# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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 \ ...
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 ...