3
votes
0answers
31 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 ...
0
votes
4answers
24 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 ...
0
votes
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 ...
0
votes
1answer
25 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\\ ...
0
votes
2answers
53 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 ...
1
vote
0answers
26 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 ...
0
votes
1answer
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.
0
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
3answers
40 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.
0
votes
2answers
49 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 ...
1
vote
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 ...
0
votes
1answer
60 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 ...
1
vote
1answer
37 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 ...
1
vote
1answer
32 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 \\ ...
0
votes
1answer
46 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 ...
0
votes
1answer
40 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 ...
1
vote
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 ...
1
vote
1answer
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 ...
1
vote
2answers
45 views

total least squares derivation with matrices

Taken from a computer vision book: "to minimize the sum of the perpendicular distances between points and lines, we need to minimize $$ \sum_i (ax_i + by_i +c)^2$$ subject to $a^2 +b^2 =1$. Now using ...
2
votes
3answers
57 views

Extrema of $x+y+z$ subject to $x^2 - y^2 = 1$ and $2x + z = 1$ using Lagrange Multipliers

Find the extrema of $x+y+z$ subject to $x^2 - y^2 = 1$ and $2x + z = 1$ using Lagrange multipliers. So I set it up: $$ 1 = 2x\lambda_1 + 2\lambda_2 \\ 1 = -2y\lambda_1 \\ 1 = \lambda_2 $$ Plug ...
0
votes
0answers
38 views

KKT Conditions and Convexity

min $x^2 -xy +y^2 -5x+6y$ subject to $1 \leq y$, $y^3 \leq 2x$, and $x \leq 8$ Write out the KKT conditions for this problem. Show that $(x,y) = (4,2)$ is a KKT point, and is therefore a global ...
1
vote
2answers
49 views

Lagrange Multiplier inconsistent

max $f(x,y,z)$ subject to $x+y+z=1$ $$f(x,y,z)= - (x+0.5y)\log(x+0.5y) -2(0.25y+0.5z)\log(0.25y+0.5z)-\log(2)(1.5y+z)$$ $$\frac{\partial F}{\partial x} = - (1+\log(x+0.5y))$$ $$\frac{\partial ...
0
votes
1answer
17 views

Substitute for constraints in constrained optimisation problem

I have to solve the following optimisation problem $$\max_{c,\ell, L} \{\ln(c) + \ln(\ell)\}$$ subject to the constraints: $c=Lw$ and $1=\ell +L$. Is it okay to solve the problem ...
1
vote
2answers
30 views

Does Lagrange multiplier have solution if functions doesn't intersect

I am trying to get intuition behind Lagrange multiplier and question that bothers me is: Does Lagrange multiplier have solution if two functions(main function and constraint) doesn't intersect. Thank ...
0
votes
1answer
45 views

Lagrangean Relaxation of quadratic assignment problem to yield $n$ knapsack problems?

Consider the assignment problem: $$ Z = \min \sum_i\sum_j\sum_k c_{jk}\cdot x_{ij}\cdot x_{ik} $$ s.t. $$ \sum_i x_{ij} = 1 \quad\forall j $$ $$ a \leq \sum_j x_{ij} \leq b \quad\forall i $$ $$ ...
2
votes
1answer
80 views

The meaning of $\lambda$ in Lagrange Multipliers

This is related to two previous questions which I asked about the history of Lagrange Multipliers and intuition behind the gradient giving the direction of steepest ascent. I am wondering if the ...
0
votes
0answers
26 views

Can someone explain Method of Lagrangian multipliers

Can someone explain Method of Lagrangian multipliers to a beginner? I need some knowledge about solving problems using this method. If someone can provide the basic details in a simple manner along ...
0
votes
2answers
78 views

Distance from Ellipsoid to Plane - Lagrange Multiplier

Find the distance from the ellipsoid $x^2 + y^2 + 4z^2 = 4$ to the plane $x + y + z = 6$. I'm trying to do it using Lagrange multipliers over the distance equation, but then it just gets ...
0
votes
1answer
28 views

Maximize/minimize $1/3 x^3 + y$ with constraint $x^2 + y^2 = 1$?

I keep running around in circles when I use the Lagrangian multiplier method getting $x = 1/y$ But then when I substitute $(1/y)^2 + y^2 = 1$ I then get $1/y^2 + y^2 = 1$ and this doesn't give me ...
3
votes
3answers
55 views

Maximize $x^2+y^2+z^2$ on $x^2+y^2+4z^2 = 1$

Hi this is a lagrangian optimization problem. Essentially as the title says, the question is asking us to maximize (if possible) $x^2+y^2+z^2$ on $x^2+y^2+4z^2=1$. I started by the standard ...
2
votes
0answers
35 views

Inequality constraints in Lagrangian

I was reading Lagrangian multipliers . In the above text I can't understand why $\lambda \ge 0 $ for $g(x)\ge0$ and vice versa . Can anyone give me the explanation to this ?
0
votes
0answers
33 views

Lagrangian Method Proof

Suppose $f(\mathbf x)$, $g(\mathbf x)$ are smooth functions where $\mathbf x^*$ is a constrained local minimizer of $f(\mathbf x)$ subject to $g(\mathbf x)=0$. If $\nabla g(\mathbf x^*) \neq 0$ and ...
1
vote
0answers
23 views

Regularized least squares

In Image Restoration, a true image f (in vector form)can be related to degraded data y through a linear model of the form $$y = Hf + n$$ where H is 2d blurring matrix and n denotes noise vector and ...
1
vote
1answer
46 views

Convex Functions and Subsets

Suppose that $f, g: \mathbb R^n \to \mathbb R $ are $C^1$ convex functions. Show that $C = ${$\mathbf x \mid g(\mathbf x) \leq 0$} is a convex subset of $\mathbb R^n$. Show that if $\nabla f(\mathbf ...
1
vote
2answers
43 views

Max and Min using Lagrange Multipliers

Suppose A is a symmetric matrix. Show that the maximum and minimum of $\mathbf x ^T A \mathbf x$ subject to the constraint $\mathbf x ^T \mathbf x=1$ are the maximum and minimum eigenvalues of A. I ...
2
votes
3answers
94 views

Lagrange Method Problem

I am from engineering background and I am currently studying calculus. I had a question from assignment to be solved from a course on coursera but I could not do it. People have posted solution in the ...
0
votes
2answers
40 views

Simple Lagrange Multiplier Problem, not working out

The question should be simple. Use the Lagrange Multiplier to maximize $f(x,y) = 4x^2 + 10y^2$ subject to the constraint $x^2 + y^2 = 4$. But when I set it up I get two different values for ...
0
votes
3answers
62 views

Use Lagrange Multipliers to find the absolute extrema

Use Lagrange Multipliers to find the absolute extrema (if any) of: $f(x,y) = 4x^2 + 9y^2$; subject to $2x +3y = 6$. Using Lagrange I end up with one point: $(\frac{3}{2}, 1)$ I'm just not sure how ...
0
votes
1answer
27 views

Can I use Lagrange Multipliers with inequality constraints?

Suppose I had a problem: Maximize $f(\bf{x}) $ subject to the contraints $g_i(\bf{x})< b$ Can I still use Lagrange multipliers? My text says that the constraints need to be equalities.
0
votes
0answers
28 views

Convex Minimization Problem with double sum

Given fixed natural number $n$ and two real numbers $A$ and $B$. I'd like to find $c_{12},\dots c_{(n-1)n}$, i.e., ${n\choose2}$ real numbers, such that $\sum_{1\le i<j\le n}^nc_{ij}=1$ which ...
2
votes
2answers
173 views

Closest distance between two quadratic curves

I'm having trouble with the following problem : "find the closest distance between $x^2+4y^2=4$ and $xy=4$" I tried to solve using the properties of ellipse and hyperbola, but the relatively tilted ...
1
vote
1answer
50 views

Lagrange multiplier - Find maximum on surface

I need someone to walk me through a 3 variable lagrange problem, since I haven't been able to find a reliable source to teach me, please. Here it is: Find the maximum of the function $F(x,y,z) = ...
9
votes
4answers
230 views

Maximum of $(1-q_1)(1-q_2)\ldots(1-q_n)$

I'm trying to find the maximum of $(1-q_1)(1-q_2)\ldots(1-q_n)$ where $n\ge 2$, on a the set $\{(q_1,\ldots , q_n) :q_1^2+q_2^2+\ldots+q_n^2=1 \ q_i\ge 0 \}$ (With the condition $q_i\ge0$ this is just ...
0
votes
2answers
41 views

Extrema of two variable function

Find extrema of $f(x,y)=x^2-xy+y^2$ from set $M=\{ [x,y] \in \mathbb{R}^2;|x|+|y|\le1\}$ I am solving this kind of problems for the first time and I am not sure what I am doing, what I have got ...
1
vote
0answers
19 views

Lagrangian with Nonholonomic Velocity Constraints

I have a Lagrangian $L$ dependent on four generalized coordinates $[\theta(t), \phi(t), l(t), h(t)]$. And I have two differential non-holonomic constraints given by: $$ Eq1: \dot l(t) = -c\dot \phi/ ...
0
votes
0answers
52 views

Solve multivariable nonlinear equation with mixed constraints?

How do I solve a multivariable ($x_1$ to $x_m$) nonlinear (in this case a quadratic) objective function with mixed (equality (my Sum) and inequality (bounded variables) in this case linear ...
1
vote
2answers
49 views

Minimizing $\sum_{i=1}^n \frac{x_i^2}{w_i}$ subject to $\sum_{i=1}^n x_i=1$

Minimize $\displaystyle\sum_{i=1}^n \frac{x_i^2}{w_i}$ subject to $\displaystyle\sum_{i=1}^n x_i=1$. The answer is $x_i=\displaystyle\frac{w_i}{\sum_i w_i}$ but I don't know why apart from ...
-1
votes
1answer
52 views

Optimization: Finding line connecting non-pareto-optimal allocation in Edgeworth Box to PO allocation

Two people, A and B, with respective utility functions of: $$U_a(X_a,Y_a) = X_a^2 Y_a\\ U_b(X_b,Y_b) = X_b Y_b^2$$ Total $X$ (that is, $X_a+X_b$) is fixed at $36$. Total $Y$ ($Y_a+Y_b$) is fixed ...
2
votes
0answers
76 views

Maximize a function subject to the constraint $x^2+y^2=R^2$

Please help me how to deal with maximization of function $$f(x,y)=1-e^{-\pi x}+e^{\pi x}\left[1-\cos(\pi y)+\sin(\pi y)\right]$$ subject to the constraint $x^2+y^2=R^2$. Using Lagrange ...