Questions tagged [constraints]

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

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Name of this parametrization of the region $\{ 0 < x \leq y \leq z \}$

I'm working on a problem where I have to assign a probability density $p(x,y,z)$ on $\mathbb{R}^3$ which is non-zero only on the region $$ R(x,y,z) = \{ 0 < x \leq y \leq z \}. $$ To this end I use ...
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103 views

10 similar quadrilaterals with 8 points.

Let 8 points define 10 or more similar quadrilaterals with no self-mapping. No self-mapping means that a square counts just once instead of 8 times. What is the maximal number of differently sized ...
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15 views

Distribution of random numbers with fixed sum

I have access to a black box function $f$ that returns 4 random integers $n_1$, $n_2$, $n_3$, $n_4$ with $4 \le n_i \le 13$ and $\sum_i n_i = 25$. Experimentally, I can see that $n_1$, $n_2$, $n_3$ ...
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32 views

Constraint Optimization and Lagrange Multipliers (Methods of Optimization)

Newbie question here. So I am starting to learn about constrained optimization in my multivariable calculus course and I was taught how to use the Lagrangian and Lagrange multipliers to solve an ...
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Drawing Level Curves and Gradients on the same graph

$g_1(x,y) = x^2 - y$ , $g_2(x,y) = y$ , and $g_3(x,y) = x$, Firstly I drawed the level curves corresponding to $g_1(x,y) = 0$, $g_2(x,y) = 0$, and $g_3(x,y) = 1$. Then, I shaded the region all points ...
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Show set of solution to constraint is compact

This is the question: Show that the function $f(x,y) = x^4 + y^4$ takes maximum and minimum values along the curve $x^4+y^4-3xy = 2$. A solution from my teacher: Since $f$ is continuous, we only ...
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72 views

Understanding the adjoint (state) method: existence and uniqueness of the adjoint equation

Dear math enthusiasts, I recently came across the adjoint (state) method in the context of sensitivity analysis of model perturbations to systems described by PDEs. I am a novice in the area so I was ...
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23 views

Loss Function vs. Penalty vs. Lagrange Multiplier

Suppose I want to maximize $u=c_1+c_2$ subject to a budget constraint $p_1c_1+p_2c_2\leq m$ and a requirement $c_2<\overline{c_2}$ that must be satisfied. The source of the $c_2<\overline{c_2}$ ...
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34 views

Constrained optimization problem with amplitude constraints

My problem is based on beamforming. I want to minimize the consequence of $\|Hw\|_2^2$, where $Hw\in \mathbb{C}^{N \times 1}$, is the power of signal coupled to receive antennas from the transmit ...
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29 views

Solving a system of equations with constraints on the values we want to find

In one paper I find these set of equations: $$ u_1 = b( \omega_1^2 + \omega_2^2 + \omega_3^2 + \omega_4^2)$$ $$ u_2 = b(\omega_1^2 + \omega_2^2 - \omega_3^2 - \omega_4^2)$$ $$ u_3 = b(\omega_1^2 - \...
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47 views

How can I mathematiically enforce a set of constraints on a range from 0 to 1?

I have a function $f(x)$ which needs to be bounded between 2 functions $g(x)$ and $h(x)$. Functions $g(x)$ and $h(x)$ are guaranteed never to intersect. I have a function $f(x) = a + bx + cx^2$ and I ...
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Transform a constrained multidimensional integral into an unconstrained one

A quantum-mechanical "density" matrix is Hermitian (self-adjoint), positive definite, having trace 1. In terms of its four ordered eigenvalues ($\lambda_1 \geq \lambda_2 \geq \lambda_3 \geq \...
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31 views

Intuition on a constrained optimization problem

I've solved a) and b), the problem I'm having is with c), I don't intuitively see why there must be an infinite number of solutions to this problem. The only thing I intuitively can see is that I ...
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47 views

Constrained minimization: characterizing derivatives of optimum with respect to parameters

Suppose I have a function $f(x,y)$ defined on $[0,1]^2$ that is equal to the value of the following constrained minimization problem: $$f(x,y) = min_{a\in [0,1-x),\ b\in[0,x)}\ \left\{ h(x,a) + g(x,b) ...
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28 views

Monotonic increasing function with constraints

I am looking for examples of monotonically increasing functions f(x) which can satisfy the following constraints: f(x=0) should be 0 and f(x>=25) should be 1. I want one or more parameters in the ...
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65 views

How to use Dirac function to represent constraint in an integral?

Hello and I want to do integral under constraint like: $ \int_{f(x1,x2...xn) = 0}g(x1,x2,...xn)\rm dx1dx2...dxn $ some book told me I can use Dirac function to represent constraint as: $ \int_{x1,x2......
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Constrained optimisation: stationary points of constrain

I'm new to optimisation and have a problem. I'm supposed to find stationary points to the following function $f$ under the constrain $g$: $$f(x,y) = xy$$ $$g(x,y) = x^4 + y^4 + 2xy - 4 = 0$$ which ...
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How to calculate how many “double-downs” can be done given a total balance and a starting bid

I want to know how can this be calculated mathematically. Given a total balance $\mathrm{B}$ and an initial bet of $\mathrm{b_{init}}$, I want to calculate the max number of double-downs ($\mathrm{X}$)...
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Ways to describe matrix whose elements are linear combinations of elements of a vector?

The motivation for this question comes from trying to solve the following system: $$B = AX: \left[X\right]_{ij}=\sum_{l=1}^k \alpha_{k}^{(ij)}c_k$$ Where $A,B \in \mathbb{R}^{N\times p}, \hspace{0.1in}...
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49 views

Constructible problems for which the solution is non-constructible?

For the sake of this question, I am using the word "constructible" in the sense of constructive mathematics: e.g. a real number is constructible if you can construct a Cauchy sequence for it ...
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28 views

Independent conditions of unitary matrix, $U U^{\dagger}=1$ and $U^{\dagger}U = 1$

I am trying to show that, for a simple 2x2 complex matrix \begin{equation} U = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \end{equation} there will be only 4 real constraints for $U$ to be ...
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82 views

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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140 views

Linear objective function with determinant as a constraint

So I have the following optimization problem $$ \begin{aligned} \max_{x:=(x_1,\dots,x_n)^T\in\mathbb{R}^n} \sum_{i=1}^n x_i \\ \text{s.t. }\quad \text{ det} G(x) = 0 \end{aligned} $$ were $G(x) = I+\...
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43 views

Solve the inequality $ a^{ab}c+b^{bc}a+c^{ca}b\leq 1-2abc$

Let $a,b,c>0$ such that $a+b+c=1$ then we have : $$ a^{ab}c+b^{bc}a+c^{ca}b\leq 1-2abc\quad (1)$$ I have a proof : I was thinking for an alternative proof considering by example Young's ...
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9 views

Build a heart like curve following some constraints

Hi it's a funny problem . Working on some curve I have found a Heart's curve let me propose it . There is four curves to define the heart's curve .First : Let $0< x< 1$ then define : $$f(x)=x^x+(...
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42 views

Maximize sum of logarithms subject to constraints

I have the optimization problem \begin{align} \: \max \: \sum_{i = 1}^{M} \log_2\left(1 + \frac{S_{i}}{N_i}\right) \\ \text {Subject to} \: \sum_{i = 1}^{M} S_{i} \leq P_T \end{align}. $N_i$ are ...
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How to linearize a non-linear constraint in an optimization problem?

I am very new to optimization, and I wouldn't know where to begin with this problem that I have so I would truly appreciate any and all help on this - I am trying to find a vector $\boldsymbol{x} = \...
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General algorithm for addition of two constrained numbers

I am looking to write a general algorithm for addition of constrained numbers and am wondering if someone could give me some guidance on where to start or if there is already a paper on strategies for ...
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Question about Constrained Differential Solution

https://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/2.-partial-derivatives/part-c-lagrange-multipliers-and-constrained-differentials/session-42-constrained-differentials/...
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38 views

How to find a closest point on a constraint to a given point?

(This is a multi-variable calculus problem, not a linear algebra one) Find the closest point to the point $(2,7,8)$ on the constraint: $$ 4x + 7y = x+3y+5z$$ And find the projection of the vector $(2,...
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How to obtain a rank-5 solution to an optimization problem

Say I want to determine a matrix solution $A^*$ to the minimization problem $$\min_{A \in \mathbb{R}^{n\times n}} f(A)$$ with the constraint that this solution $A^*$ must have rank $k$, where $k \ll n$...
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83 views

Find the maximum value of $x^2y$ given constraints

Find the maximum value of $${ x }^{ 2 }y$$ subject to the constraint $$x+y+\sqrt { 2{ x }^{ 2 }+2xy+3{ y }^{ 2 } } =k$$ where k is a constant. I tried it by substituting value of x and then ...
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Expressing inequality constraints as equality

Is it possible to express inequality constraints as equalities? I have a system of linear equations that I am trying to solve where the system is subject to a set of inequality constraints. The ...
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32 views

How to constrain a rectangle within an arbitrary 2d polgyon?

I am working on a graphics project wherein I need to keep a rectangle (assume it has a fixed but arbitrary size, rectWidth * rectHeight) constrained inside an arbitrary polygon. The polygon is ...
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26 views

How to solve an objective function with inequality constraint?

I have an objective function as follows, $\displaystyle\min_{\mathbf{A},\mathbf{B}}\lVert \mathbf{X}-\mathbf{ABZ} \rVert_F^2$, where $\mathbf{X},\mathbf{Z}\in \mathbb{R}^{p\times n}$, $\mathbf{A}\in ...
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35 views

constrained optimal control excluding the optimal state

I don't know if this is a very basic question: Let's say there is the typical optimal control problem with the cost function $$J = \int_{0}^{T} \mathcal{L}(x(t),u(t),t)\mathrm{d}t + \Psi(x(0),t=0)$$ ...
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42 views

Closed form for the maximum of the two-variable function $(a x + (1 - a) y) (b (1 - x) + (1 - b) (1 - y))$

I'm trying to characterize the maximum of this function within the unit interval ($x,y\in [0,1]$): $$f(x,y)=(a x + (1 - a) y) (b (1 - x) + (1 - b) (1 - y))$$ for $0 < a < 1$ and $0 < b < 1$...
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Efficient Method to Find A Root and/or Minima of A Multivariate Polynomial With Constraints on the Variables

I have a polynomial $P(x,y,z,...)$ which has integer coefficients and real roots. I want an efficient method that is guaranteed to find a root (I only need one), and if possible, one that finds a ...
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98 views

Identify boolean function that satisfies some constrains

The problem I want to find a boolean function $f(x,y):\{0,1\}^n \rightarrow \{0,1\}$, where $x=\{x_i\}_{i=1}^{m}$ and $y=\{y_i\}_{i=1}^{k}$ are $m$ and $k$ boolean variables, such that: $m,k \ge 1$ (...
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29 views

If evey linear program can be transformed to an unconstrained problem, then the optimum is unbounded because the objective is linear?

Since optimization problems with linear equality constraints can be converted into an unconstrained problem this should apply for linear programs in standard form, right? But doesn't this mean that ...
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53 views

Maximization of quadratic form subject to a set constraint

Given a quadratic form $x^tAx$ where the matrix $A$ is symmetric and $x^tx$ = 1, we can deduce that the maximum of the quadratic form is the first eigenvalue of the matrix $A$ (also, the first ...
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37 views

What constraint am I missing for this SAT optimization problem?

I'm trying to solve a variation of the bin-packing problem. I have a certain floor space and I wish to place as many boxes as I can without stacking, i.e. if the floor is a 4 x 4 grid and I have one ...
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19 views

Alternative definition of an active constraint

In constrained optimization, an active constraint is generally taken to mean one whose inequality sign can be changed into an equality sign without affecting the optimum. However, I would like to have ...
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84 views

Lagrange Multipliers and quasi-Newton methods

Consider an optimisation problem of the form $$ \begin{aligned} &\min f(x)\\ &\text{s.t. } g(x) = 0 \end{aligned} $$ with $f,g: \mathbb{R}^n \to \mathbb{R}$ convex and twice continuously ...
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Optimization constraints for types of oil

I'm looking at a lecture online which shows the following problem They have two types of oil and they want to produce standard and premium gasoline. However i'm confused how they've arrived at the ...
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36 views

Safe packing Constraint satisfaction problem - is it optimal?

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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12 views

Efficient intersection of linear subspaces? How to solve a big nonlinear least squares with linear constraints?

Problem: I want to solve a big linearly constrained nonlinear least squares problem. The number of unknowns is between millions and billions. In terms of cost functions, the Hessian would be pretty ...
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20 views

How to verify whether a optimization problem with a L2 constraint (quadratic constraint) and a linear constraint has its closed-form or not?

I have already known that the optimization with a quadratic constraint can be solved using Rayleigh quotient, and with a linear constraint can be solved by Lagrange method. But what if the two ...
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19 views

Existence of solutions to constrained inequality

Consider a system of $N$ points in 3D Euclidean space (although it would be cool if it can be analyzed in general dimensionality) with weights $0 < w_i \leq w^{\bf{max}}$ and pairwise separations $...
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35 views

Find a vector that is as proportional as possible to a given vector under a set of linear constraints

Let $d\in \mathbb{R}^n, \ b\in \mathbb{R}^n, \ A \in \mathbb{R}^{m\times n},\ \lambda\in \mathbb{R}$. Let $x=\lambda d+\varepsilon $, where $\varepsilon\in \mathbb{R}^n$. Let $E_\lambda =\left \{\...

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