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Constraint programming is a particular form of optimization modeling that tends to be well-suited for combinatorial models like scheduling and planning.

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Create a closed tour visiting the numbers. The tour must start and end at the same cell. During the tour, you can only visit the neighboring cell (up,down,left,right). The tour cannot contain a number ...
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I'm figuring out how to fill a task schedule with some very specific constraints. I live with 9 flatmates (myself included). To keep our home clean and everyone happy, we have a very specific cleaning ...
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Let's first consider a simple, unconstrained example with two nodes {0,1}. Each node has two actions: {stay, transfer}, and all transition probabilities equal 1. Specifically, $P(0|0, stay) = 1,~P(1|0,...
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For a continuous linear map $f$, I have the following problem $$ \max_{\mu} \int f(\mathbf{x}) \;\text{d}\mu(\mathbf{x})\\ \text{s.t.} \; c(\mathbf{z}) = \int f(\mathbf{x}) \text{d}\mu(\mathbf{x}|\...
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I am a network engineer who is currently doing some network optimization problem. In my application, there is a requirement for the network delay to be bounded in some interval once some boolean flag ...
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I'm trying to model a linear job scheduling optimisation problem. There is a single machine and N jobs $J_1, J_2, ..., J_N$. Each job consists of one step with processing time $p_1, p_2, ..., p_N$. ...
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I've been trying to develop the most efficient algorithm to solve a Sudoku puzzle. The one that I've developed isn't able to solve certain kinds of puzzles without having to use backtracking. One such ...
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I play video games, and it sounded like a fun exercise to try to find the optimal order in which to complete quests: There exist multiple quest trees There are some quests with inter-dependencies ...
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I am considering a weighted least square problem with data $X \in \mathbb{R}^{n \times p}$, (diagonal) weight matrix $W \in \mathbb{R}^{n \times n}$ and responses $y \in \mathbb{R}^n$, i.e. finding $$\...
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I have below objective function $S = \lvert 18 - a - b \rvert + \lvert 15 - a - 2b \rvert + \lvert 11.1 - a - 3b \rvert + \lvert 7 - a - 4b \rvert + \lvert 3.4 - a - 5b \rvert + \lvert -1.5 - a - 6b \...
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Let a, b, c be three real number constants satisfying $a^2 + b^2 + c^2 \leq 1$. Define the function $f(x, y, z) = \frac{x^2 + y^2}{2(1+z)}$ under the constraints $(x-a)^2 + (y-b)^2 + (z-c)^2 \leq \mu^...
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I have an optimisation problem that I thought should be in the form, \begin{align} \mathrm{maximise}_{x\in\mathbb{R}^p} & f(x) + \lambda\|x\|_1 \\ \mathrm{subject~to~~~~~~~} ...
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In a simplified version of the ILP problem I am trying to formulate, I have the following: A set of elements $A_{i,j} \in \mathcal{A}$. Each element $A_{i,j} \in \mathcal{A}$ has an associated ...
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I am quite new to the field of optimization and currently having a problem of formulating a constraint with binary variable. For each value of $b$, if there exists one value of k such that $z_1[b, k]$ ...
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Let $\mathbf{B}$ be a definite positive square matrix of size $n \times n$, and $\mathbf{b}$ an $n$-sized vector. It can be shown that the solution of $\arg\min_x \left(\mathbf{x}^T \mathbf{B} \mathbf{...
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Place the numbers 1 to 9 into the nine positions in such a way that, the 6 equations are valid. Each position must have a distinct value. Multiplication and division have priority over addition and ...
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I am working on a problem where I have a set of 2D vertices and a test point. I want to check whether the test point lies inside the polygon formed by the set of given vertices. I am trying to model ...
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For this discussion, I will be considering polynomials over multiple complex variables, and a system of polynomial constraints, where the constraints on the variables can be written as a set of ...
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I need to write a program that sorts people into groups. To give a little context: The aim of the program is to create an equitable distribution of tasks and people for a school trip. Every day the ...
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I have a problem setting similar to bin packing but not exactly. I have to put the few boxes of certain dimensions in a square area besides each other. Like a grid. The boxes should be placed around ...
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A squared rectangle is a rectangle dissected into squares. squared rectangles are called perfect if the squares in the tiling are all of different sizes and are positive integers. The smallest perfect ...
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I am stuck at a mixture of a combinatorial and maximization problem and don't know how to proceed. Hopefully someone has an idea that can bring me further. Imagine that we have a sequence of $n$ coins....
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Given an arbitrary vector $\mathbf{y}\in \mathbb{R}^{n}$, I would like to find $\mathbf{x^*} \in \mathbb{R}^n$ which is $$ \text{argmin}_\mathbf{x} \|\mathbf{x}-\mathbf{y} \|_F^2$$ s.t. $$ \mathbf{x}\...
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This is a mathematical and algorithmic question, so I hope it is not flagged for failing to be a pure mathematical question. The puzzle "Two not touch" (or Star Battle) consists of a $10 \...
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Given the following three equations, assuming all unknowns are integers: $$ x y =\left(x + y - z \right) a \\ x z =\left(x + z - y \right) b \\ y z =\left(y + z - x \right) c $$ And the ...
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There are $n$ seats in a row. $p$ people (where $p<n$) can seat anywhere as long as long as they sit at least one seat apart due to personal relationships. This statement is part of a larger ...
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Trying to solve system $(A+\lambda{W})x=b$ (derived from the method of Lagrange multipliers) where $\lambda \in \mathbb{R}$ - the Lagrange multiplier. $A$ - symmetric non-singular matrix. $W$ - ...
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I have the following optimization problem that I want to solve (note that the $X$ variable consists of a binary matrix subject to a single constraint). First, some definitions: My $X$ variable: $$ X = ...
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I'm trying to print out nets of a cube on a sheet of paper, and I'm hoping to fit as many as I can on single sheets. The squares that make up the net are $\frac{1}{2}$ an inch wide, and I'm printing ...
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(Originally posted here https://stackoverflow.com/q/72687231/10291218) Suppose I have an integer array A of size n with two ...
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I've got a CSP which is based on constraining bit vector variables. It is explained below through an example, followed by the full definition. So, what I'm concerned about is if you have some idea if ...
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While working on the algorithm, I need to solve the following problem $$ \min_{x \in \mathbb{R}^n} \| x \|_1 + \frac{\alpha}{2}\| x - y \|^2 \\ \mathrm{s.t.} \ \| x - s \|^2 \le r$$ where $y,s \in \...
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I am having trouble understanding the following question. Question. Take the following non-linear programming problem. \begin{equation*} \min f(x_1,x_2,x_3) = x_1^2-3x_1x_2+x_2^2+x_3^2 \\[.15cm] \text{...
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I’m looking for good heuristics for finding at least one (of a probably large set, although possibly none) high dimensional ($|v|>5000$) binary vector that satisfies a set of non-linear/non-...
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I'm trying to prove $-\ln(1-\|x\|^2_2)$ is self-concordant function. I think 1-dim case is easy to prove, but I cannot prove multi-dim cases. It's really hard to use definition because heavy Hessian. ...
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Suppose we have a set of tasks $T = \{t_1, t_2, \dots, t_n\}$ with durations $\{d(t)| t \in T\}$, that need to be executed on some machine such that their execution times do not overlap. We can ...
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I want to find a minimum threshold value for a constraint, such that if this constraint is satisfied, the next one must be satisfied. For example, given two inequations $f_1(X)\geq a$ and $f_2(X) \geq ...
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I am new to mixed-integer programming and I am confused about how to approach this if-then condition. How do I the following constraint in mixed-integer programming: if Dm +t < Dn + then Zmn=1, 0 ...
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I have the following constrained linear system: $$ Ax = y \\ Cx \ge b $$ where $$ y\in \mathbb{R}^3 \\ x \in \mathbb{R}^n \\ b \in \mathbb{R}^m \\ $$ Also, $n$ and $m$ are typically greater than 3, e....
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The Skyscraper puzzle has the goal of positioning Skyscrapers of $n$ different heights on an $n \times n$-Field so that the following requirements are met: Every field contains a skyscraper. All ...
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I have few reference I'm currently reading through but I still don't quite get the difference between optimising a function over a manifold and simply use constrained optimisation. Do the algorithms ...
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I'm trying to solve a problem for $x$ which is a vector of length $n$ with only binary elements, i.e. each $x_{i}$ is either $0$ or $1$. There are two constraints on $x$, one quadratic and one linear: ...
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Let $x_j$ denote the ratio of assets to allocate to investment option $j$, where $j=1,...,n$ Let $c_j$ denote the annual expected rate of return on investment option $j$ How do we write the following ...
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p.s. I already know few things, but it is interesting to listen to different perspectives. what I know: the augmented Lagrange method combines the penalty method and the method of Lagrange multipliers....
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I am not sure if I am asking on the right place. But given a set of independent variables $X_i$ and the dependent variable $Y_i = f(X_i, b) +c$, how can I estimate the regression equation given a set ...
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Consider the following linear programming (hereafter, problem [1]) $$ \max_{y\in \mathbb{R}^J}c'y\\ \text{s.t. } b_t' y \leq a_t \text{ }\forall t\in \{1,...,T\} $$ where $c$ is a $J\times 1$ vector ...
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Problem: You need to pack several items into your shopping bag without squashing anything. The items are to be placed one on top of the other. Each item has a weight and a strength, defined as the ...
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I am looking for a analytical function that applies to the constraints: \begin{equation}f(x) = \begin{cases}1 \text{ if } x=1 \\ \alpha \text{ otherwise }\end{cases}, \text{where } 0 \leq \alpha \leq ...
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I would like to add a new constraint to a standard Knapsack problem by introducing groups. My variables are $x_c \in \mathbb{Z}^+, c\in \mathbb{C}$. Where $\mathbb{C}$ is the set of all items. Each ...
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We have four variables: a, which represents numbers from 0-999 b, which represents a1000 (<...
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