Questions tagged [constraints]

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

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Least Squares Solution of Minimal Norm when $A^{*}b = 0$

Suppose, given a matrix $\textbf{A} \in \mathbb{C}^{m \times n}$ and a vector $\textbf{b} \in \mathbb{C}^{n}$, I want to find the minimal norm solution of $$\min_{\textbf{x}}\|\textbf{A}\textbf{x} - \...
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132 views

Binary variables in time series: integer linear programming

I'm working on a problem and I can't seem to find an easy solution to it. It's about an optimization problem, concerning a time series. I have a binary variable $\alpha_t$ for $t \in [0, 24[$. I ...
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18 views

Conditions for two B-Splines to represent the same curve

What will be the weakest constraint for two B-Spline curves $S_1(t)$ and $S_2(u)$ to represent the same curve in space? Assume that their orders are $k_1$ and $k_2$ respectively, knot vectors are ($...
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64 views

Combining inequalities to not have coefficients

We have in input an inequality cons and a set of inequalities C and we want to find a way to sum them and simplify in a way that no variable has a coefficient and that the number of constraints that ...
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0answers
22 views

$L_{2,0}$ constraint in the optimization problem

I am trying to solve a minimization problem where my constraint is $$||W||_{2,0} = 1$$where $W \in R^{k \times k}$. The constraint is used in such a way that only one column of $W$ matrix is non-zero. ...
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1answer
25 views

Minimise $C(x,y)=11x+3y$ subject to the constraints.

Minimise $C(x,y)=11x+3y$ subject to the constraints $ g(x,y)=-3x^2-3y^2+10xy $ and $x\geq 0, y\geq 0$. I started solving using this Lagrange multiplier, but the constraint set is not compact, right? ...
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15 views

Sampling from joint discrete distribution

I have a set of items $a_1, a_2, \dots, a_n$. My aim is to generate from this set of items, a list of item tuples $\{(a_i, a_j), \dots\}$ such that $a_i\ne a_j$. The constraints are as follows. The ...
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1answer
28 views

Fulfilling conditions of Inverse function(?)

Let's assume that $$h(g(t))=t$$ What conditions ae needed to say that $$g(h(t))=t$$ Is satisfied too? (Provided that $h$ and $g$ are continuous and derivatable, but not knowing whether they have ...
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2answers
54 views

Calculus 3: Lagrange Multipliers

Find the minimum and maximum of $f(x,y,z) = x^2+y^2+z^2$ subject to two constraints, $x+2y+z=8$ and $x−y=5$. Looking at the equation, it's clear that there is no maximum. After working this problem ...
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1answer
36 views

Existence of Positive Solutions to Constrained Linear Elliptic Second-Order PDE

Consider the elliptic second-order PDE on some bounded domain $D$ (in any dimension) $-\Delta_D u + \alpha u = 0$ subject to the constraint $\gamma^i \nabla_i u + \beta u = 0$, where $\Delta_D$ is ...
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42 views

Imposing a given constrain (f(x,y)>0) in a variational problem

My problem I am trying to solve a chemistry problem stating it as a constrained variational problem. I am learning this subject by myself and I have problems imposing a non-integral constrain. One ...
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2answers
16 views

Solving an expression with three unknowns (Constraints?)

I have the volume of a cuboid, and I want to find all of the possible dimensions that would evaluate to a cuboid of that volume. The volume is defined as xyz = 5000...
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41 views

Follow a circumference path with a constraint

I have this problem. I tried a lot of options before to post this question. By referencing the above, I have an object (the blue point) that is moving towards $a)$ where it finds a value of $n = 40$. ...
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13 views

Constrained Linear System (X in SO(3)) [closed]

I'm trying to figure out what is the minimum number of rows for matrix A in the system: $AX=b$ Where X is a vector of 9 elements of a rotation matrix ($R \in SO(3)$). Since a rotation matrix can be ...
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79 views

Maximizing Profit - Linear Programming

I'm trying to formulate a linear program for this problem: There is a blacksmith who can produce $n$ different alloys, where alloy $i$ sells for $p_i$ dollars per unit. One unit of alloy $i$ takes $...
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60 views

Solve Least Squares with Constraints

I have a problem where I am supposed to solve a system of equations in matrix form. The system is 4x4 with 4 unknows. The matrix comes from the least squares method. However, I have a constraint. It ...
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59 views

Schedule feasibility graph

I have a scheduling problem of this type: a set $E$ of events $e_1, e_2, \dots, e_n$, I have to determine if a schedule $T : E \rightarrow \mathbb R^+$ exists with constraints of both type $T(a)-T(b)\...
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1answer
45 views

Sum of m dice rolled n times where sum of each dice is lower than some value

Let there be $m$ dice (not neccessarily all same-sided, but even when they are I don't have a solution). Each dice is rolled $n$ times. The sum of $n$ rolls of all dice needs to be $T$. The sum $X_i$ ...
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1answer
89 views

finding extreme points for Lagrangian with multiple inequality constraints

I am trying to find maximum of \begin{equation} f(x, y) = x^2 - xy + y - 4x \end{equation} \begin{equation}\label{constraints} \text{s.t. } 0 \leq x \leq 2 \text{ and } 0 \leq y \leq 1 \end{...
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linearization of constraints

I am dealing with an optimization model where my binary variables xi have to follow this type of constraint (in the attached link): ...
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50 views

trust region optimization conjugate gradient steihaug subproblem

I'm trying to solve by hand the trust region optimization conjugate gradient - Steihaug method from the book Numerical Optimization by Nocedal and Wright - Algorithm 7.2 as shown below. I'm struggling ...
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18 views

Approximation with inequality constraints

Suppose $\mathbf x = [x_1\; x_2\; \cdots\; x_n]$ is a discrete approximation of a function at $n$ points. I want to get another approximation of this function at $n/2$ even points, say $\mathbf y = [x'...
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30 views

Constrained complex least squares

Apologies if this is an ignorant question as I am new to this field. Assuming that we are solving a problem of the form Ax =b where A and b are complex valued and known. The optimal solution in a ...
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26 views

Fluid mass balances

A tank is fed a solution of potassium sulfate ($10%$ by weight) in water at a rate of $0.5$ liters per minute. Assume that the density of the solution is the same as pure water. The outlet flowrate is ...
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2answers
101 views

How to take the partial derivative of $f(x,y) = x\ln(x) + y\ln(y), x + y = 1$?

Let $f(x,y) = x\ln(x) + y\ln(y)$ be defined on space $S = \{(x,y) \in \mathbb{R}^2| x> 0, y > 0, x + y = 1\}$. My question is, how do I take the partial derivative for this function, given ...
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0answers
23 views

When will $l_1$-relaxation give the same answer as the $l_0$ problem?

The $l_0$-relaxation problem $$\min_x \Vert x \Vert_0 \text{ subject to Ax = b}$$ is non-convex. On the other hand, the $l_1$ problem $$\min_x \Vert x \Vert_1 \text{ subject to Ax = b}$$ is ...
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131 views

Doubt about definition of infinite limit and limit as x tends to infinity

In the limits and continuity chapter in my textbook, the following definitions are given: (1) For limit of f(x) as x tends to infinity: We say that $ƒ(x)$ has the limit $L$ as $x$ approaches ...
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31 views

How to perform constraint optimization with undefined parameters in WolframAlpha?

I'm having trouble finding how to do a constraint optimization with undefined parameters in WolframAlpha. For example, how do I solve a standard Cobb Douglas problem such as: maximize $A x^a y^b$ on $...
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45 views

Finding the optimal solution of a specific parametric performance index with constraints

Consider the following performance index: $$J=\cfrac{1}{2}u_1^2+\cfrac{1}{2}u_2^2+\cfrac{1}{2}u_3^2+p_1\cdot u_1+p_2\cdot u_2+p_3\cdot u_3$$ Suppose $u_1$, $u_2$ and $u_3$ are the design variables and ...
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1answer
19 views

Search a smooth periodic function based on critical values and derivatives

I am looking for a $\mathcal{C}^\infty$ periodic function $f: [0,\ 2\pi]\rightarrow [-1,\ 1]$ with the following properties, for an arbitrary $\epsilon \in \mathbb{R}^+$: $f(0) = -\epsilon$, $f'(0)=0$...
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0answers
14 views

Constrained Raleigh Quotients

I've recently come across some engineering papers on constrained Raleigh Quotients, for example this one: https://www.cis.upenn.edu/~jshi/papers/supplement_nips2006.pdf Has anyone ever seen research ...
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1answer
443 views

Can any optimization problem be expressed as one without constraints?

For the optimization problem defined as $p^* = $ min$_{x \in \mathbb{R}^n} f_0(x)$ with constraints $f_i(x) \le 0, i = 1, ..., n$, can the problem be expressed as one without constraints? I think ...
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3answers
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Number of subsets with conditions [closed]

How can I calculate number of subsets with conditions of their cardinality? for example: Let $A=\{1,2,3,4,5,6\}$ How many subsets of $A$ there are with at least $3$ elements? How many subsets of $A$...
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1answer
59 views

Rank-dependent constraints in linear programming

I have a typical linear optimization problem: $c'x \to \max_x$ s.t. $~~l \leq Ax \leq u,$ where $x = (x_1, ...,x_n)'$, $A - k \times n$ matrix of constraints coefficient, $l$ and $u$ are $k \times ...
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62 views

Solving/minimizing $Fw=b$ with $F$ and $w$ unknown and $b$ known

I have a system $Fw=b$ that can be underdetermined, square or overdetermined. The matrix $F$ contains known scalar functions $f_i(t)$ evaluated at unknown values $t_j$. The vector $w$ contains unknown ...
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1answer
194 views

Optimization under constraints - unique solution or not

Say we have a problem such as minimize $f(x)$ such that $h(x)=0$ and $g(x) \leq0$. Let the minimum achieved under these constraints be $f(x^*) = p^*$. My question is: If $f(x)$ is convex, are $p^*$ ...
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2answers
53 views

Given some $n ∈ ℤ$ what conditions must $v$ satisfy for $n \left \lfloor {v} \right \rfloor $ = $\left \lfloor {n v} \right \rfloor $

I'm probably overthinking this. What constraints must you place on $v\in \mathbb R$ : $n \left \lfloor {v} \right \rfloor $ = $\left \lfloor {n v} \right \rfloor $ if $n$ is an arbitrary integer? I ...
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1answer
116 views

Convert Quadratic to Conic Constraint

Per Wikipedia, a quadratic constraint of the below form $$x^TA^TAx+b^Tx+c\leq0\tag{1}$$ can be written as the following equivalent conic formulation $$\left \|[(1+b^Tx+c)/2,~Ax ]\right \|_2\leq(1-...
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0answers
26 views

Concave/Convex Function rate of increase

I have the following constraints: $$\sum_{i=1}^2 p_i =2P$$ $$\sum_{i=1}^2 \gamma_i =2\Gamma$$ and the function $R$ which is strictly concave w.r.t. $p$ and strictly convex w.r.t. $\gamma$. How do I ...
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44 views

Minimize distance given other distance constraints

I want to minimize the Euclidean distance between a pair of points $\mathbf{P}_1, \mathbf{P}_2$ by finding a new point $\mathbf{P}_{2*}$, with the inequality constraints that for each $R_i$ of a set ...
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2answers
430 views

Given that $|3x|+|2y| \leq 1$ then the maximum value of $9x+4y$ is ?.

Suppose that $|3x|+|2y| \leq 1$ then the maximum value of $9x+4y$ is?. I thought of drawing the region satisfied the constraint given on the $xy$ plane. here is the region enclosed the lines - I ...
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146 views

How does one minimize/maximize the Lagrangian if its gradient is non-linear?

If one is trying to maximize(or minimize) the Lagrangian $$\mathcal{L}(x,y,\lambda) = f(x,y) - \lambda \cdot g(x,y)$$ its fairly straightforward that this is achieved by solving: $$\nabla_{x,y,\...
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1answer
76 views

Maximize a table

I am studying the max-sum algorithm to solve Distributed Constraint Optimization Problem. I have a very basic doubt about the maximization of a function w.r.t. a single variable. Consider the ...
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1answer
116 views

Log transformation in constrained optimization

In constrained maximization, should I log-transform the constraint if the objective function is log-transformed? $$\max_{x, \ y} \,\, x^\alpha y^{1 - \alpha} \qquad \text{s.t.} \qquad p_x x + p_y y \ ...
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1answer
66 views

Equivalence and convexity in optimization problems with restrictions

Let $\mathcal{X} = \{x_1,\dots,x_N\}$ be a finite set in $\mathbb{R}^d$ and let $S$ and $D$ be two subsets of $\mathcal{X}\times \mathcal{X}$, with $S \cap D = \emptyset$. If $M$ is a positive ...
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1answer
40 views

Find a values of the equation that will be satisfy the constraints on that values

I want to find coefficients in the equation that will be satisfy the constraints lying on them or find that this is impossible: $$14.31818 = \frac{48}{Div_1} \times \Big( Div_2 + \frac{FRACN}{2^{13}} ...
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1answer
71 views

Optimization problem with inventory rollover

I need help with a constraint related to inventory rollover. Say you own a flower shop and the demand for flower seed packs is such: January: 200 February: 300 March: 500 April: 800 So you need to ...
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40 views

Constrained Extrema on a compact set

Let Let $S:=\{x\in \mathbb R^{3}:\vert\vert x \vert\vert_{2}^{2}=\frac{1}{4}\pi^2\}$ and $f:S\to \mathbb R$, $f(x,y,z)=\sin(x)+\sin(y)+\sin(z)$ Task: Find the global maximum and minimum of $f$ ...
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1answer
113 views

Multivariate Derivative with constraint [Implicit Function Theorem/Chain rule Application]

I'd love to get some help understanding this question. Let $ w=f(x,y,z) $ with the constraint $g(x,y,z)=3.$ At point $P(0,0,0)$ we have $df = <1,1,2> , dg=<2,-1,-1>.$ Find the value of P ...
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0answers
18 views

Acumulation constraint in a non linear optimization problem

I'm no expert on non linear optimization, so I have been having trouble finding a way to code the following problem $$ \begin{aligned} & \underset{P_i}{\text{max}} & & \sum_{i=1}^n \big[ -...