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

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1answer
14 views

How to interpret multiple critical points (from Lagrange multipliers) that all give a maximum value

If I have 6 critical points, 3 of which give the same maximum possible value of a function f(x,y,z), subject to a constraint g=c, is there something more to say about this solution -- or we just ...
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2answers
50 views

Solve this set of Lagrange multiplier equations,

I'm trying to solve $$(yz,xz, xy) = (\lambda\frac{2x}{a^2},\lambda\frac{2y}{b^2},\lambda\frac{2z}{c^2})$$ with the constraint equation $$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}=1$$ ...
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0answers
34 views

Calculus of variations of multiple integrals and multiple constrains

Let $q(x_1,\dots, x_6)$ be a arbitrary multivariate density that we choose beforehand. What I want is to calculate a new multivariate density $p(x_1, \dots, x_6)$, which is obtained by minimizing the ...
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1answer
12 views

Why is the lagrange dual function concave?

In a book I'm reading it says I'm struggling to understand the last sentence. Why can one conclude concavity from having a pointwise infimum of a family of affine functions?
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2answers
29 views

Using Lagrange's Method in Finding Extreme Values (New to This Method)

Did I do this hw question correctly (at least in theory, I do not expect anyone to check my algebra work)? In particular, did I solve for lambda and plug lambda back into my equations for x,y, and z ...
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2answers
28 views

Lagrange Maxima

I am sorry to post this again, but I am still confused. Alice has only $24$ hours to study for an exam, and without preparation she will get $200$ points out of $1000$ points on the exam. It is ...
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1answer
26 views

LaGrange question 2

You are building a barn, with no floor, in the shape of a rectangular box with a square base. The roof material costs $\$19/\text{m}^2$, the sides and back material costs $\$13/\text{m}^2$, and the ...
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3answers
41 views

Lagrange Question

Alice has only $24$ hours to study for an exam, and without preparation she will get $200$ points out of $1000$ points on the exam. It is estimated that her exam score will improve by $x(50−x)$ points ...
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0answers
48 views

On lagrange multipliers and constrained optimisation

I have a real-valued function of two real-valued variables $f(x,y)$. The global maxima is sough by solving the system $$ f_x(x,y)=\frac{\partial f(x,y)}{\partial x}=0\\ f_y(x,y)=\frac{\partial ...
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0answers
31 views

How do I solve a under-determined quadratic multi-variate system?

I have the following equation: $$ Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_3 + \beta_{11} X_{1}^2 + \beta_{22} X_{2}^2 + \beta_{33} X_{3}^2 + \beta_{12} X_{1} X_{2} + \beta_{23} X_{2} ...
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3answers
43 views

Shortest distance from a point

Find the shortest distance from the point $(0,b)$ to the parabola $y=x^2+8$. Express your answer in terms of $b$. (Comment: If $b \le \frac{33}{4}$ then the answer is just $|b|$, so assume that $b ...
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1answer
17 views

Use lagrange multultipliers to find the indicated extrema

maximize $f(x,y,z)=x+y+z$ subject to $x^2+y^2+z^2=1$ I do not understand this at all or where to go from here would appreciate some insight
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1answer
86 views

Is this a known result?

I heard the following result and I am wondering if anyone can verify its correctness and also provide a source to cite. If the Lagrangian $L(x,\lambda)$ is convex in $x$ at the optimal Lagrange ...
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0answers
11 views

How can we constrain lagrange multipliers in svm dual by adding constraints in primal problem?

Consider svm-dual,i.e., \begin{align} &\text{maximize} \sum_{i=1}^n \alpha_i-\frac{1}{2\lambda} \sum_{i,j=1}^n \alpha_i \alpha_j y_i y_j K(x_i,x_j)\cr &\text{subject to, } 0\leq \alpha_i ...
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2answers
35 views

Lagrange multiplier understanding problem

I do have a problem with the lagrange mutiplier method. I understand how it works for something like: maximize $f(x,y)$ subject to $g(x,y)=c$. But how do I handle something like: Maximize f(x,y) ...
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1answer
40 views

An optimization problem, in the form of a word problem,

The manager of a $1000$ seat concert hall knows from experience that all seats will be occupied if the ticket price is $50$ dollars. A market survey indicates that $10$ additional seats will remain ...
2
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1answer
32 views

Lagrange minimisation

I'm really struggeling with this obviously easy Lagrange question. Been at it for two days and don't really get the hang of where to start.. can someone pointme in the right direction? A consortium, ...
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0answers
18 views

Existence of lagrange multipliers with polyhedral constraints

I am working with a paper (Exact regularization of polyhedral norms, Schöpfer 2012) which states as a well-known fact that, if $f$ is a polyhedral norm, then for some $\mu^* > 0$ \begin{equation} ...
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0answers
39 views

Shannon Entropy Maximization with Constraints

I have got a cumulative distribution function $F_X(x)=Pr(X<=x)$. This distribution is described by 2 parameters $\alpha, \beta$. We define $F_k$ as follows: $\forall k<=n_k, ...
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1answer
22 views

Lagrange multipliers method question.

On the wikipedia page and indeed in my own opinion the method of Lagrange multipliers as applied to an equality constraint function is as follows, Extremise $f(x,y)$ subject to $\phi=c$ for some ...
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4answers
87 views

Why does taking derivatives of $L$ in Lagrangian multiplier problems let me find solutions to optimizations problems?

Consider the problem Maximize $f(\mathbf{x})$ subject to $g(\mathbf{x})=c$ Using the method of Lagrangian multpliers, I would set up a Lagrangian like $$L = f(\mathbf{x})-\lambda ...
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0answers
30 views

optimization of a function with inequality constraint

I have a function to be maximized subject to constraints. I can write the primal Lagrange function as the following: (objective function WITH two constraints in the last two terms) $$L_P = ...
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1answer
36 views

Distance between two points using Lagrange [closed]

Set up the system of equations required according to Lagrange in order to minimize the square of the distance between P1 and P2. $${M_{1}=‎\lbrace(x,y)\in R^2} \mid x^2+\frac{9}{4}(y-1)^2=9‎\rbrace$$ ...
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1answer
20 views

Lagrange multiplier derivative condition

For the Lagrangian $\mathcal{L}(\mathbf{x}, \lambda) = f(\mathbf{x}) + \lambda g(\mathbf{x})$, I read that $\partial\mathcal{L}/\partial\lambda$ must equal $0$. Could someone please explain why?
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1answer
36 views

Local extrema in special directions

I am looking for the extrema of a function $G(y_1,y_2,y_3,y_4)$ subject to the constraint $y_1 = y_4 + y_2y_3.$ We know that $G$ is defined if $(y_2,y_3,y_4)$ is in the cylinder $\mathbb{D} \times ...
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4answers
86 views

Minimization on compact region

I need to solve the minimization problem $$\begin{matrix} \min & x^2 + 2y^2 + 3z^2 \\ subject\;to & x^2 + y^2 + z^2 =1\\ \; & x+y+z=0 \end{matrix}$$ I was trying to verify the first ...
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0answers
48 views

Confusion over Lagrangian function

When forming the Lagrangian of an optimization problem, why don't we include all constraints in the Lagrangian? For example, the optimization problem $$ \begin{align} \max_{x}&\quad f(x)\\ ...
0
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1answer
41 views

Minimization involving equality constraints

I am trying to find closed form solution to following problem \begin{equation} \begin{array}{c} \underset{\mathbf{x},\mathbf{y}}{\text{minimize}} \hspace{4mm} \big(\left( \mathbf{y}^T ...
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0answers
25 views

Lagrange multipliers, once I use the constraint equation, do I have to worry about it again later?

I am solving $ grad [f(x,y,z)]$ = $\lambda$grad[g(x,y,z)] I have then three equations, one involving x's and lambdas, another involving y's and lambdas and a third involving z's and lambdas. I then ...
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1answer
22 views

For given mean $\mu$ of random variable X in [0,1], what is the probability distribution function $p(X)$ that makes $VAR(X)$ maximum?

Given the conditions $\int_{0}^{1} p(x)dx=1$, $\int_{0}^{1} xp(x)dx=\mu$ and $p(x)\ge0$ for $\forall x \in [0,1]$, What probability distribution function $p(x)$ makes $Var(X)$=$\int_{0}^{1} ...
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0answers
25 views

How to use find the Lagrange Multipliers in multidimensional Calculus of Variations

Suppose I wish to minimise the integral $$I = \int_{s_0}^{s_1}\int_{t_0}^{t_1}F\, dt ds$$ Where $F$ is a function of the six variables $x(s,t)$, $y(s,t)$, and their four partial derivatives, ie $$F ...
0
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1answer
74 views

Lagrange multipliers problem

I have a two variables function: $f(x,y)=3x+y$ and I wish to find its minimum and maximum values with the constraint $\sqrt{x} +\sqrt{y} =4$. According to the answer, there is a minimum and a maximum. ...
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4answers
38 views

Closest point to $(2,0)$ on with a hyperbola as a constraint

I'm looking to find a point on the hyperbola $y^{2}-x^{2}=4$ which is closest to $(2,0)$. As far as I know I need to find the distance formula and use lagrange multipliers.
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2answers
28 views

Optimization problem: smallest euclidean distance with positive entries constraints

Suppose there is the simple function: \begin{align} f(x,y,z) &= (x-a)^2 + (y-b)^2 + (z-c)^2 + (x+y-S-z - d)^2 \end{align} where $a,b,c,d$ are nonnegative constants, and $S$ is an integer. I ...
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1answer
46 views

Finding min and max under constraints

I have a two variable function: $f(x,y)=4x^2-y^2-xy-2x+6y$. I need to find its absolute minimum and maximum under the constraints: $y=4-2x$, $x \geq 0$, and $y \geq-2$. I am not sure how to do it, ...
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0answers
20 views

Taking the partial derivative of a Lagrangian with square roots

I have a nasty function that is simplified from an even nastier function: $F(x,y,z,\lambda) = \frac{-0.0129x-0.0051y-0.0066z}{\sqrt{.44^2x^2 + .15^2y^2 + .44^2z^2 + 0.05808xy + .139392xz - ...
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0answers
15 views

Lagrangian multiplier vectorial form

When i have this problem: $min f(x)$ $h{_i}(x)=0, i=1,\dots,m$ $g{_j}(x)<=0, j=1,\dots,p$ I can use the Lagrangian multiplier to write function in: $L(x,\lambda,\mu)=f(x)+\sum_{i=1}^{m} ...
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1answer
45 views

Lagrange multipliers problem with two constraints

Hi guys I am working with the following polynomial and I am trying to find the $\lambda , \mu$. I have a polynomial and I am trying to do Lagrange multipliers. Here is what I have. $f(x,y,z)= a ...
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0answers
21 views

How to obtain the optimal lagrange multiplier vectors if the globally optimal solution for a nonconvex QCQP is found.?

I am using a blackbox solver to solve the following nonconvex QCQP to global optimality. $$ \min_x x^TQ_0x + c^T x \\ s.t. \quad x^TQ_1x+c_1^Tx=b_1 \\ Ax=b \\ l\leq x\leq u $$ where $Q_0$ is ...
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2answers
67 views

Lagrange multiplier optimizing a 3-D ellipse with respect to the origin

I cannot solve this question: The plane $x+y+2z=2$ 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. My ...
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4answers
24 views

extrema with constraints (lagrange?)

I'd like to find the point of $E: 2x+3y+z = 14$ which has the smallest distance to the point of origin (0,0,0). I think I have $ d(x,y,z) = \sqrt{x^2+y^2+z^2}$ with constraint $2x+3y+z = 14$. What ...
2
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1answer
18 views

Lagrange Multiplier - equation system

I'm trying to get the extrema of a function $x + y²$ with a constraint $2x² + y² = 1$ using Lagrange multipliers. The Lagrange function is $x + y² + \lambda (2x² + y² - 1)$. I have three partial ...
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3answers
57 views

find extreme values of $\cos(x)+\cos(y)+\cos(z)$ when $x+y+z=\pi$

How can I find the maximum and minimum of $\cos(x)+\cos(y)+\cos(z)$ if $x,y,z\geq0$ such that they are vertices of a triangle with $x+y+z=\pi$. I don't know how to start, but I feel like the Lagrange ...
0
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1answer
49 views

Gradient of a Lagrange dual function

Consider: $$\min_{x \in \mathbb{R}^n} f(x)$$ $$\ \ \ \ \ \ \ \text{s.t. }\ h(x) \leq 0$$ Lagrangian:$\ \ \ L(x,\lambda) = f(x) + \lambda h(x)$ Suppose $x^* = \arg\min_{x} L(x,\lambda)$ ...
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Solving a nonlinear constrained optimization involving CDF and expectation of normal distribution

I would like to know if it is possible to solve the following nonlinear constrained optimization problem and find how the optimal solution varies with $C$ and $\beta$: $\max_{x,y}\beta ...
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2answers
36 views

Find the maximum and minimum of the function $f$

Find the maximum and minimum of $f(x, y)=xy-y+x-1$ at the set $x^2+y^2\leq 2$. I have done the following: Since the region $x^2+y^2\leq 2$ is closed, $f$ has a maximum and a minimum, which is ...
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0answers
29 views

Find the extremas of the fuction $f$

I have to find the extremas of $f(x, y)=3x+2y$ subject to $2x^2+3y^2 \leq 3$. Since the region $2x^2+3y^2 \leq 3$ is closed, $f$ has a maximum and a minimum, which is either at the boundary or at ...
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1answer
47 views

Theorem of Lagrange multipliers - Extremas of $f$

I have to find the extremas of $f(x, y, z)=x+y+z$ subject to $x^2-y^2=1$, $2x+z=1$. I have done the following: We will use the theorem of Lagrange multipliers. The constraints are ...
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1answer
21 views

Help with Lagrange multipliers on an intresting function

Hi guys I am trying to do Lagrange multipliers to figure out $\lambda$ $$F=a \log(x^2-y)+b\log(x^3-z)-\lambda (x^2-y+x^3-z -1)$$ Where a and b are constants and we have the constraint $x^2-y+x^3-z ...
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2answers
34 views

Lagrange Multiplier Method On Linear Equation Set

I am trying to perform a Lagrange constraint problem for a simple set of linear equations (I realize this can be solved by substitution) but I'm curious why/how the Lagrange method is failing and I'm ...