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Questions tagged [constraints]

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

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express constraint violation

A very simple question, what's the mathematical symbol (expression) that represents constraint violation. Specifically, we have a set of constraints R each of which taking three variables, two sets of ...
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1answer
60 views

How can I show that these two problems have the same optimal solution?

How can I show that these two problems have the same optimal solution: $$\inf \{ x^TAx + b^Tx : 1-x^Tx \ge 0,\ x \in \mathbb R^n\}$$ $$\inf \{ x^TAx + b^Tx : 1-x^Tx = 0,\ x \in \mathbb R^n\}$$ when ...
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740 views

Fit an ellipse with constraints

I'd like to fit an ellipse with the equation of $ x^2 + ay^2 + bx + c =0 $ This is basically the equation of an ellipse with no tilt and with its center on the horizontal axis. I have some ...
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809 views

How to mathematically describe a loop over a set with two indexes.

I have a set of sets $G = \{D_{0,0}\,D_{0,1}\,D_{0,2}\,D_{1,0},...,D_{n,0}\,D_{n,m}\} $ What I know want to express is a constraint that for each set in $ G $, if $ x \in D_{0,0} $ then the statement ...
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54 views

How to mathematically describe the number of Element x in a set

I am trying to formulate the following. I have a Set A={x, y, z}, I also have a Set B, C and D, which all are subsets of A. It is not exactly defined which elements are in B, C and D. I only want to ...
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369 views

Linear Programming Problem - Looking for an Explicit Solution

How can I solve a linear program of the form: $$\min c^Tx\\ \mathrm{s.t.}\ Ax=b\\ x\geq0\\$$ where $c$ is fixed. In the specific case I am looking at, $$x \in R^n$$ $A$ is an $m\times n$ ...
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626 views

Constrained Optimizatoin: The Frank-Wolfe Method

A general convex optimization problem is framed as such: $$\min f(x) : x \in \Omega$$ where $\Omega$ is convex. The Frank-Wolfe method seeks a feasible descent direction $d_k$ (i.e. $x_k + d_k \in \...
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115 views

Every straight line in the complex plane can be represented $\overline a z+a \overline z+b=0$

I'm trying to solve the following problem: "Show that any straight line in $\mathbb R^2$ can be represented via the complex equation $\overline a z+a \overline z+b=0$ ; $a\neq 0 \in \mathbb C,b \in \...
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1answer
96 views

unnecessary constraint in optimization problem

I have some optimization problem (optimizing parameter $\alpha$)with those constraints: $$\alpha_i\ge0$$ $$\sum\limits_i \alpha_i y_i =0$$ and a third constraints: $$w-\sum\limits_i \alpha_i y_i x_i = ...
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3k views

Notation: is a factor of

How can one write $x$ is a factor of $y$ (as a constraint)? I am also not sure what else to add to meet the question quality requirements.
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Curiosity - maximising a product with a constraint

I have integers greater than 4, for instance $i_1$, $i_2$, $i_3$, ..., $i_n$. We have to change the greatest of these integers (for instance $i_1$ if they are ranked by descending order) by adding to ...
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52 views

restricting number of zeros in a vector

I need to make an M dimensional vector and restrict it to have R entries as zeros. Is there any expression or condition in vector form that can ensure this ?
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How to maximize area of two circles inside a rectangle without overlapping?

Two circles have to be drawn inside a rectangle of dimensions $W\times H$ such that the area of both circles is to be as large as possible without overlapping. Let the radii of the circles be $r_1$ ...
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2answers
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Constrained Newton-Raphson method

Peace be upon you, I want to solve a system of two equations in which the existence of $ln\left(\frac{\alpha}{\alpha+\beta}\right)$ function makes some limitations in iterations of the Newton-Raphson ...
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1answer
6k views

How to linearize a quadratic objective function with linear constraints?

I have an optimization problem that I'm working on. The objective is defined as follows: $Maximize: c_i\cdot w_i \cdot x_i - d_i \cdot y_i \cdot \delta_i $ subject to some linear constraints where $...
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570 views

Critical points, minima and maxima of a constrained optimisation problem

$f(x_1,x_2,x_3) = x_1 x_2 + x_2 x_3 + x_3 x_1$ with the constraint $x_1+ x_2 + x_3 = 1$. Now I test the critical points by taking $\nabla f = 0$, hence at $(x_2 + x_3, x_1 + x_3, x_2 + x_1) =0$. This ...
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477 views

Trace minimization subject to constraints

I have seen in an article that $ \min_{\mathbf{K}} \hspace{0.2cm} tr[\mathbf{K} \Sigma \mathbf{K}^T]$ s.t. $ \mathbf{KH} = \mathbf{I} $ where $\mathbf{H}$ is of full column rank yields, $\tilde{\...
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142 views

Closest Positive-Definite Matrix Subject to a Contraint

Given a positive, semidefinite, real 2n by 2n matrix $A$, is there a formula or an algorithm that finds the closest (in some sense, preferably Frobenius distance) positive, semidefinite, real 2n by 2n ...
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Solution of a DAE system of two ODE of second degree

I should solve the following DAE system: $$\ddot{x}(t)=-\alpha y(t)$$ $$\ddot{y}(t)=\beta x(t)$$ with the conditions: $x(t)\ge0$, $y(t)\ge0$ and $x(t)+y(t)=N$ with $N\gt 0$. I'm able to solve the ...
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180 views

Quadratic Optimization Problem with Box Constraints

I want to solve a problem of form $$\min_x x'Ax + b'x \;\;\mbox{ s.t. } l\leq x \leq u$$ where $A$ is a positive semidefinite matrix, thus the function I'm optimizing should be convex. However the ...
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Strong duality in trace maximization

I'm working on understanding the derivation of the solution for principal components analysis. Let $\mathbf{S} \in \mathbb{R}^{p \times p}$ be a positive semi-definite matrix with rank $d < p$. ...
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2answers
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Can we solve $q(x)p'(x)+2p(x)q'(x)=0$ given constraints?

If we suppose that we want $-p'(x)q(x) = f(x)$ for a given $f(x)$, and $$q(x)p'(x)+2p(x)q'(x)=0$$ Can we get $p(x)$ and $q(x)$?
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ODE with constraints

Given the ODE system $$\dot{x} = y \\ \dot{y} = \frac{1}{\alpha} (z - y)$$ where $\alpha > 0$ is a constant. How can I find a bound for $z$ depending on $x$ such that $\forall t ~x(t) \geq 0$ under ...
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1answer
22 views

Path consistency without consistency

My math teacher has asked us to find a (not trivial) problem that is path consistent, but not consistent. I have found ones that are arc-consistent and not consistent, but I have not been able to find ...
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1answer
176 views

Optimization problem with a minimization sub-problem as a constraint

I have a problem, for predefined $x_0,z\in\mathbb{R}$, which looks like $$\min_{\alpha,x} \sum_{i=1}^n \alpha_i f_i(x_i,z) $$ subject to \begin{align} \sum_{i=1}^n \alpha_i &= 1 \\ \sum_{i=1}^n \...
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1answer
53 views

Computation time

I am implementing a mixed-integer linear programming problem, and I am dealing with an huge number of constraints. Does anyone know what the linear relation is between the number of constraints of ...
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2answers
2k views

Max and Min using Lagrange Multipliers

Suppose A is a symmetric matrix. Show that the maximum and minimum of $\mathbf x ^T A \mathbf x$ subject to the constraint $\mathbf x ^T \mathbf x=1$ are the maximum and minimum eigenvalues of A. I ...
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323 views

Optimization problem (total distance from point on sphere to other points)?

Let $M_i(x_i, y_i, z_i)$ be a set of $n$ fixed points. Given their coordinates, find a point $M(x, y, z)$ which is on the sphere $x^2 + y^2 + z^2 = 1$ and has the minimal sum of distances between ...
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1answer
119 views

Impact of constraints on convex optimization

Let me start by saying I know almost nothing about optimization so please bear with me. Basically, I am wondering whether it is possible to solve a problem with two constraints by solving the problem ...
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1answer
50 views

What values or form of values can we get for these multiplications modulo a prime?

If we have four complex values, all of the form $a + b i$, for integers $a$ and $b$, we can label them $c$, $d$, $e$ and $f$. Now if we want to find $g$ and $h$ such that $$g \equiv ce \equiv df \...
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71 views

Is a given point P outside a given bounding box, in Ax < b form

Given a point $x$ and a bounding box $B$ - let's say we have the unit normals $N_i$ of the sides (pointing inwards) and one point on each side $P_i$ - we can check if $x$ is inside $B$ as follows: $\...
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How can we prove that we can get any $f$ modulo a prime $p$, that satisfies these equations?

We can suppose that we have four naturals not equal to zero: $a, b, c, d$. Further, we're working modulo a prime $p$. Now if we find $a, b, c, d$ that satisfy: $$f \equiv a \cdot c \equiv b \cdot d ...
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1answer
1k views

Projected gradient method with multiple constraints (simplex)

I am trying to minimize convex objective $f(X)$, for matrix $X$ s.t. $X\ge 0$ component-wise, and $X1^T = 1^T$. I want to use projected gradient descent. However, I only know how to project on ...