1
vote
0answers
46 views

Application of a general “Weierstrass theorem”

http://books.google.at/books?id=9OSrV73a40gC&pg=PA45&lpg=PA45 gives a general Weierstrass theorem. Are there notable applications of this theorem, say in the calculus of variations? (I could ...
4
votes
1answer
68 views

Division of plane into equal area regions

We divide a plane ($\mathbb{R}^2$) into infinite number of regions each of area equal $1$. We can use only (one-dimensional) curves which may meet at points. Fix a point $p$ on a plane and consider ...
1
vote
1answer
49 views

Variational Methods, why KL divergence is the difference between true distribution and approximating distribution.

Likelihood = $L(\textbf{w}) = P(V\mid \textbf{w})$. $$\ln P(V\mid \textbf{w}) = \ln \sum_H P(H,V\mid \textbf{w})$$ $$= \ln \sum_H Q(H\mid V)\frac{P(H,V\mid \textbf{w})}{Q(H\mid V)}$$ $$\geq ...
1
vote
0answers
23 views

Euler Lagrange equations

I need to minimise $$\int\limits_\Omega|\nabla H_\epsilon(\phi)|\,dx\,dy$$ with respect to $\phi$. Where $H_\epsilon$ is the regularised Heaviside function, so that it is differentiable. This can be ...
1
vote
0answers
34 views

Non-convexity of an energy functional

How would I go about showing that the following Mumford Shah functional is not convex? $$E_{MS}(u,C)= \int_{\Omega} |u_{0}(x,y) -u(x,y)|^{2}\ dx\ dy + \mu \int_{\Omega \backslash C}|\nabla ...
1
vote
1answer
20 views

minimizing a function involving exponential term

Let $w\ge e$ . I want the following $$ \min_{r\geq0} r(e^r-w) $$ Is there any way to find it. Thanks.
1
vote
0answers
38 views

Existence of a Minimizer $h(\rho) = c \|\rho\|_{3} - \int_{\mathbb R^3} \, dx \frac{\rho(x)}{|x|} $

given the following functional $h(\rho) = c \|\rho\|_{3} - \int_{\mathbb R^3} \, dx \frac{\rho(x)}{|x|} $ with $\rho>0$ , $\|\rho\|_1 = 1$ and obviously $\rho\in L^1(\mathbb R^3)$. Can I see ...
3
votes
1answer
168 views

Finding the shortest path length on a curved surface(hyperboloid)

I wish to find the minimum path length between two points $P_1(\sqrt2,0,-1)$ and $P_2(0,\sqrt2,1)$ on a hyperbolic surface $S =\{(x,y,z)\in R^3\ |\ x^2+y^2-z^2=1\}$ I faintly recall studying ...
2
votes
0answers
88 views

Calculating the maximum of a function

How can one determine $$\max_{f_0,f_1}\frac{\int_{\mathbb{R}}f_1(y)^xf_0(y)^{1-x}\log\left(\frac{f_1(y)}{f_0(y)}\right)\mbox{d}y}{\int_{\mathbb{R}}f_1(y)^xf_0(y)^{1-x}\mbox{d}y}$$ given ...
2
votes
0answers
42 views

What methods are available for this optimization problem?

I have an intermediate knowledge of the calculus of variations: I can handle constraints in functional or integral forms and extrapolate to multiple variables and functions. If I dig in my notebooks I ...
2
votes
0answers
91 views

Calculus of variations for implicitly defined functional

I would like to minimize a functional of the type: $$L[\gamma]=\int_a^b F(T(\gamma(t))dt$$ on the space of paths $\gamma$, where $T=T(\gamma,t)$. Now, usually I would simply apply Euler-Lagrange's ...
2
votes
1answer
84 views

Extremum of functional of a complex function

consider functional $E$ defined by $$E[z]=\int F(x,z(x))dx$$ where $F$ is a complex-valued nonlinear function. How can we find the function $z(x)$ so that $$G=|E|^2=EE^*=\iint ...
4
votes
1answer
32 views

Optimization in $L_1$, does this make sense?

I'd like to find a probability distribution $f(x)$ on the unit interval $[0,1]$ that obeys a given set of moment constraints, e.g. $\int_0^1 xf(x) dx = \mu_0$ for some given $\mu_0$, and so forth. ...
3
votes
2answers
56 views

Minimizing a Functional with a Path Length Constraint

Say you have some functional of the form $\int_0^{t_f} L(x,\dot{x},y,\dot{y},z,\dot{z}) dt$ that you're trying to minimize. Normally one can solve this using the Euler-Lagrange equations, and when you ...
1
vote
2answers
150 views

Maximum area under a curve by calculus of variations

I am asked to find the function that has the maximal area for a given length L when x runs from -a to a. I calculated the integral to be varied as follows: $$ \int_{-a}^{a}\ y + \lambda \sqrt{1 + ...
0
votes
1answer
63 views

Is it possible to solve or approximate this second order nonlinear system of differential equations.?

Given initial values $d[0]$ and $k[0]$, I would like to solve for the initial rate of change, $\dot d[0]$, and compare this value against some data. I have the following profit function, which I ...
28
votes
5answers
987 views

Can this ant find its way back to the nest?

So the puzzle is like this: An ant is out from its nest searching for food. It travels in a straight line from its nest. After this ant gets 40 ft away from the nest, suddenly a rain starts to ...
2
votes
1answer
107 views

How can Hotelling reduce the Euler-Lagrange equation in his calculus of variations mine problem?

In a 1931 paper Hotelling gives the discounted profit of a mining operation as: $$P=\int_{0}^{\infty} \dot{x} p(x,\dot{x},t) e^{-rt} \:\:dt$$ Note that this is, for the most part, a typical calculus ...
0
votes
2answers
123 views

Calculus of variations, what is a functional

I'm reading a bit about the calculus of variations, and I've encountered this bit: Suppose the given function $F(.,.,.)$ is twice continuously differentiable with respect to all of its arguments. ...
0
votes
0answers
33 views

Maximize an integral with variable end points

How can I find optimal $s(\cdot)$ in this problem (I have little knowledge about calculus of variation) Thanks. $$ \large{\max_{s(\cdot)} \int_0^{\theta^*} ...
14
votes
1answer
252 views

Fastest curve from $p_0$ to $p_1$

I'm trying to solve a problem in path planning: Given points $p_0$ and $p_1$ and vectors $v_0$ and $v_1$, find a function $p(t)$ st. $p(0) = p_0$, $p(T) = p_1$, $p'(0) = v_0$ and $p'(T) = ...
1
vote
0answers
30 views

Optimal form a differential equation

Suppose I have an objective function: $min(-\int_0^T((C*x_2 + f(x_1))*V) dt$ with the constraints $\dot x_1 = x_2 $ and $ \dot x_2 = -{C\over m} *x_2 -{f(x_1)\over m}-A $ The standard base ...
1
vote
0answers
41 views

Regularity questions in constrained variational problem

Consider the problem of minimizing $$ I(u) = \int_a^b F(t,u(t),u'(t)) d t $$ over, say, $W^{1,\infty}(]a,b[)$. Then regularity theory tells us that if $F$ and $F_{\dot q}$ are $C^k$, and in addition ...
1
vote
0answers
26 views

Maximum value for $F(y)=\int_0^1[y'\sin(\pi y)-(y-t)^2]dt$?

This is what I have done thus far: $F(y)=\int_0^1[y'\sin(\pi y)-(y-t)^2]dt=-\frac{1}{\pi}\int_0^1[(\cos(\pi y))\frac{d}{dt}]dt-\int_0^1(y-t)^2dt$ (as $-\frac{1}{\pi}[(\cos(\pi ...
0
votes
0answers
53 views

Optimal parametric curve via calculus of variations

Assume we have $N+2$ points in $\mathbb R^2$, $(x_0,y_0),(x_1,y_1)\dots(x_N,y_N),(x_{N+1},y_{N+1}),$ where $(x_0,y_0)=(x_{N+1},y_{N+1}).$ We would like to find a pair of functions \begin{cases} x = ...
0
votes
1answer
47 views

how to solve this optimization problem with functions?

Suppose $\theta\in[0,1]$ and $s(\cdot)$ is strictly increasing in $\theta$ with $0\leq(0)<s(1)\leq1$ and $s(0)+s(1)>1$ How could I solve the program, $\max_{s(\cdot)\in ...
1
vote
0answers
105 views

A maximization problem in Sobolev space

For $k>0$, let $f_k$ be a sequence of positive functions in $H_N^1(0,1)$, where $H_N^1(0,1):=\{u\in H^1(0,1)|u^{'}(0)=0=u^{'}(1)\}$, $H^1(0,1)$ is the usual Sobolev space consisting of $L^2(0,1)$ ...
0
votes
0answers
115 views

Linear programming with countably “infinite variables” and “finite constraints”!

Is it possible to do a linear programming with countably "infinite variables" and "finite constraints"? If not, what do you purpose? (Example Link): Maximum and minimum of an integral under integral ...
4
votes
0answers
486 views

Maximum and minimum of an integral under integral constraints.

Find the maximum and minimum of the following integral in terms of $f(x),a,C$: \begin{align}I=\int_{0}^{a} \frac{x}{f(x)}p(x)dx \end{align} s.t.: 1) $\int_{0}^{a} p(x)dx=1$ 2) $\int_{0}^{a} ...
2
votes
1answer
162 views

Maximize a functional

Please help me how to deal with maximization of functional like this: $$F\{a(s)\} = \int\limits_0^t \left( g(a(s)) - \alpha\, v(s)^2 \right) ds, \ a(s) \in \left[0, \infty\right)$$ where $g(x) = x ...
2
votes
1answer
101 views

Constructing shortest interpolation curve from points in $\mathbb R^2$ with parametric equations.

Assume we are given a set of $n$ points from $\mathbb R^2$, $(x_1,y_1),(x_2,y_2)\dots(x_n,y_n)$. We want to construct a path connecting all these points using a pair of parametric equations ...
0
votes
1answer
98 views

Condition or Proof: Minimizer of one function is maximizing another function

I have two real functions $f(X),g(X)$ where the argument $X$ is a real matrix. The solution $X^*$ for the problem of minimizing $f$ is ending up maximizing $g$ as well. I am looking for a way to prove ...
2
votes
1answer
104 views

Does setting derivative to zero suffice always for minimization of convex functions?

I have this convex function in $X$, given by $Trace(AX^TBX)$ where $A$, $B$ are p.s.d and all entries are real. Now if I had a linear function $l(X)$ that prevents a trivial zero-matrix solution for ...
2
votes
0answers
61 views

Find u that minimizes the integral mean

How do I find $u : [0,\infty) \to \mathbb{R}^m$ that minimizes \begin{equation} J(u(\cdot)) = \lim_{t \to \infty} \frac{1}{t} \int_0^t L(x(\tau),u(\tau)) d \tau, \end{equation} subject to ...
1
vote
1answer
46 views

Maximization of the product of two inner products

I have an optimization problem of the form $\max_\gamma \langle f\circ\gamma,w_f\rangle\langle g\circ\gamma,w_g\rangle$ where $f\circ\gamma$ is the composition of $f$ and $\gamma$ and the inner ...
2
votes
1answer
78 views

Local and global extremes

I Wrote problems and solutions, I need just few explanations. 1.Let $$J(x)=\int_{0}^{1}x'^{2}dt,\quad x(0)=0, x(1)=1. $$ Find the extrema value for $J$. I'm doing this using Euler equation ...
3
votes
1answer
242 views

Derivation of Euler-Lagrange equation

Here is a simple (probably trivial) step in the derivation of the Euler-Lagrange equation. If we denote $Y(x) = y(x) + \epsilon \eta(x) $, I want to know why is $\dfrac{\partial ...
4
votes
1answer
298 views

Optimizing a functional with a differential equation as a constraint

I am working on solving the following optimization problem. I think it is well-poised but, if not, please give me some pointers that could make the question make more sense. We have a parametric ...
2
votes
2answers
494 views

satisfy the Euler-Lagrange equation

Two circles of unit radius, each normal to the line through their centers are a distance d apart. A soap film is formed between themas shown below; energetic considerations require the filem to ...
4
votes
2answers
162 views

Finding an explicit expression for a minimizer

Suppose $f$ is a continuous function on the interval (0,1). We consider the energy functional $F(u) = \int^1_0\frac{1}{2}((u')^2+u^2)\,dx - \int^1_0fu\,dx$ which is well defined for continuously ...
8
votes
1answer
297 views

Minimizing Lagrangian with two functions

I read this problem where I have to minimize a functional $E[L]$ using calculus of variations, but I'm not sure what is the procedure to follow. The functional is the expected loss: $$E[L] = ...
1
vote
0answers
65 views

Sufficient conditions for Hessian definiteness for critical points of functionals

Let $C$ be the set of smooth curves from the unit interval into $\mathbb{R}^n$. Let $f : C \rightarrow \mathbb{R}$ be a functional on these curves given by $f(x) := \int_0^1 L(x,\dot{x}) dt$. Define ...
1
vote
0answers
191 views

How to find $\kappa$ to minimize integral $I = \frac{1}{\kappa}\int\limits_{0}^{T} \mathrm{exp}\left(-f(\kappa,x)\right) \mathrm{d}x$

I am trying to find such value $\kappa \in (0,1)$ that would minimize the integral \begin{equation} \begin{aligned} I = \frac{1}{\kappa}\int\limits_{0}^{T} \mathrm{exp}\left(-f(\kappa,x)\right) ...
1
vote
0answers
65 views

integer transform

Let be $X$ the following finite set: $X=\{0,1,2,\ldots,63\}$. I want to find two function $f$ and $g$ , where $f,g:X \times X \to Z$. We define $x'=f(x,y)$ and $y'=g(x,y)$. We impose the following ...
0
votes
0answers
30 views

Rate of convergence for a many point object ( function or surface ).

I am aware of the definition of a Rate of convergence for iterative methods involving a single point $$ x_{n+1} = \phi(x_n) ~~~;~~\lim_{n \rightarrow \infty} {{ |x_{n+1} -\alpha|} \over{|x_{n} ...
1
vote
1answer
39 views

Approximation in $L^2$

Let $G$ be a domain assumed smooth enough. I want to show that the mean value $m$ is minimizing $ m \rightarrow \| f-m\|_{ L^2(G)} $ for $ f \in L^2(G)$. Is it unique? Is it allowed to derive under ...
12
votes
1answer
342 views

Arnol'd's trivium problem #68

I came across this blog that says that its French version has answers to most of Arnol'd's trivium problems, and I figured I'd try my hand at some of the ones they don't have. Number 68 raised my ...
1
vote
1answer
117 views

how to obtain Euler equation for smoothing spline minimization problem?

The question might be trivial, but I don't understand why this minimization problem in Sobolev space $$ \min_{g}\int_{0}^{1}\left\{ f(x)-g(x)\right\}^{2} dx+\lambda\int_{0}^{1}\left\{ ...
2
votes
0answers
41 views

Gradient/sub-differential of a minimum of a function

Suppose I have a function $F(x,D) = ||y-Dx||_2^2$, such that $x^{*}(D)= \displaystyle arg \min_{x} F(x,D)$ (that is given $y$ and for a fixed $D$) and subject to some constraint $h(x) <\epsilon$, ...
1
vote
2answers
181 views

Intuition of 'Transversality conditions' needed

I have read different matrials on Calculus of Variations, but I still do not grasp the intuition of transversality condition. From textbooks, I can only roughly get an idea that with a transversality ...