Questions tagged [constraints]

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

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Constrained Optimization Geometry Confusion

In a constrained optimization problem, let's consider the example $$\begin{cases}f(x,\ y) = yx^2\ \Tiny(function\ to\ be\ maximized) \\ g(x,\ y) = x^2 + y^2 = 1\ \Tiny(constraint)\end{cases}$$ why ...
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Count the number of unique elements in a vector by linear constraints (ILP)

Let $\mathbf{x}\in \{0,1\}^n$, be the objective variables of an ILP. Further, let $\mathbf{a} \in \mathbb{N}_{\geq 0}^n$ be a given random vector and $\mathbf{w} = \mathbf{x} \odot \mathbf{a}$ where ...
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Show that $C=\{x \in \mathbb{R}^4 \mid x^TAx \geq 1\}$ is not empty?

Consider the set $C=\{x \in \mathbb{R}^4 \mid x^TAx \geq 1\}$ where $A \in \mathbb{R}^{4 \times 4}$ is a symmetric matrix with two distinct positive eigenvalues and other eigenvalues of $A$ are ...
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System of ODEs with integral constrains

Can someone point me in a direction to solve this kind of integral constrained system of ODEs. As far as I know, there are no analytic methods that can solve this. So I will resort to numerical ...
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78 views

constrained rank approximation

I'm trying to solve a problem similar to this problem. Instead of requiring the diagonals to be 0, I'd like to require columns of the low rank approximation to decrease in value while going down the ...
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25 views

McCormick envelope of two variables which are also defined in terms of an envelope

I have a equation which is defined as $\langle\langle x_ix_j\rangle^M\langle \cos(\theta)\rangle^C\rangle^M$ where $M$ is the McCormick envelope of product of variables $x_ix_j$ and $C$ is defined as ...
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54 views

Optimal transport with relaxed constraint on terminal distribution

I have read the topic on relaxing constraint on relaxing marginal constraints Optimal transport with relaxed constraint on marginals, where the constraint is expressed as the difference of initial and ...
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1answer
136 views

The shape of a feasible region with equality and inequality constraints

I was wondering if anyone can help me with this (probably basic) question. I want to know how the following feasible region looks like if we have thousands of variables. The constraints are linear. ...
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Solving for integrand from integrated quantities.

Given equations of the form: $A(r) = \int_{t_{1}}^{t_{2}}F(r,t)dt$ $B(t) = \int_a^b F(r,t)r^2dr$ where $A(r)$, $B(t)$, and all of the limits on the integrals are known, is there enough information ...
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291 views

Norm constrained least square minimization with an additional single linear equality constraint: A quadratically constrained quadratic program (QCQP)

Given is the following QCQP problem: \begin{align} &&\min_{\mathbf{x} \in \mathbb{C}^n} &\|A \mathbf{x} - \mathbf{b}\|^2,\tag{1}\label{1}\\ \text{ subject to:}&&&\\ &&\|...
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Number of independent components of a vector satisfying a differential constraint?

Edited question Consider a vector field $\vec{A}(\vec{x})$ such that in one case $\nabla\cdot\vec{A}=0$. It looks like that this condition gives rise to a differential equation constraint $$\...
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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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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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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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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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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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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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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
34 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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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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40 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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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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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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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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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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83 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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64 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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75 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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47 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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111 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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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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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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24 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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151 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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38 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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46 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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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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491 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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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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71 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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1answer
257 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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55 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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138 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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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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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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172 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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173 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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74 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 ...