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Questions tagged [lagrange-multiplier]

For questions on Lagrange multipliers, a strategy to solve constrained optimisation problems.

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Lagrangian Multipliers of Two Equivalent Constrained Problems

Given an convex optimization problem with multiple linear inequality constraints. Applying a strictly increasing function to both sides of the constraints yields an equivalent problem. If both ...
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Will lagrange multiplier method find all stationary points, or just minima and maxima?

Let $f,g : \mathbb{R}^n \to \mathbb{R}$ be smooth functions and $U_c = g^{-1}(c)$ for each $c \in \mathbb{R}$. For each $\lambda \in \mathbb{R}\setminus\{0\}$, consider the equations $$ \nabla g = 0 \...
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(Strong) Duality for the integer programming for $\text{gcd}(c_1, c_2, \ldots, c_n)$

It is known that (quoted from CLRS, 3rd edition) If $a$ and $b$ are any integers, not both zero, then $\text{gcd}(a, b)$ is the smallest positive element of the set $\{ax + by: x, y \in \mathbb{Z}\}...
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Finding the maximum and minimum values of a function in 3 variables subject to a given constraint using Lagrange multiplier

Question: Use Lagrange Multiplier to determine the maximum values of $f(x,y,z) = x^2 + y^2 + z^2$ subject to constraint $xyz = 4$. I do not know how to solve this, I got the expression $x^2 = y^2 = z^...
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Minimizing sum of reciprocal of quadratic functions

Given a set of constants $a_1,\ldots,a_n$, I want to solve the following single-variable optimization problem: $$\min_x \sum_{i=1}^n \frac{a_i^2}{x(2a_i-x)}, \quad s.t. \quad 0\leq x \leq 2a_i, \...
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*Recovering* Lagrange multipliers

I am currently reading this paper : https://www.di.ens.fr/~fbach/skm_icml.pdf about Multi-instance kernels, and in section 2.1 (page 2) it is written the coefficients $\eta_j$ are recovered as ...
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Proving the AM-GM Inequality with a given fact

Given $x + y + z \geq 3$ for all $(x, y, z) \in \mathbb{R}^{3}$ such that $x,y, z > 0$ provided $xyz = 1$, show that $$\frac{a_1+a_2+a_3}{3} \geq \sqrt[3]{a_1a_2a_3}$$ holds. I'm not really sure ...
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Taking variational vs partial derivative Lagrange Multipliers

So I'm trying to solve the optimization problem shown in the picture below. The functional then becomes $$P = \int_0^L \frac12 E A \epsilon^2-pu+\sigma \left( \frac{du}{dx}-\epsilon \right)dx = \text{...
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Can I solve the following problem in integration along with constraints? [closed]

max $\int (x-f(x)dx$ such that $f(0) =a$ $f(1) = b$ $f'(0) = c$ $f'(1) =d$ $f''(0.5) = 0$ $f'(x) >0 $ $ \forall x \in[0,1] $
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Brachistochrone step from Advanced classical mechanics (Bagchi)

I'm not sure as to how the book got from: $dx=\frac{a+b}{2}\int{\sqrt{\frac{1+\cos(\theta)}{1-\cos(\theta)}}\sin(\theta)d\theta}$ to: $x=\frac{a+b}{2}(\theta-\sin(\theta))+constant$ where $a$ and $...
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Find the minimum of $\space\frac{1}{x}+\frac{1}{y}+c\cdot xy\space$ subject to $\space x+y-c=0$

Let $f(x,y):\mathbb{D}\rightarrow\mathbb{R}$ be the function: $$f(x,y)=\frac{1}{x}+\frac{1}{y}+c\cdot xy\space\space|\space\space c\in(0,\sqrt[4]8)\text{ $\space$constant}$$ $$\mathbb{D}=\{(x,y)\...
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Lagrange multipliers don't seem to work

Consider the following constrained optimization problem: $$ \min x^3+y^3 \\ \text{s.t.} x^2+y^2\leq 1 $$ From plotting this, the minima seems to be at $(-1,0)$ and $(0,-1)$. Now, the KKT conditions ...
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How to find the shortest distance from $(1,0)$ to $y^2=4x$?

I need help with this problem: Find by the method of Lagrange multiplier the shortest distance from the point $(1,0)$ to the parabola $y^2=4x$. Check your answer by a method of substitution. ...
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Gradient Descent vs Lagrange Multipliers

I'm bit confused between Gradient descent and convex optimization using Lagrange Multipliers. I know that we use Lagrange multipliers when we have an optimization problem with one or more constraints. ...
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Optimization : is it possible to replace 2 Lagrange multipliers by a single one.

The question is simple: "find the point closest to the origin that is on the intersection line of $y+2z=12$ and $x+y=6$." Normal method would use two Lagrange multipliers and get the point $(2,4,4)$....
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Lagrange multiplier without implicit function theorem

Here is a proof of the Lagrange multiplier method from Calculus Early Transcendentals by James Stewart (8th ed). It does not rely on the Implicit Function Theorem like all other "rigorous" proofs seem ...
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Find the maximum values of the function $f(x,y,z)=x^2y^2z^2$ subject to the constraint $x^2+y^2+z^2=289$.

I have already figured out the majority of the problem. I took the derivative of $x$ and got: $x=0,y=0,z=0$. Initially I put that there were infinitely many solutions, but that is not true. I do not ...
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Why does the Lagrange multiplier $\lambda$ change when the equality constraint is scaled?

Consider the problem $$\begin{array}{ll} \text{maximize} & x^2+y^2 \\ \text{subject to} & \dfrac{x^2}{25} + \dfrac{y^2}{9} = 1\end{array}$$ Solving this using the Lagrange multiplier method,...
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Why does the Lagrange multiplier method not work to minimize $f(x,y) = x$ subject to $x^3=y^2+x^4$?

Why does the Lagrange multiplier method not work to minimize $f(x,y) = x$ subject to the equality constraint $x^3=y^2+x^4$? How does one use the 2nd derivative test to classify the critical point? And ...
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3 variables multiplication with 1 constraint lagrange multiplier

The question states that : Suppose we want to optimise the function $f(x,y,z) = ax^2 + by^2 + cz^2$, where $a$, $b$, $c$ are positive constants, subject to the constraints $x^3 + y^3 + z^3 = 1$ and $x,...
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Differentiating the Lagrangian function for $ \min_{x_t,y_t}E_0\sum^\infty_{t=0}\beta^t \left(\frac{1}{2} ( x_t^2 + \alpha y_t^2 )\right) $

Consider this maximization problem:$$ \min_{x_t,y_t}=E_0\sum^\infty_{t=0}\beta^t \left(\frac{1}{2} ( x_t^2 + \alpha y_t^2 )\right) $$ $$ \text{s.t. }\text{ } x_t=\beta E_t[x_{t+1}]+ky_t+u_t $$ where $...
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Find point closest to origin on ellipse

How do I find the point closest to origin on the eclipse: $$x^2 + 4y^2 = 4 $$ I tried using the Lagrange multiplier method, by using $$x^2 + 4y^2 - 4 = 0$$ as a constraint, and using $$f(x,y) = x^2 ...
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Solving for global Maximum and minimum on a interval

We need to determine the global maximum and minimum of: $f(x,y)=y^2-16x^2$ on the interval of: $\{(x,y) : y ≤ 1−x^2,y ≥ 0\}$ My initial thought was that I could use extreme value theorem, later ...
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Optimization with Multi variable Partial Derivatives

You’re assembling the perfect fruit salad and need to determine the optimal number of fruits to add. You have the option between Strawberries, Grapes, and Mangoes, which you buy in quantities S, G, M,...
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Lagrange multiplier term in Hamiltonian

My question is about a step in this paper: PhysRevB.65.165113 (X.G. Wen) or arxiv page 6. Or alternatively: PhysRevB.90.174417 or arxiv page 3. All papers on spin liquids and the projective ...
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What to do when Euler Lagrange Equation is highly nonlinear ode

In $\mathbb{R}^3$, suppose there is a curve on X-Y plane $y(x)$ defined on $x\in [-a,a]$ satisfying: $y(x)\geqslant 0$; $y(-a)=y(a)=0.$ Rotate $y(x)$ along x-axis in $\mathbb{R}^3$ and get a solid ...
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Why is the Lagrange multiplier considered to be a variable?

I'm learning the math behind SVMs (Support Vector Machines) from here. I understood the intuition behind Lagrange multipliers. But what I don't understand is that, why Lagrange multiplier is ...
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Lagrange multipliers with inequality constraints

When solving non-linear optimization with inequality constraints, one method seems to be to divide the problem in two parts and solve it inside the boundary and on the boundary. My question is: why do ...
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Find a point on an ellipse closest to a fixed point inside the ellipse

I want to find out ${(u,v)}$ on the ellipse $$\frac{u^2}{a^2}+\frac{v^2}{b^2}=1$$ for a point ${(x,y)}$ inside ellipse, which will denote shortest distance from $(x,y)$ to the ellipse boundary. I ...
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How do I maximise the following integral?

I am trying to maximise the function $F(x) = -\int_0^{\infty} p(x)\log(p(x))dx$ across all functions $p: \mathbb{R}_{\geq 0} \rightarrow \mathbb{R}_{\geq 0}$ such that $p(x) = p(1/x)$ for all $x \in (...
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feasible set of Lagrangian with respect to the original problem

Can we say that in general, for any constrained optimization problem, the feasible set of the original problem is a subset of the feasible set of the Lagrangian (even for the case where we take the ...
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Lagrange multipliers and Maxima CAS

I am considering the system eqns : [a11*q1+a12*q2+a13*q3-q1=0, a21*q1+a22*q2+a23*q3-q2 = 0, a31*q1+a32*q2+a33*q3-q3 = 0, a11+a21+a31-1 = 0, a12+a22+a32-1 = 0, a13+a23+a33-1 = 0, 2*a11*(1-q1)*q1-...
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Lagrange Multiplier quadratic with positivity constraint

Is it possible to solve the following problem using Lagrange Multipliers? If not Lagrange multipliers, what is the best way to approach this? Maximise $(p - \frac{1}{4})^2 + (q - \frac{1}{4})^2 + (r -...
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Find the point on the plane $x+2y+3z = 1$, which is nearest to the point $(-1,0,1)$ by Lagrange’s multiplier method.

Find the point on the plane $x+2y+3z = 1$, which is nearest to the point $(-1,0,1)$ by Lagrange’s multiplier method.
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Finding a constraint on one variable of a multivariable function to constrain the entire function

I have a function. Now I want to let my variables only take values between 0, and 1. The problem is as follows. For what values of Y, is L(x,y) < 0. That is, without putting a further constraint ...
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Finding the maximum entropy.

I'm trying to solve the following question: Here is my attempt using Lagrange multipliers: $L=-x_{1}lnx_{1}-x_{2}lnx_{2}-\cdots -x_{n}lnx_{n}+\lambda (x_{1}+\cdots +x_{n})$ $0=\frac{\partial L}{\...
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Optimal Value of a Cost Function as a Function of the Constraining variable

Consider the optimization problem : $ \textrm{min } f(\mathbf{x}) $ $ \textrm{subject to } \sum_i b_ix_i \leq a $ Using duality and numerical methods (with subgradient method) i.e. $d = \...
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Solution to maximum entropy of the gravity model

I need some algebra help to come up with a solution to the maximum entropy formulation of the gravity model. The problem: $ \hspace{35pt}\\ max \ H =-\sum_{i=1}^I \sum_{j=1}^J T{_i}_j\ln( T{_i}_j) \\ ...
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In this constrained minimization problem, should the Lagrange multipliers be positive?

Consider the following (real, block ?) matrix $Z_{n\times k+1}=[1_{n\times 1},X_{n\times k}]$. Note how $z\equiv v^TZZ^Tv$ can be written as: $v^T11^Tv+v^TXX^Tv=v^TJ_nv+v^TWv$, where $J_n$ is a unit ...
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What is proof of direction of Change in position vector $d \vec r$

How can I prove that the direction of an infinitesimal change in position vector $\mathrm d\vec r$ is the same as that of the instantaneous velocity $ \vec v=\mathrm{d}\vec r/\mathrm{d}t$? What I ...
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How to set grad[ ] in nlopt?

I would like to ask you about NLopt as follows below. Question1: If the number of constraints is bigger than the number of variables, how can we set “grad[ ]” in the “myconstraint”? Is there any (...
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Initial conditions on Lagrange Multipliers in variational problems

If one wants to extremize the integral $\int_{x_0}^{x_1} F(x, y_i, y_i')dx$ subject to constraints $\phi_j = g_j(x,y_i,y_i')=0$, using the calculus of variations, then one can generate the Euler-...
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Do Lagrange's multipliers fail in this case?

Question: Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint. (If an answer does not exist, enter DNE.) f(x,y)= y^2 - x^2 and (1/4)x^2 + ...
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Maximizing entropy

Let $v_1<\cdots<v_n$ and $\mu\in(v_1,v_n)$ be real numbers. Consider set $$X=\left\{(p_1,\ldots,p_n)\in[0,1]^n\ |\ \sum_{i=1}^np_i=1,\ \sum_{i=1}^np_iv_i=\mu\right\},$$ which is convex (easy) ...
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Minimizing using Lagrange Multipliers

I was given the question: Consider the point $P = (3, 4, 0)$ and the cone $z^2 = x^2 + y^2$. Determine the point on the cone that minimizes the square of the distance between $P$ and the cone. Am ...
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Why do derivations for SVM not consider slack variables for inequality constraints?

(This is related to a question I asked a few days ago) I've been through a few SVM derivations and the ones I follow are this Caltech lecture and this MIT lecture. However, with both of them the ...
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how is regularization equivalent to constrained optimization

In machine learning, a common theme is to minimize a function of the form $$\sum(W \cdot x - y)^2 + \lambda \sum_i w_i^2$$ Most books simply state that this is equivalent to minimizing $$\sum(W \...
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For a Fixed Variance, Gaussian Distribution Maximizes Entropy?

I was reading this paper. In page 5, second column, they mention that, $$h(Q) + h(P) \ge log(e \pi) => \sigma(Q) * \sigma(P) >= \frac{1}{2}$$ Where entropy $h$ is defined in the following way:...
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How to solve this constrained maximization problem?

Does anyone have an idea of how to tackle the following maximization problem? Maximize the function $ f(x,y,z) = x - y - \alpha z^2 $, $ \alpha > 0 $, under the following constraints: C1:...
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Do we maximize the Lagrangian function or minimize it for the SVM derivation? And why?

These are the resources I've looked into: MIT lecture Math.stackexchange question Slide from MIT Stanford lecture Caltech lecture The professor in (1) at 25:47 says that "I don't care if we find the ...