Questions tagged [constraints]

In mathematics, a constraint is a condition of an optimization problem that the solution must satisfy.

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How does this pair of Multivariable Calculus problems relate to the concept of duality?

Consider the following pair of a priori independent problems... A manufacturer's revenue is $\$100s^{\frac{1}{3}}h^{\frac{2}{3}}$, where $s$ is the number of tons of steel they purchase and $h$ is ...
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Question about Lagrange multipliers and combining constraints

Hi I have this question about Lagrange multipliers and specifically when there are 2 constraints given. The standard answer to this question uses the lagrangian and 2 constraints with 2 extra ...
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Error of Constraint Optimization Problem

My problem concerns the big-O error after transforming a constraint optimization problem. I'm not really familiar in this field so sorry for any incorrect notation/logic in my reasoning. I will first ...
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Reformulation of Convex Constraints

I am trying to reformulate the constraints $$ \alpha^\intercal L \beta + \|L^\intercal \alpha\|_{2}^{2} \leq \rho, $$ where $\alpha\in\mathbb{R}^{n},\beta\in\mathbb{R}^{m}$ and $\rho\in\mathbb{R}$ are ...
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how do I write a constraint in the form $g(x, y) = 0$ for Lagrange multiplier [closed]

I understand that the constraint of an optimisation question has to be written in the form $g(x, y) = 0$. How do I write a constraint such as $x, y, z$ are all greater or equal to zero as $g(x, y) = 0$...
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Maximum of a polynomial function on the unit sphere in $\mathbf R^3$

Given three positive real numbers $a,b,c>0$, consider the following function, defined on $\mathbf R^3$ $$ f(x,y,z) := y^2(a^2x^2+b^2y^2+c^2z^2). $$ Question. What is the maximum of $f$ on $\mathbf ...
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How do I determine the maximum subgraph that avoids certain edges?

I have a graph where I want to select a subset of the nodes subject to a particular constraint: An edge between nodes A and B indicates that I cannot select BOTH A and B to belong to the subset. I ...
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Advice on solving Lagrange Multiplier equations

I am trying to solve this equation: Find max and min values of $$ f(a,b,c) \left (\frac{1}{abc}\right ) $$ Subject to: $$ a^2 + b^2 + c^2 = 1\ $$ I calculated the system of equations $$ \frac{1}{a^...
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Linearization of nested absolute value objective $|a-b-|c||$

I am trying to define an optimization problem that minimizes the distance between $a(x)$ and $b(x)$, where I need to adjust $b(x)$ downwards using the cost function $c(x)$ (hence, the cost must always ...
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Simplifying the set of constraints of an optimization problem

I’m currently working on a constrained optimization problem (the problem comes from the European Power Market) where the constraints define a solutions space which forms a complex polytope with many ...
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Minimize the $L^1$-norm with linear equality constrain [closed]

How can I solve efficiently the problem: $$\min_x \| A x \|_1 \quad \text{subject to} \; {b}^{T} x = c$$ for $A \in \mathbb{R}^{m \times n}, b \in \mathbb{R}^{n}, c \in \mathbb{R}$. The matrix $A$ ...
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How to linearize or reformulate an implication constraint that implies that a decision variable belong to an interval?

I am an electrical engineer who is working in computer network and I need to model my delay with respect to a binary variable $x$ as folow $\left\{ {\begin{array}{*{20}{c}} {x = 1 \Rightarrow \left( {...
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Optimal control problem with inequality constraints at discrete times

I am interested in solving the following optimal control problem $$\min_{\mathbf{u}(t)} \ \phi(\mathbf{x}(t_f),t_f)+\int_{t_0}^{t_f}\mathcal{L}(\mathbf{x}(t), \mathbf{u}(t), t)\, \mathrm{d}t$$ ...
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Prove optimality of constrained convex optimization problem analitically (using KKT conditions)

I'm trying to prove that the constrained convex minimization problem with decision variable $\boldsymbol{x} \in \mathbb{R}^{n}$ given by $$ \min_{\boldsymbol{x}} \Vert \boldsymbol{x} \Vert_{2} \text{ ...
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Proof of solvable constrained optimization with a subset of a larger set of constraints

Way out of my comfort zone here so apologies if I'm not providing enough or correct information. I work with an application of constrained optimization to assemble test forms (automated test assembly)....
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Deriving correct integral constraint equation for calculus of variations problem

I have the following calculus of variations problem: $$\mathcal{L}=-2X'(t)\ln{x'(t)}-2Y'(t)\ln{y'(t)}$$ where $x(t)$ and $y(t)$ are the functions I'm interested in, and $X(t)$ and $Y(t)$ are given as ...
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How to add discrete function constraints to a calculus of variations problem

Suppose that $L$ is the Lagrangian of a system and $f$ is a function of $x$. The objective is to find a function $f$ that optimises: $J[f] = \int{L(x, f, f')}dx$ How do you fix the value of $f(x)$ at ...
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Gradient vector explanation (please help)

I'm trying to explain how to use Lagrangian multipliers, through an example. I start in this way: Understanding how Lagrangian multipliers work can be done through a simple example. Consider the ...
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How to prove that optimization problem solution is/is not differentiable with respect to constraint variable?

Assume you have the following simple constrained optimization problem: $$h(c)=\min_{g(x)=c} f(x)$$ where $f,g$ are both differentiable. What are the standard way to show that the problem solution $h(c)...
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Solving a matrix optimization problem (part 2)

Problem definition This post is a follow up of this previous one. Consider the following $m\times(n+1)$ matrix \begin{equation*} B(\sigma)\triangleq \left[\begin{array}{ccc} b_0^n(s_1) & \cdots &...
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How to reformulate linear constraint into a geometric programming constraint? [closed]

Arcording to this post, smart manipulation of difficult non-linear constraints by a change of variable can lead to a formulation of Geometric Programming. $\begin{array}{*{20}{c}} {\min }&{\...
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Can we assume for free the upper bound for a variable in the minimization problem $\min_{(x,t) \in \mathbb{R}^n\times \mathbb{R}} t$?

Problem: Let us consider the following minimization problem \begin{align*} &\min_{(x,t) \in \mathbb{R}^n\times \mathbb{R}} t\\ \text{s.t }& \Vert x-q_i\Vert^2 \le r_i^2 + t,\ \forall i = \...
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Minimising inconclusive range for binary classification with two thresholds

Background I have an optimisation problem related to a binary classification tasks. I have arrays of probabilistic predictions, $x$, and true binary labels, $y$, i.e. $x_i\in [0,\,1]$ and $y_i\in \{0,...
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Constraint Integer Linear Programming

I have an integer linear programming problem where i want to maximize over $\{0,1\}^n$, so i have the problem $$\max_{x \in \mathbb{R}^n}c^Tx, \text{ subject to } x_i \in \{0,1\} \text{ for all } i ...
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Pump filling constrained discrete optimization

Given three tanks $A, B, C$ with capacities of $12,8,5$ litres of water each and initial condition tank $A=12$ we want to fill containers $A$ and $B$ to have equal amount of water. Allowed actions are ...
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How to extract a connected tree of nodes from undirected graph with certain attributes?

I am working on a problem where I need to extract a connected tree of nodes based on certain attributes while optimizing for the minimum number of nodes. Some attributes of the nodes are known in ...
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How to understand asymptotic normality under constraints?

Consider the following constrained maximum likelihood problem: \begin{align*} \min\limits_{\theta \in \mathbb R^d}~ & -\log p(x_{1:n};\theta) \\ {\rm s.t.} ~~& f(\theta)=0. \end{align*} Let $F(...
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How to avoid a solution of zeros in a quadratic program?

I want to find optimal $\beta \in \mathbb{R}^r$ that solves the following quadratic program: $$ \min_{\beta \in \mathbb{R}^r}\;\; \left(G\beta\right)^{\prime}W\left(G\beta\right)\;\;\; \text{s.t.}\;\; ...
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Different Approaches for Proving Kantorovich Inequality

Here is a statement of the famous Kantorovich inequality. Thoerem (Kantorovich). Let $A$ be a $n\times n$ symmetric and positive matrix. Furthermore, assume that its eigenvalues are $0 < \lambda_1 ...
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How to reformulate or linearize the phrase "become redundant" or "not needed"?

I am an electrical engineer and currently I have to deal with an optimization problem with a very specific requirement: $\begin{array}{*{20}{c}} {\mathop {Min}\limits_x }&{f\left( x \right)}\\ {{...
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What are the requirements for a diagonally dominant matrix to have a diagonally dominant square?

Given a diagonally dominant matrix, i.e., a matrix $A$ whose entries satisfy $$|a_{ii}| \geq \sum_{j\neq i}|a_{ij}|\quad\forall\enspace i,$$ what other constraints are required to guarantee that $A^2$ ...
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Example of function which behaves linearly on $[0,1]$ and admits an asymptote as $x\to \infty$

Problem : Let $f(x)$ be a positive differentiable function on $x>0$ such that : Let $x\in[0,\varepsilon]$,$0<\varepsilon<1$ such that : $$f(x)\simeq ax+b$$ And : $$\lim_{x\to\infty}f(x)-cx-d=...
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How to add Linear Matrix Inequality (LMI) constraints to a Semidefinite program (SDP) in standard form

Given an SDP problem with $m$ equality constraints and one Linear Matrix Inequality (LMI) in standard form: $$ \begin{align} \min \quad & \mathbf{F}_0 \bullet \mathbf{Y} \\ \text{s.t.} \...
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Constraint forcing maximum parameter value to constant

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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How to numerically optimize a function with 'discontinuous' box constraints?

First of all: I am an engineer and no mathematician, therefore please excuse my 'lazy' form or prensenting the problem. I want to solve a nonlinear and constrained optimization problem: \begin{align} ...
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How to incorporate Lagrangian multipliers into matrix system when constraint is highly nonlinear

I have a dynamical system which I am modelling using Lagrangian analytical mechanics. The Euler-Lagrange equations for my $N$ generalized coordinates can be arranged into a (linearized with respect to ...
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Relaxing a binary variable in a Mix Integer Programming problem

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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Find the minimum and maximum value of a for $ 9\sum_{\mbox{cyc}} a^2 +10(ab+ ac+ad+bc+bd+cd) = 16$. [closed]

With ordered 4 real values satisfying the constraint as below, find the possible range of the largest one (i.e., the allowed interval for the variable $a$): Let $a \geq b \geq c \geq d$ be reals ...
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Proving that a set of quadratic constraints have no solution

I want to prove that a given set of quadratic and linear inequalities has no solution. The set size is of 13 equations, or, including the positiveness constrain, 21 equations. I already used a python ...
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Minimizing a functional subject to boundary conditions

I want to find a smooth function $y : [0,1] \to \mathbb{R}$ that minimizes $$ S = \int_0^1 (y(x)y'''(x) + 3 y'(x) y''(x))^2 dx $$ subject to the constraints / boundary conditions: (i) $y(0)=1$, (ii) $...
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Optimization constraint

I'm learning optimization and I came across equality constraints. Lets say we have an objective function J defined over $V$ with values in $\mathbb{R}$, where $V$ is the normed vector space in which ...
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Optimisation Problem Setup and KKT conditions

I am trying to setup an optimization problem with equality and inequality constraints. I want to estimate a specific variable call $\chi$ subject to a minimisation problem -- guidance on the setup and ...
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Point isolation through linear constraints

I consider a set of $n$ points in $\mathbb R^d$: $X=${$x_i$}$_{i=1}^n$. I would like to know for each point $\tilde x\in X$ if there exists a dimension $j\in${$1, ..., d$} and $k$ linear constraints (...
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Discrete point inside a polygon formed by set of vertices

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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Same sign constraint in linear optimization problem

I'm trying to find a way to force a group of variables to take the same sign. Either positive or negative. Clearly, the fact that an OR statement is required implies that the problem becomes non-...
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Proving a Constrained Ineqality

Suppose I have the following pairs of variables: $a$ , $\hat{a}$ $b$ , $\hat{b}$ $c$ , $\hat{c}$ $d$ , $\hat{d}$ With the following constraints such that $a, \hat{a}, b, \hat{b}, c, \hat{c}, d, \hat{...
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Primal Interior-Point Methods

It is common that when one solves the nonlinear inequality constrained problem $$ \min_{x\in\mathbb{R}^{n}}f(x) \\ \text{ such that } c_{i}(x)\geq 0,\,\,\,i=1,2,\dots,m $$ that one introduces slack ...
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Opinion on Barrier/Penalty method for box constraints in Optimization

I had the idea to use an additional term in an numerical optimization problem for a box constraint. It is somewhat a mix between penalty and barrier function and I am wondering what the drawbacks are ...
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Can the implication $(x_1 = 0) \rightarrow (p(x_1,...,x_n) = 0)$ be encoded in a system of polynomial constraints in $\mathbb{C}[x_1,...,x_n]$?

Consider a set $S$ of polynomials in $\mathbb{C}[x_1,x_2,...,x_n]$, the polynomial ring of $n$ variables over the complex numbers. The set $S$ can then be interpreted as a system of constraints on the ...
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Uncertainty analysis in maximum likelihood estimation under constraint

I'm not from a statistical background so you might have to excuse me for my somewhat inaccurate (or even erroneous) phrasing, I'll try the phrase my problem as I understand it. The maximum likelihood ...
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