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Questions tagged [lagrange-multiplier]

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

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Question about Lagrange Duality and saddle points

Consider two Hilbert spaces $\mathcal{H}$ and $\mathcal{G}$ and an extended real valued function $f:\mathcal{H}\to\mathbb{R}\cup\{+\infty\}$ with $f\in\Gamma_0(\mathcal{H})$ the set of proper, lower ...
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Extreme values of ${x^3 + y^3 + z^3 - 3xyz}$ subject to ${ax + by + cz =1}$ using Lagrange Multipliers

If ${ax + by + cz =1}$, then show that in general ${x^3 + y^3 + z^3 - 3xyz}$ has two stationary values ${0}$ and $\frac{1}{(a^3+b^3+c^3-3abc)}$, of which first is max or min according as ${a+b+c>0}$...
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Stationary values of ${x^3 + y^3 + z^3 - 3xyz}$ when ${ax + by + cz =1}$

If ${ax + by + cz =1}$, then show that in general ${x^3 + y^3 + z^3 - 3xyz}$ has two stationary values ${0}$ and $\frac{1}{(a^3+b^3+c^3-3abc)}$, of which first is max or min according as ${a+b+c>0}$...
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Finding absolute minimum and maximum using lagrange mutlipliers?

I have this function $f(x,y)=3y^2-12x^2+1$ with the constraint $h(x,y)=x^2+y^2-x-2y+\frac54=0$. First thing I do is write $$3y^2-12x^2+1-λ(x^2+y^2-x-2y+\frac54)$$ which expands to $$3y^2-12x^2+1-...
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Use Lagrange multipliers to find all extrema of exponential function [answered]

Using Lagrange multipliers, find the extreme values of $U(x,y)=e^{x+y}$ on the surface $x^2+xy+y^2=1$ $\nabla U(x,y)=\lambda\nabla f(x,y) $ where $f(x,y)=x^2+xy+y^2-1$ (the constraint surface). $$(e^...
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Do I really need to solve 5 nonlinear equations in this Lagrange multiplier problem?

I need to solve the following optimization problem Let $X=\left\{ x_{i}\right\} _{i=1}^{n}$ be an independent sequence of $k$-face die rolls. Where for $j\in\left[k\right]$ we have $p\left(x_{i}=j\...
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Usage of Lagrange Multipliers in multivariable calculus

I just learnt about Lagrange Multipliers & am confused about why they are useful. Why can we not just check for critical points by checking if the gradient vector of the objective function $f$ is $...
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Find the point that is at distance $1$ from $(0,0,0)$ and at distance $3$ from $(1,2,3)$ that is closest to $(5,-2,4)$.

I have this question : Find the point that is at distance $1$ from $(0,0,0)$ and at distance $3$ from $(1,2,3)$ that is closest to $(5,-2,4)$. Here is my failed attempt. I used Lagrange ...
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Hydro Turbine Optimization

So I was doing the project related to Lagrange multiplier called hydroxyzine optimization recently and I have encountered problem on question4-6, which required me to plot a graph and find the optimum ...
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Inconsistent lagrange multiplier

so I have a function $f = 2\pi r h$ with $r, h$ as incognites. I want to minimize it. The restriction $g = \pi r^2 h-0.25$ The problema is that when I do the method I get an inconsistency like: $...
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Lagrange Mutiplier for inequality constraint

I'm a bit confused about Lagrange multipliers. I know it works wonders if I only have equality constraints. Whenever I have inequality constraints, or both, I use Kuhn-Tucker conditions and it does ...
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How do we know we're maximizing the Lagrangian objective function in PCA?

In Principal Component Analysis, we start with $m$ observations $x_1,\dots,x_m$, each of which is an $n$-dimensional vector. Assume we have centered the data; that is, we have subtracted the variable ...
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System of equations for Lagrange multipliers

I'm looking for all the critical points of $f(x,y,z) = 2xy + 2yz -2x^2 -2y^2 - 2z^2$ constrained to the surface of the unit sphere in $\mathbb{R}^3$. Thus far I've set up the Lagrange multiplier: $$\...
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Use Lagrange multipliers to find the max and min of the function $f(x,y)=xe^y$ subject to the constraint $x^2+y^2=6$.

Use Lagrange multipliers to find the max and min of the function $f(x,y)=xe^y$ subject to the constraint $x^2+y^2=6$. My attempt: I tried to find both partials and set them equal to $\lambda$ times ...
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Finding the maximum of a function on a triangle

I want to find the maximum of $f(x,y) = x^ae^{-x}y^be^{-y}$ on the triangle given by $x\geq0$, $y\geq0$, and $x+y\leq1$ in terms of $a$ and $b$ such that $a,b>0$. I can see that the vertices of ...
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Constraint Qualification in Lagrange

$f(x,y)=x^2-y^2$ subject to single constraint $g(x,y)=1-x-y=0$ For this question, I understand that the Constraint Qualification holds, since, rank of $D(g(x,y))=1$ everywhere. Solving Lagrange would ...
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Use Lagrange multiplier to find extrema of $f(x,y,z,t)$ subject to the stated constraints.

Use Lagrange multiplier to find extrema of $f$ subject to the stated constraints. $f(x,y,z,t)=xyzt$ $x-z=2$ and $y^2+t=4$ My attempt: Let $g(x,z)=x-z-2=0$ and $h(y,t)=y^2+t-4=0$ $\nabla ...
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Maximum of $ab+2bc+3ca$ with $a^4+b^4+c^4=1$

Let $a,b,c\in \mathbb R^+$ with $a^4+b^4+c^4=1$. What is the maximal value $ab+2bc+3ca$ can take? I tried using Cauchy-Schwarz several different ways and the best upper bound I got was $\sqrt{14}$, ...
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How to take derivative of integral of function?

I'm reading a textbook where it forms a Lagrangian function $$ L = \int_0^1 f(x)^{1 - \frac{1}{\alpha}}dx - \lambda\int_0^1 g(x) f(x) dx$$ But how do you take the derivative of this thing? The ...
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Solve optimization problem using KKT conditions

I'm trying to understand the solution to Boyd and Vandenberghe Problem 5.30: Boyd and Vandenberghe Problem 5.30 The Lagrangian is $$L(X,\nu)=\text{tr}X-\log\det X+\nu'\left(Xs-y\right),$$ so the ...
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Non-linear constrained problem transformation to equivalent un-constrained problem

I have the following non-linear optimization problem: min $f(x, y, z) = x + y + z$ s.t. $x^2 + y = 3$ $x + 3y + 2z = 7$ Is there a way to transform this problem to an equivalent minimisation ...
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How does the constraint change when the Lagrange multiplier changes?

Now I have a convex function $f(x)$, $x\in \mathbb{R}^n$, consider the minimization problem: $\min_x f(x)+\lambda x^Ts$, where $s$ is a positive real vector and $\lambda$ is a parameter, I am ...
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Optimization with an Ellipse, Lagrange Multipliers

The plane $ x + y + 2z = 4$ intersects the paraboloid $z=x^2+y^2$ in an ellipse. Find the points on this ellipse that are nearest to and farthest from the origin. From this, I thought that $x^2+y^2+z^...
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Why this constraint is crucial in a Lagrange multiplier problem?

Consider we need to find the max. and min. values of this function $ f(x,y)=xy $ subject to the constraint $x^2-y=12$.Why it is necessary to assume that $ y\leq0$?
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Why does the method of Lagrange multiplier fail?

Question Let $\mathbf{x}$ and $\mathbf{y}$ be two vectors in $\mathbb{R}^2$, and $f$ be a function defined by $f(\mathbf{x},\mathbf{y}) := \mathbf{x} \cdot \mathbf{y}$. Can this function be ...
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Lagrangian: $\min_{\mathbf{X} \in \mathbb{R}^{N \times K}} \left\|\mathbf{Y}-\mathbf{X}\right\|_F^2$ s.t. $\mathbf{A}\mathbf{X} = \mathbf{B}$

Let us say that the optimization problem can be posed in the matrix form as given below $\min_{\mathbf{X} \in \mathbb{R}^{N \times K}} \left\|\mathbf{Y}-\mathbf{X}\right\|_F^2$ s.t. $\mathbf{A}\...
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Minimize $x^2+6y^2+4z^2$ subject to $x+2y+z-4=0$ and $2x^2+y^2=16$

Minimize $x^2+6y^2+4z^2$ subject to $x+2y+z-4=0$ and $2x^2+y^2=16$ My try: By Lagrange Multiplier method we have $$L(x,y,z,\lambda, \mu)=(x^2+6y^2+4z^2)+\lambda(x+2y+z-4)+\mu(2x^2+y^2-16)$$ For $$...
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A Problem of Lagrange Multiplier

The problem is find the minimum value of $x^2+y^2+z^2$ subject to the condition $x+y+z=1$ and $xyz+1=0$. Let $f(x,y,z)=x^2+y^2+z^2$, then after some calculation I got this two equations: $...
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Proof of AM GM theorem using Lagrangian

Given: $\prod_{i=1}^n x_i = 1$ leads to constraint function $G(x_1,x_2,...,x_n)=\prod_{i=1}^n x_i-1$ ($\prod_{i=1}^n x_i =x_1 x_2...x_n$) Task is to to find the minimum using conditional extrema of ...
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Cannot solve Lagrange multiplier for 2 dimensional problem in cartesian coordinates

My Q is this : A rod of fixed length (20m) has its base sliding along the X axis and its top slides up the y axis simultaneously. The rod will hit a 5m by 5m box positioned at the origin of the X-Y ...
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Why it is necessary to have $y\le 0$ in the given problem?

Find the maximum and minimum values of $f(x,y)=xy$ subject to the constraint $x^2−y=12$. Assume that $y≤0$ for this problem. Why is this assumption needed? NOTE: I just want to know the necessity of ...
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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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Calculus 3: Lagrange Multipliers Issue

Find the point (x0, y0) on the line 12x + 12y = 2 that is closest to the origin. I set ∇f to ∇g to get: <2x,2y> = λ<12,12> I then found λ = x/6 and λ = y/6, meaning that x = y Once I ...
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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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making sure we found all the extremals

I am curios to know whether there is anyway to be sure that we found all the stationary points using Lagrange multiplier method.? Thank you.
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Finding the maximum and minimum of $2x-y-5z=k$

Problem: Find the maximum and minimum of $2x-y-5z$ about $$ x,y,z \in \mathbb R$$ that satisfy Conditions $$x^2+y^2+z^2=9$$ $$x-y-z=1$$ $$2x+y+2z\ge0$$ I can solve the problem without the third ...
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Prove the optimal value of this minimum is not smaller than the optimal value of this maximum

$ min \langle c, x \rangle$ s.t. $Ax \geq b, x \geq 0$, where $x$ is an integer and $A$ has integer entries. Show the optimal value of this is no less than the optimal value of this maximum: $max \...
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Minimizing Jensen-Shannon Divergence with constraints

I am trying to minimize the following function : $$J(p) = JSD(p_u || p)$$ with constraints : $$ \int p = 1 $$ $$ p(x) \geq \hat{\pi} p_p(x) $$ where $JSD$ is the Jensen Shannon Divergence, $p_u = \pi ...
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How can I solve this system of equations (Lagrange multiplier problem)?

A rectangular box with no top is to have a surface area of 16$m^2$. Find the dimensions that maximize it's volume. Here is what I have: Objective function: $f(x)=xyz$ Constraint function: $2xz+2yz+...
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Equivalence of two Lagrangian Optimization Problems with equal restrictions

Assume we have given two strictly convex functions $f_1$ and $f_2$ that map from $\mathbb{R}^p$ to $\mathbb{R}$. We want to find the vector $X\in \mathbb{R}^p$ that optimizes them. Both functions ...
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Formulate dual problem using Lagrange multipliers

The problem asks for me to minimize $f(x) = -8x_1 + x_2$ subject to $x_2 \leq 8$ and $(x_1-4)^2 - x_2 \leq 8$. I found $L(x_1, x_2, \lambda) = -8x_1 + x_2 + \lambda_1(x_2-9) + \lambda_2((x_1-4)^2 - ...
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How do I determine the maximum and minimum points for this problem using the Lagrange multiplier approach?

Here is the original problem: Find the extrema of $f(x,y)=xyz$ on the unit ball $xyz$ on the unit ball $x^2+y^2+z^2 \le 1$. Here is what I got: $f_x = yz, f_y = xz, f_z = xy$ $g_x=2x, g_y=2y, g_z =...
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How can I solve this particular non-linear system of equations?

Here is the original problem: Find the extrema of $f$ subject to the stated constraints. $f(x,y) = x - y$, subject to $x^2-y^2=2$ I'm solving a problem involving Lagrange's multipliers, and I've ...
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Finding Max and Min values using Lagrange Multipliers

So I am given the function $x^2 + y^2$ and the constraint $x^2 -2x +y^2 -4y=0$. Must find max and min values (answers are $f(0,0)$ is a min and $f(2,4)$ is a max.) I use $\nabla f = \lambda \nabla g$. ...
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How to find a global(local, if global is impossible) minimizer under some equation constraints in a reasonable time?

I'm looking for an algorithm that finds a global minimizer of a $C^1$ class function $f: \mathbb R^n \to \mathbb R$ under the constraint $x\in S$, where $S = \{x\in\mathbb R^n | g(x)=0, x_i\geq 0$ for ...
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Lagrange Multipliers and Unfeasible Constraints

Given a point $\mathbf{a} \in \mathbb{R}^{3}$ and a unit vector $\boldsymbol{n} \in \mathbb{R}^{3}$. A unique plan $P$ goes through $\mathbf{a}$ and possesses a normal $\boldsymbol{n}$. A surface $S$ ...
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Lagrange Multiplier to find maximum and minimum

Question: Find the maximum and minimum of the function $f(x,y,x)=3x^2 + 2xz + 5y^2 +3z^2$ on the sphere $x^2 + y^2 + z^2 = 25$ Attempt: This seems that it should be solvable using the Lagrange ...
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Building Dimensions (Lagrange Multipliers) (Optimization Problem)

A rectangular building is to have a volume of $8000ft^3$. Annual heating and cooling costs will amount to $\$2/ft^2$ for its top, front, and back, and $\$4/ft^2$ for the two end walls. What ...
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Distance between a line given by planes and origin by Lagrange's multiplier.

Find the minimum distance of the line given by the planes $3x+4y+5z=7$ and $x-z=9$ from the origin, by the method of Lagrange’s multipliers. As- Is it ok to take the Lagrange functions as given ...
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Lagrange multipliers over two closed and bounded constraints

Using Lagrange Multipliers, find all extremum points and the extremum values of the function $$f(x,y,z) = x^2-y^2-z^2$$ subject to the constraints: $$x^2+y^2=4$$ $$y=\sqrt{3+z^2}$$ So far I've ...