)$ is the $n$-th Harmonic number. How...
View ArticleNeed help with integer programming exercise
This is an exercise from Wolsey that I can't solve. Show how to go from Equivalence (1) to (2) and from Equivalence (2) to (3):$$ \begin{align}X &= \{ x \in \{0, 1\}^4~\mid~97x_1 + 32x_2 + 25x_3 +...
View ArticleBenchmark problems for Benders Decomposition
We are implementing a scheduling model using Benders Decomposition. Does someone know of any existing implementation of Benders or any repositories that contain continuous or integer problems solved...
View ArticleContinuous optimization with a Euclidean TSP objective
I am trying to solve a problem of the form $$\min_{x_1,\dots,x_n} f(x_1,\dots,x_n)$$ subject to a constraint that $\mathrm{length}(\mathrm{TSP}(x_1,\dots,x_n))\leq c$, where $x_1,\dots,x_n$ are all...
View ArticleCPLEX MIP warm start seems slow down the program?
I have been working on a combinatorial optimization problem which can be modeled as an integer linear programming. I implemented it as a c++ project in visual studio 2017 and CPLEX1271. With the hope...
View ArticleWarm starting ideas for iteratively solving a model with a few additional...
I'm trying to solve a MILP model iteratively, and at each iteration a few constraints are added to the problem that cut off the previous optimal solution. I'm trying to figure out ways to implement a...
View ArticleWhat is the suitable optimization method for this case?
What is the best optimization method to solve a large-scale problem (about 300 thousand variables)?The problem is nonlinear, nonconvex, involves only binary variables, and is unconstrained. The...
View ArticleClik here to view.
Constraints to avoid disjointed solutions in a MIP
Given an directed graph $G= (N,E)$, where $N$ is the set of nodes and $E$ is the set of all edges, each associated with a direction. $G$ is a connected graph but not necessarily a complete graph.A path...
View ArticleSolving a weighted XOR-SAT problem
I want to solve a variant of the weighted XOR-SAT problem. Concretely,Given $n$ boolean variables $x_1,\ldots,x_n$ each of which is assigned a non-negative cost $c_1,\ldots,c_n\in\mathbb{R}_{\ge 0}$...
View ArticleGood encoding for grid layout problem
How would you encode the problem given by https://oeis.org/A337663 with an off-the-shelf solver?You need to lay out $n$ ones and $2, 3, ..., m$ on an infinite grid, for $m$ as large as possible. For...
View ArticleMaximizing sum of probabilities with variable distributions
Suppose $\\{X_i\\}$ are binary decision variables and $\\{A_j\\}$ are Skellam random variables with $(\mu_1, \mu_2) = (\sum_i b_{i} X_i, c_j)$. Here, $b_i, c_j \in \mathbb{R}^{\geq 0}$ are constants....
View ArticleScheduling for the shortest days using ILP
I've tried Or-Tools and MILP solvers a couple of different ways on this, but they take a surprisingly long time to realize that the solution they generated fairly quickly is in fact minimal. Is there a...
View ArticleAllocating credit card points
I’m interested in the idea behind this in general, so I thought this would be the best place to post, though I have a practical and semi-urgent need of allocating the points on my credit card towards...
View ArticleProblems with Big-M Constraint
I have the following constraints for my roster optimisation problem:\begin{align}&(1-r_{i,t})\le \sum_{j=t-\chi}^{t-1}sc_{i,j}\quad &\forall i\in I, t\in...
View Article
