# Questions tagged [constraints]

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

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### minimizing an integral with a constraint

how can I determine the minimum value that the following integral can take, knowing that y is not singular in $x=0$ and that $y(1)=y'(1)=1$ $$J=\int_0^1[{x^4(y^")+4x^2(y')^2}]dx$$
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### Is the sum of KKT multipliers strictly positive?

In a given constrained optimization problem, the objective is convex and the constraints are strictly convex. I know that at least one of the constraints is binding. The Karush-Kuhn-Tucker multipliers ...
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### Constrained optimisation of function with nonzero gradient

Suppose a continuous function $f(x,y,z):\mathbb{R}\longrightarrow\mathbb{R}$ is such that $\nabla f(x,y,z)\neq\mathbf{0}$ for every $(x,y,z)\in [0,1]^3$. A property of the gradient is that it is zero ...
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### MLE of $\theta$ when $f(x, \theta) = c \theta^c x^{-(c + 1)}$, $x \geq \theta$; $c$ constant > 0; $\theta > 0$

Let $X_1, \ldots, X_n$ denote a sample from a population with density $\theta$ when $f(x, \theta) = c \theta^c x^{-(c + 1)}$, $x \geq \theta$; $c$ constant > 0; $\theta > 0$. I came up with a ...
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### Linear Programming: Either OR constraint non-binary decision variables

I'm working on a production problem where I'm producing a number of products. My decision variables indicate quantity levels of production across a range of prices. My current LP solves for the ...
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### How to construct a set of variables that automatically satisfies certain constraints

I ran into this problem when trying to do a project on one-electron reduced matrices of fermions. The math can be formulated as following: Let $\{a_i\}_{i=1...m}$ be a set of variables with additional ...
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### Why use interior point methods when ineqality constraints can be turned into equality constraints?

It is relatively easy to perform Newton steps with equality constraints, solving for the KKT system. As I understand, when dealing with inequality constraints, the complementary slackness conditions ...
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### Existence of a set of binary vectors which fulfill certain constraints

The following problem seems somewhat abstract but it would be nice to get it (dis)proven. Assume we are given a set $\mathcal B$ of $n$ binary vectors $\boldsymbol b_1,\dots,\boldsymbol b_n$, each of ...
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### Neural networks versus linear regression for optimization with nonnegative constraint

Let's say, I have a simple machine learning task to train weights $\vec{w}$ based on measurements matrix $X$ and known labels $\vec{b}$. I know that there are not nonlinearities in the model so I can ...
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### Quasi-Monte Carlo sampling with a summation constraint

The problem Given arbitrary integers $n$ and $d$, I want to sample $n$ data points, each of dimension $d$. An additional constraint should be satisfied, namely that the values of each individual data ...
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### How would I constrain a regression model with two covariates so that the coefficient B2 is equal to 1/2B1?

If the original model is y = B0 + B1X1 + B2X2 How could I manipulate the equation or the data so that the coefficient B2 is equal to 1/2 B1? y = B0 + B1X1 + 1/2B1X2 In theory, how could I do this with ...
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### Implication of a set of inequalities on the coefficients of a quadratic polynomial

Consider the set of inequalities \begin{equation} \sum_{i=1}^{n} c_{i} x_{i}^2 \geq 0, \quad \forall x \in \mathbb{R}^{n} : Ax = b, \end{equation} for some matrices $A \in \mathbb{R}^{m \times n}$ and ...
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### Relations between KKT necessary conditions

I am trying to understand the relationships in the KKT theorem between being a maximizer, satisfying the first order conditions (FOCs) and complementary slackness (CSC), and the linearly independent ...
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### Generate random points satisfying linear constraints

In my problem, I have a vector x of len N. Where each element xij is the price of the product i in the country j. Let's say that I have 100 products and 20 countries, so N=100x20=2000. The solution of ...
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### Find the values of unknowns

I have an inequality of the kind $\dfrac{(1-a)}{b} > \dfrac{c+d}{c~ d}$ subject to the condition that $0 < c\leq 1, 0 < d \leq 1, a > 0 , b > 0$. How could I possibly find the values ...
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### Reformulation of a non-convex QCQP with norm objective and constraints

There is a set of $N$ points $S$ in a 3D space and a set of vectors $V$, where each $s_i$ is allowed to translate along vector $v_i$. I want to minimize the total displacement of the points in the ...
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### Maximization of function with multiple variables and constraints

I have a function with 6 variables (just for information I report it below, but I am satisfied with a theoretical answer):  a+ (a*b)/(1-b) + (a*c)/(1-c) + (a*d)/(1-d) + (a*e)/(1-e) + (a*f)/(1-f) + ...
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### writing a constraint for a maximisation problem [closed]

There are $n$ seats in a row. $p$ people (where $p<n$) can seat anywhere as long as long as they sit at least one seat apart due to personal relationships. This statement is part of a larger ...
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I'm trying to solve this question: Сonsider the following Polyhedron which has five edges $BCD, BCEO, BDFO, CDFE, EFO$. Write down a canonical set of constraints that define the Polyhedron. The ...