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

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proof of log-sum giving maximum value given equality constraint

How to prove the following equation: $$ -\log\sum_{k=1}^K f_k=\min_{\bf{u}}-\sum_{k=1}^K u_k \log(f_k) +\sum_{k=1}^K u_k \log(u_k)\\ s.t.\ u_k \in (0,1), \sum_k u_k=1 $$ using Lagrangian multiplier? ...
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1answer
34 views

Optimization with Lagrange multipliers

I am new to Lagrange multipliers. Could some one show me how to minimize the following function: \begin{align} f(x,y)=ax+by-\sqrt{cxy} \end{align} subject to: \begin{align} 0 &\le x\\ 0 &\le y ...
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23 views

When does a polynomial have finitely many critical points on a level set of another polynomial?

Suppose I have two polynomial functions $f$ and $g$ and I am interested in the critical points that $f$ has on a level set of $g$, i.e. $\{x\in \mathbb R^n : g(x)=a_1\}$ for some $a_1\in \mathbb R$ . ...
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72 views

Maximize $\sqrt{x^2+y^2+z^2}$ subject to several conditions on $x,y,z$. [on hold]

Find the maximum value of the following expression, $$R(x,y,z) = \sqrt{x^2 + y^2 +z^2} $$ if we are given that: $$\begin{align} y+4^{y+z} & \leq x + 4^{x+z} \\\ x^2 + y^2+8x & = 0 \\ 4 ...
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2answers
64 views

System of equations in Lagrange multiplier problem

Continuing from Confounding Lagrange multiplier problem: I'm having trouble solving the system of equations below arisen from a Lagrange multiplier problem where we are to optimize $f(x,y,z) = 4x^2 + ...
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41 views

Confounding Lagrange multiplier problem

Optimize $f(x,y,z) = 4x^2 + 3y^2 + 5z^2$ over $g(x,y,z) = xy + 2yz + 3xz = 6$ According to the theorem the gradients must be parallell, $\nabla f = \lambda \nabla g$, so their cross product must ...
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3answers
42 views

Finding the minimum distance from the origin to the surface $xyz^2=2$

This was an old exam question I was looking at for a friend, although it's been a while since I've done this stuff: Q. Find the shortest distance from the origin to the surface $xyz^2=2$. I ...
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21 views

Find the extreme value of $f(x,y,z)=x(y+z)$ on the curve of intersection of the right circular cylinder $x^2+y^2=1$ and the hyperbolic cylinder $xz=1$

Using the concept of Lagrange multipliers, we can treat the circular cylinder as $G$ and assign it $\lambda$, and treat the hyperbolic cylinder as $H$ and assign it $\mu$ Then $\Delta$f=$\lambda G + ...
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1answer
25 views

How to deal with non-existent derivatives in Lagrangian?

I am stucked at a detail in a constrained optimization problem: Question Assume that the objective function is continuous on its domain $D$, but at some points $Z \subseteq D$ it is not ...
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1answer
105 views

Clarification on optimization problem, continued

Background This is a follow-up to this question. The problem statement is the same: Maximize $$f(\alpha_1, \dots, \alpha_5) = \sum_{1 \le i < j < k \le 5} \alpha_i \alpha_j \alpha_k$$ ...
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32 views

Constraint optimization with lagrangian

I am having trouble regarding the general steps one needs to take in order to solve an constraint optimization using Lagrangian. More specifically, I want to maximize objective equation $f(x,y,z,w)$ ...
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2answers
24 views

Having a bit of trouble with min/max distance from sphere to point

The sphere is $x^2 + y^2 + z^2 = 81$ and the point is $(5,6,9)$ I am using Langrane multipliers , but the answers I am getting are so far off. I will post my system of equations soon. I found ...
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37 views

Strong duality: When does the optimal primal variable coincide with the primal variable giving the dual function.

I'm considering the inequality-constrained optimization problem of finding $$ x^{\star} = \arg \min_{x} f(x) \;\; \text{s.t.} \;\; h(x) \le 0 $$ which is assumed to have a unique minimizer. The ...
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4answers
29 views

Help with Lagrangian Constrained Optimisation

Question: Maximise f (x, y) = x2y, where (x, y) ∈ R2 given the constraint that all (x, y) are points on a circle with radius √3 around origin (0, 0). Solution: f (±√2, 1) = 2 is the maximal value ...
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Determine $P_2 = f(0.7)$ when Neville's method is used to approximate $f(0.5)$

Let $f(x) = \ln(x + 1)$. Neville's method is used to approximate $f(0.5)$, giving the following table. $$x_0 = 0 - P_0 = 0$$ $$x_1 = 0.4 - P_1 = 2/8 - P_{0,1} = 3/5$$ $$x_2 = 0.7 - P_{2=?}- ...
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1answer
41 views

a question about Lagrange multiplier with higher order constrain term

If we want to minimize $f(x, y)$ subject to condition $g(x, y) = c$. Then we treat $\lambda$ as a new variable and consider minimizing $$h(x,y,\lambda)=f(x,y)+\lambda (g(x,y)-c)$$ Question: can ...
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1answer
42 views

Matrix optimization

I'm trying to minimize over $U$ the objective $\|X^{\top}UU^{\top}UU^{\top}X\|_F^2 = \text{trace}(X^{\top}UU^{\top}UU^{\top}XX^{\top}UU^{\top}UU^{\top}X)$ subject to $U^{\top}U = I$, where $X \in ...
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1answer
29 views

How to computer the Lagrange multipliers associated with an optimal solution

Suppose I have a solution $x^*\in\mathbb{R}^n$ to the following problem \begin{align*} \text{minimize}_{x}& \sum_{i=1}^n f_i(x)\\ \text{subject to}\quad &g_i(x) = 0\,\,i=1,\ldots,m\\ ...
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2answers
35 views

using lagrange multipliers to fit a curve through a point

So this is part math/ part statistics. I have a set of data I'm fitting a 2nd order curve through using least squares method (matrix form). However, I've been given the requirement to pass the curve ...
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2answers
32 views

How to find the absolute extrema of a function on an elliptical cylinder using Lagrange multipliers?

Optimize the function $ f(x,y) = x^2y $ on the elliptical cylinder $ \ x^2 \ + \ 2y^2 \ \le \ 6 \ $ using Lagrange Multipliers. Well, from what I know that I have to find the gradient then to ...
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24 views

What is the meaning of positive lagrange multiplier?

I'm handling a maximization problem with a constraint and there is a sentence "but, the lagrange multiplier is positive". I can't understand why that sentence is needed. Is there any difference ...
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2answers
149 views

Maximizing Area of Triangle in Circle

I was playing around with another example that I made up where I am trying to maximize the area of a triangle inscribed in a circle of radius. I want to do the problem using the method of Lagrange ...
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3answers
73 views

Related Methods: Lagrange Multipliers

It really pains me to ask this question, but I was working on an optimization problem and wanted to show a friend how we could also use Lagrange Multipliers to solve it. I was considering the ...
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2answers
59 views

Optimizing elementary symmetric polynomial on the unit sphere

I'd like to optimize $x_1 x_2 x_3 + x_1 x_2 x_4 + x_1 x_3 x_4 + x_2 x_3 x_4$ on the unit 4-sphere. I'm thinking I should do lagrangian optimization, but I'm having trouble solving the resulting ...
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2answers
27 views

Tips on resolving a Lagrange Multipliers equation system

I'm having a very hard time resolving the system of equations after using the Lagrange Multipliers optimization method. For instance: The plane $ x + y + 2z = 2 $ intersects the paraboloid $ z = x^2 ...
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32 views

maximum and minimum values of a function

HI! I am currently working on some calc3 online homework problems and this one is giving me a bit of tough time. I found the gradient of f to be <16x,10y> and the gradient of g to be <4,20>. I ...
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40 views

Maximization of Function with two restrictions.

Maximize $$f(x,y,z)=xy+z^2,$$ while $2x-y=0$ and $x+z=0$. Lagrange doesnt seem to work.
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33 views

Help getting a closed-form solution to a maximisation problem

I'm working through a maximisation problem that I can't seem to get a closed-form solution to. It may be the case that there is no closed-form solution, but I would like a second opinion, since I've ...
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1answer
24 views

Unable to solve system of equations in Lagrange multiplier problem.

The problem: Find the right triangular prism of given volume and least area if the base is required to be a right triangle. As for parameters of the right triangular prism, $V$ is volume, $A$ is ...
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12 views

Finding extremes on set with one constraint

I have $f(x,y)=x*y*e^{-x^2-y^2}$ and I have set $A=\{[x,y]\in \mathbb{R}^2,x^2+2y^2\ge2\}$. I have to find extremas on set A. How do I do it? It is first time when I am encountering problem with only ...
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42 views

find extrema of $2-\left(z-\sqrt{x^2+y^2}\right)^2+\left(z-\sqrt{x^2+y^2}\right)^3$

$$f(x,y,z)=2-\left(z-\sqrt{x^2+y^2}\right)^2+\left(z-\sqrt{x^2+y^2}\right)^3$$ Find maximum and minimum of the function.
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2answers
56 views

Method of Lagrange multipliers to find all critical points of a function

I am having difficulties in understanding the steps/method required to find the critical points of a function using the method of Lagrange multipliers. I have read through my text book and tried my ...
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1answer
39 views

extrema of funcion

$f(x,y,z)=x+2z$ and $M=\{[x,y,z]\in\mathbb{R}^3:x^2+2y^2=4,z+y\le 1\}$. I found out that M is not bounded from below so it does not have minimum or infimum. But how do I find maximum? I tried to use ...
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2answers
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In regards to lagrange multipliers, Confusion about derivation.

In my calculus III textbook, the following sentence is causing trouble for me and preventing me from understanding the theory behind Lagrange multipliers. "Since the gradient vector for a given ...
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1answer
62 views

Finding max/min through lagrangian

I am trying to solve this problem, but I am doing something wrong: $$f(x,y,z)=x^2-y^2,M=\{[x,y,z]\in\mathbb{R}^3:x^2+y^2+z^2=9,x+z\ge1\}$$ And let $g(x,y,z)=x^2+y^2+z^2-9$. Set M is closed and ...
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1answer
43 views

Explain Lagrange multipliers?

I am having serious issues with comprehension of this method. In particular, I don't understand the conditions. Thus far, I think it's something like; Given an objective $f: A \to \mathbb{R}^1$ and ...
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1answer
43 views

Lagrange's Multiplier Method

I need to find the distance between the ellipse $\frac{x^2}{4} + \frac{y^2}{9} = 1$ and the line $y = 10 - 2x$ using Lagranges' Multiplier Method. So far I know how to find the minimum distance ...
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1answer
38 views

Use Lagrange Multipliers to determine max and min

Using Lagrange Multipliers, determine the maximum and minimum of the function $f(x,y,z) = x + 2y$ subject to the constraints $x + y + z = 1$ and $y^2 + z^2 = 4$: Justify that the points you have found ...
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19 views

Lagrangian method for Numerical Optimization

I know of a technique, but I don't know of its name and I don't have any real literature on the technique. On the wikipedia page for Lagrange multipliers a method is provided to convert a Lagrangian ...
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1answer
33 views

Optimization of a convex target function with inequality constraints

I want to solve the following optimization problem: \begin{equation} \begin{split} \text{maximize} &\;\;\; \ln x_1+\ln x_2+\ln x_3+\ln x_4 \\ \text{s.t} &\;\;\; x_4\le4 \\ ...
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1answer
48 views

Maximizing the volume of a box using Lagrange multipliers

We are given a box of surface area $64$. As such, I wish to maximize $f(x,y,z) = xyz$ subject to $g(x,y,z) = 2(xy+xz+yz) - 64$. If I have understood in correctly, I am to find the critical points of ...
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Can Lagrange multipliers be used to give a good bound on the number of critical points?

I will explain my problem by illustrating a simple case. Easy question: Let $f(x,y)$ be a "generic" polynomial in two variables, of total degree $\le D$. What's a good upper bound for how many ...
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1answer
43 views

How to interpret Lagrangian function (specifically not Lagrangian multiplier)

I am reading the following tutorial on Lagrangian multipliers (http://www.cs.berkeley.edu/~klein/papers/lagrange-multipliers.pdf). My goal is to gain an intuitive understanding of why the Lagrangian ...
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1answer
92 views

Eigenvalues of a symmetric matrix with Lagrange multipliers

Problem: Using Lagrange multipliers, prove that all symmetric matrices $A \in \mathbb{R}^{n \times n}$ have all real eigenvalues. Proof: Consider $f: \mathbb{R}^n \rightarrow \mathbb{R}$ defined by ...
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78 views

The shortest path connecting three points

I have 3 points X,Y,Z, lets call them buildings. I need to find the shortest amount of path that connects the 3 buildings, these buildings can be in any sort of shape and any distance from each ...
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1answer
49 views

Maximum distance from the origin to the surface

I am having trouble getting the maximum distance from the origin to the surface $$ \frac{x^4}{16} +\frac{y^4}{81} + z^4 = 1 $$ Knowing I have to maximize $x^2 +y^2+ z^2$ and that the constrain ...
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1answer
68 views

Finding the shortest distance between two planes using Lagrange multipliers

A problem (among a list of Lagrange multipliers problems in Earl Swokowski's Calculus) states as follows: find the shortest distance between $2x+3y-z = 2$ and $2x+3y-z=4$. I can see that the ...
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1answer
77 views

First derivative of Lagrange polynomial

Given the Lagrange basis polynomial as: $L_i(x)= \prod_{m=0, m \neq i}^n \frac{x-x_m}{x_i-x_m} $ is there a generic equation for the first derivative ${L_i}'(x)$ for any order,t hat is for any $n$?
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22 views

Question about vector optimization using Lagrange Multiplier

I try to find the vector $x = (x_1, \cdots, x_n)$ to maximize function $f(x)=f(x_1, \cdots, x_n)$ subject to the constraint $x_1^2 + \cdots +x_n^2 = a$, where $a$ is a positive constant. I use ...