0
votes
0answers
7 views

Minimization Problems & Collective Behavior

I am having some trouble with this problem: http://imgur.com/MX1aPzT Could anybody help me understand the concepts behind the question and how I should go about it? I have been reading through some ...
3
votes
3answers
34 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 ...
1
vote
0answers
25 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
7 views

Optimizing single element of vector wrt. second order cone constraints

Can anybody put me on the right path to solving the following problem analytically: Given a vector $\bf{x}=(x_1,...,x_n)^T$, how do I find the bounds for a single element subject to second order cone ...
0
votes
0answers
29 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
17 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
41 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
31 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 ...
0
votes
1answer
25 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
2answers
32 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
20 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
19 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
137 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
38 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) = ...
8
votes
4answers
226 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
38 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
15 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
40 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
45 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
42 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
54 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 ...
1
vote
1answer
32 views

How do I setup the lagrangian for this problem?

I have a function $y(x)$, that I would like to maximize, subject to two constraints. It is given by: $$ \max_{x} \ y(x) = a \ cos(x) + b \ sin(x) \\ \text{subject to:} \\ x \geq 0 \\ x \leq ...
0
votes
1answer
32 views

Optimization using Lagrange Multipliers for conditions with different codomain

I'm trying to maximize the trace of $X^TAX$ subject to the columns of $X$ being orthonormal, where $A$ is a diagonal matrix and X is not necessarily square, but does not have more columns than rows. ...
2
votes
3answers
78 views

Using LaGrange multipliers to solve for minimum

I am having troubles with one part of this homework problem. Hopefully somebody can help me out: Find the minimum and maximum values of the function $f(x,y)=x^2+y^2$ subject to the given constraint ...
3
votes
1answer
61 views

Minimize Frobenius norm with unitary constraint

I am trying to find a unitary tramsformation, $M$, that minimizes $\Vert MA-B \Vert_F^2$ where $A$ and $B$ are $N\times L,\;L\ge N$. I know how to solve it without the unitary constraint. I thought ...
0
votes
1answer
31 views

Lagrangian Multiplier Question

I can do question 2 easily but I'm running into some problems proving 1 rigorously. No idea how to go about doing it at all.
0
votes
1answer
27 views

Solving lagrange multiplies

The problem is: $$ \begin{align} \operatorname{max} & \quad ax+by \\ \text{subject to} & \quad x+y=m. \end{align} $$ The Lagrangian is: $$L(x,y) = ax+by−λ(x+y−m).$$ And so far I have: $$ ...
0
votes
2answers
75 views

Lagrange Multipliers have no solution

$f(x,y)=2x+y$ subject to constraint $x+y=m$. $(2,1)=\lambda(1,1)$ but this does not satisfy $x+y=m$ So there are no solution?
2
votes
1answer
69 views

Using Lagrange multipliers to maximize function

Use Lagrange multipliers to maximize function $$f(x,y)=6xy,$$ subject to the constraint $$2x+3y=24.$$ $$F(x,y,\lambda)=6xy+\lambda(2x+3y-24)$$ $$F_{x}=6y+2\lambda=0$$ $$F_{y}=6x+3\lambda=0$$ ...
1
vote
1answer
38 views

How can I calculate the maximum and minimum of this function?

This is incorrect and I have a feeling I am not doing this correctly.
2
votes
0answers
69 views

Find maximum of a double integral over a region

I have a region given by $$R = |{ax}|+|{by}| \le 1$$ and $$f(x,y) = \iint\limits_{R}{(ax-by)^2*(3ab^3+12a^3b-6a^3b^2)*\sin^2({\pi ax + \pi by}})dxdy$$ I need to find the values of $a$ and $b$ that ...
0
votes
2answers
110 views

Lagrange multipliers - finding maximum/minimum

I have solved the question, and obtained the critical points, but don't know how to show its a maximum or minimum of a function. I don't understand other answers because symbols confuse me so much and ...
0
votes
0answers
29 views

constrained minimization in N dimensions

I am looking to create an algorithm to minimize an N dimensional problem. I am unsure how to write it in its generic form, so I will show it in 1, 2 and 3 dimensions Minimize $ \sum_{i} x_i\left [ ...
2
votes
2answers
53 views

Lagrange multiplier. What happen when gradient of boundaries is null

Suppose that you have to maximize the function $f(x)$ ($f : \mathbb{R} \rightarrow \mathbb{R}$), continuous and differentiable for each $x \in A = \left\{ x |g(x) =c \right\}$, where $g : \mathbb{R} ...
1
vote
1answer
31 views

Help to clarify procedure to solve constrained optimisation problem

The following are excepts from my textbook: In the first box, it says that point $P_0$ may be a point such that grad g$(P_0)=\vec{0}^{\,}$ if $P_0$ is the max/min of $f$ subject to the constraint ...
1
vote
2answers
47 views

Finding the minimum of a function of two variables

Find the smallest value of $\displaystyle \sqrt{49+a^2-7{\sqrt{2}}\ a}+\sqrt{a^2+b^2-{\sqrt{2}}\ ab}+\sqrt{50+b^2-10b}\quad \quad$ for $a,b$ real and positive. What I've done so far: Let ...
2
votes
1answer
49 views

Optimal Price for Monopolist

I want to find the total demand for good r and the optimal price p for a monopolist using the following information: Marginal cost per good = $c$ (constant) All consumers have the following utility ...
0
votes
1answer
41 views

Constrained maximization of Leontif utility function $\min(x_1, x_2)$

The maximization problem is: Maximize $u(x_1, x_2) = \min[a_1x_1, a_2x_2]\; \ \text{s.t.}\;\; p_1x_1 + p_2x_2 \leq$ $w$, in which $x_i, p_i$ is the amount and price of good $i$, $w$ is the ...
1
vote
0answers
50 views

a question about Lagrange multiplier?

Q)Given $x_1+x_2+...+x_n=a$ where $a>0$, find the extremum value of $f(x_1,x_2,...,x_n)=x_1^k+x_2^k+...+x_n^k$ Also, find the range of $k$ in which the extremum value of $f$ is a maximum ...
3
votes
2answers
56 views

Help with Optimization Problem: Matrix Calculus

Can someone please help me with this problem? I am clueless :( $$ \left\{ \begin{array}{rclrcl} \min f(u) &=& u^tAu\\ \text{s.$\,$t.} \sum_{j=1}^n u_j &=& 0,& ...
0
votes
0answers
21 views

how to check slater condition for a constrained optimization problem?

Given any optimization problem that you suppose to solve with Lagrange by thrusting strong duality, you need to be sure the Slater Conditions. And I guess there is no algorithmic way to solve for all ...
1
vote
1answer
82 views

Minimization problems (using Lagrange Multipliers/Directional Derivatives)

The first problem asks to find the minimum cost of a rectangular box if the bottom costs \$2, the sides \$5 per square foot, and the top costs $7 per square foot. The volume of the box is given as 20 ...
1
vote
3answers
37 views

Constrained optimisation question

Since $f$ has a local extremum at $x_1$, then surely the LHS of equation (3) always zero? If so, then isn't lambda always simply zero too? But this cannot be, otherwise the last sentence of the ...
0
votes
0answers
56 views

What numerical methods could I use for this argmin problem?

I wish to solve the following using Numerical Methods: $$ \bar{m} = \underset{m \geq 0}{\text{argmin}} \left( \int_a^b \left( \frac{1}{\left(\sum_{i=1}^M \left(c_i^\alpha \cdot n^2 y^{-m-1} \cdot ...
1
vote
1answer
46 views

Lagrange multipliers - perturbation of constraints

I have been spending some time learning about Lagrange multipliers lately. Something is puzzling me though. Reading around (also on Wikipedia) I saw multiple time the interpretation that lagrange ...
0
votes
0answers
40 views

How do I find out if a critical point of a function is a maximum or a minimum?

If I've found the critical point of a function defined in some constraint (perhaps using Lagrange multipliers and the like); how do I find out if it's a relative/global maximum/minimum of a function ...
0
votes
1answer
38 views

What is the shortest distance from the origin to the intersection of $xyz=a$ and $y=bx$?

Constraints: $a,b>0$ Here is what I have so far: In order to get the shortest distance from the origin, we set $f(x,y,z)=x^2+y^2+z^2$ subject to the constrains $xyz=a$ and $y=bx$. By Lagrange ...
1
vote
1answer
43 views

Finding maximum of a function with an ellipse constraint

I'm trying to find the maximum of a function $f = a^T\mu$ where a is a vector when we have a constraint of the form: $$g(\mu) = n\mu^T S^{-1}\mu - C = 0$$ where C is a fixed constant, $S^{-1}$ is a ...
0
votes
2answers
78 views

$f$ does not have extrema at the Lagrange multiplier value

I have the function $2x^2+y$ and one constraint $x-y^2=1$ and want to find maximum value by lagrange multiplier. Intuitively, I see the point $(2,1)$ satisfies $c$ and have value of $f(2,1)=9$. ...
0
votes
0answers
31 views

Can one solve every nice maximization problem without taking a monotonic transformation?

Is there a way to solve the following maximization problem directly with lagrange multipliers without doing the trick of setting up the Lagrangian with $\log u$ instead (which is a valid procedure ...