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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Generalization of the method of Lagrange multipliers

In my research, I obtain an unconstrained optimization problem with objective function $$ L : R^n \times R^k \times R^\ell \to R, $$ $$ L(x,\alpha,\beta) = f(x) + \alpha^T g(x) + \beta^T h(x) + \alpha^...
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Solving optimization problem graphically

Solve the following optimization problem graphically, draw the tangent cone and determine the linearized feasible directions. $$\min_{x\in\mathbb{R}^2}-x_1$$ $$s.t. x_2-(1-x_1)^3\leq 0, -x_1\leq0, -...
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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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How do stationary points change on adding redundant constraints?

If $x_0$ is a first-order stationary point of the unconstrained optimization $\min_{x \in \mathbb{R}^d} f(x)$, is it a first-order stationary point of the constrained optimization problem $\min_{x \in ...
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Non linear constrained optimization

I have an optimization problem I'm struggling with and hoping someone can point me in the right direction. $\min\ \sum_{i=0}^n x_ip_i$, $ s.t$ $x_i = 0\ or \ q_{\min} \leq c_1(x_i) \leq q_{\max}$, $\...
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Algorithm for a minimization problem of a function that is not differentiable under a constraint

I'm trying to understand an algorithm for a minimization problem but it is unclear. Here is the function we consider $\lVert Y - X\beta\rVert_{2}^{2} + \lambda\lVert\beta\rVert_{1}$ where $Y\in\mathbb{...
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Conditional statement in mixed integer linear programming

I have been trying to enforce the following conditional statement in a MILP: If $X_1 + 2(X_2 + X_3) = 4$, then $X_4 = 1$. where $X_1, X_2, X_3, X_4$ are binary. How can I write this in conventional ...
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Do i have to add an auxiliary variable when adding a new equality constraint at a LP?

For example I have the following problem: \begin{align} &\textrm{min z} = -2x_1 -x_2 +x_3 \\ &\textrm{s.t.} \\ & \qquad x_1 +2x_2 +x_3 \leq 8 \\ & \\ &\quad -x_1 +x_2 -2x_3 \leq 4 ...
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How to map a higher dimensional vector to a lower dimensional subspace that satisfies multiple constraints?

I'd like to perform constrained gradient descent in a high-dimensional space. I'm planning to compute the gradient in the high-dimensional space and then project it to a lower-dimensional space that ...
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Showing max and min of function with constraints

The problem is to find max and min of the function: $$f(x,y)=-16x^6+24x^4-4y^4$$ with constraint $$\sqrt{x^2+y^2} = 1$$ I have been able to transform the problem into a single variable one where $$x=\...
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How to maximize scalar product under constrains?

I have a multidimensional unitary vector $\vec n$. I need to find another unitary vector $\vec r$ which maximizes the scalar product: \begin{equation} (\vec r, \vec n) = r_1 \cdot n_1 + r_2 \cdot n_2 +...
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How does SLSQP(Sequential least squares programming) algorithm work?

I am implementing scipy.optimize.minimize package to minimize a function. It has been learnt that for constrained minimization, the scipy library uses SLSQP(Sequential least squares programming) by ...
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Checking if a set of axis-aligned hyperrectangles span another hyperrectangle with a violation cost

Consider hyperrectangles (box constraints) of the form $Hx<=b$ where $x \in \mathbb{R}^n$ and hence $H = [-I ; I], b \in \mathbb{R}^{2n}$ where $b$ specifies the lower and upper bounds for each ...
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Solving $f = Ax + b$ when x is saturated but want to preserve directionality

I have a given equation of the form $f_{tot} = A x^\prime + \beta \tag{1}$ where A is a $R^{nxm}$ and we can assume m>n. Suppose that I want to produce a desired vector $f_{des}$, but I want to ...
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In this specific example, Would it be okay to assume that the function will never attain a minimum and will only attain a maximum on the constraint.

Hello, I am not sure if I can assume if this function will attain a global extrema or not. I know that if a function is compact and continuous then it will for sure attain a global maxima by the ...
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optimizing the weights in a system of linear equations with constraints

For reference, I am solving a system of linear equations for a science project. The system is of the form $G\ m=d$ where $d$ are data points, and $m$ are model elements. I have an additional number of ...
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Lagrange Multipliers. What do I do if the Lagrange multiplier is 0 or the gradient of the constraint is 0?

Consider the example below. Here, We found that $\nabla g$ is only zero at $(0, 0)$ but we see that the point $(0,0)$ does not satisfy $g(x,y) = 0$. But I don't understand why we need to ensure that ...
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How can I solve this optimization problem with integer variables?

I was recently watching a video game "speedrunner", who was trying to earn $1 billion of in-game currency in the shortest possible time. He identified 3 different actions ($a$, $b$ and $c$) ...
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How Optimizers Vary when Changing Parameters in the Constraints

I am trying to solve the following comparative statics problem. Assume I have four choice variables, $x_{1}, x_{2}, y_{1}$ and $y_{2}$, and two parameters $t_{1}$ and $t_{2}$. I have the following ...
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Number of states constrained by sum

Assume $n$ players have between $0$ and $k-1$ chips each. The total number of states for the game is then $N = k^n$, since there are $k$ options per player. Now assume the number of chips is constant ...
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Determine if any points violate the non-degenerate constraint qualification of the Kuhn Tucker theorem

$$\mathrm{Find \;}\mathrm{max}(xy^4) \\ \mathrm{With\; constraints}\; xe^y\leq 3e^2,c\geq y, x\geq 0, y \geq 0$$ where $c=0$ I calculated the Jacobian matrix $\nabla J$ which has the gradient of each ...
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Solving $b=Ax$ with matrix $A$ unknown

I have the following problem: Let $a \in \mathbb{Z}^{n}$ and $b \in \mathbb{Z}^{k+1}$ with known integer values and $b(k+1)=1$. Note: If it makes it easier, let $k<n$ I need to find a matrix $X$ ...
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What am I misunderstanding about Lagrange multipliers here?

Consider a constrained optimization problem over some euclidean space, that is for some real valued functions $f,g$ on that space we seek $$ \underset{x}{\arg \min} \ f(x)\\ \text{subject to} \ g(x) =...
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Find the extreme values of $f(x,y)=e^{-xy}$ on the region described by $x^2+25y^2\leq 4$

I have been stuck on this question for a very long time. I have tried to use lagrange multipliers but the equation seems nearly impossible to solve as the derivative of $f$ with respect to $x$ and the ...
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Number of iteration is exponential with bland's pivoting rule

Consider the following: $max$ $2^{n-1}x_1 + 2^{n-2}x_2 + \cdots + 2x_{n-1} + x_n$ s.t. $x_1 \leq 5$ $4x_1 + x_2 \leq 25$ . . . $2^nx_1 + 2^{n-1}x_2 + \cdots + 4x_{n-1} + x_n \leq 5^n$ $x_1, ..., x_n \...
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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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Analytical Solutions to Linear Programming with linear and quadratic constraint

Consider the following minimization problem $$ \min_x c' x $$ such that \begin{align*} Ax &= 0 \\ x' x &= 1 \end{align*} where $x \in \mathbb{R}^n$, $c \in \mathbb{R}^n$ and $A \in \mathbb{R}^{...
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How to find Marshallian Demand?

If the individual's utility function is given by: $U(x,y)=(X)^{1/2}+(Y)$ With constraint: $M=p_1X+p_2Y$ Find the Marshallian Demand functions for this individual. So far I can: Set the lagrangian: $...
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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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How the following simplification is made?

I am trying to understand how eq. (1) is simplified to eq. (2), but not getting it clearly. $ U_P(X,P) = \sum_{j}p_{cj}+\sum_i \sum_j x_{ij}p_{ij} -\psi \sum_i \sum_j \sum_{m \in \mathcal{B}} x_{ij}p_{...
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Gradient of extremum with respect to constraints

I have a smooth convex function $h:\mathbb{R}^n\to\mathbb{R}$ that I would like to minimize. But I would like to restrict the domain of optimization to a constrained minimizer of a different function. ...
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Frank-Wolfe direction for minimization constrained by l2-norm

I have the following problem: Describe how to calculate the Frank-Wolfe direction for the problem $$min_{x\in D}f(x)$$ where $$D=\{x\in \Bbb R^n: ||Qx||_2 \le 1\}$$ with $Q\in\Bbb R^{n\times n}$ - ...
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Concisely writing multiple similar constraints for an optimization problem

I am currently studying the linear optimization problems and would like to know if it is possible to make the notation for multiple similar constraints more concise. For example, I know how to do that ...
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Sum of integers closest to a given number

I have the following problem at hand: Given an odd number $n > 7$, find a set of non-negative integers $m_7$, $m_8$, ..., $m_{13}$ and $m_{14}$, such that the sum $m_7\cdot 7 + m_8\cdot 8 + ... + ...
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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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Find the canonical set of constraints that define the Polyhedron

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 ...
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