Questions tagged [variational-analysis]

Variational analysis is a branch of mathematics that extends the methods arising from the classic calculus of variations and convex analysis to more general problems of optimization theory, including topics in set-valued analysis, e.g. generalized derivatives.

Filter by
Sorted by
Tagged with
1
vote
1answer
124 views

what is a variational form in optimization?

This is probably a pretty basic question, but I can't figure out what people mean when they say "variational forms" in optimization. For example, in this paper I'm reading, the variational form of a ...
0
votes
1answer
173 views

Reparameterization Trick in VAE

I was reading on variational auto-encoders https://wiseodd.github.io/techblog/2016/12/10/variational-autoencoder/ and am unable to understand how the function below is generated. Based on my limited ...
3
votes
1answer
413 views

Finding extremal of the functional

Find the stationary point of the functional $$ J[y]=\int \left( x^2y'^2+2y^2 \right) dx $$ where $y(0)=0, y(1)=2.$ My Solution: E-L equation: $x^2y''+2xy'-2y=0.$ This is also Cauchy-Euler equation. ...
1
vote
1answer
107 views

Finding stationary point of the functional

Find the stationary point of the functional $$ J[y]=\int \left( x^2y'^2+2y^2 \right) dx $$ where $y(0)=0, y(1)=2.$ My Solution: E-L equation: $x^2y''+2xy'-2y=0.$ This is also Cauchy-Euler equation. ...
1
vote
1answer
741 views

What are prerequisites to Terry Tao's An Introduction To Measure Theory?

I am an economics student and want to study mathematics, variational analysis in particular, with measure approach but since I am ignorant of measure theory I decided to try this book but I still find ...
4
votes
0answers
306 views

Looking for a rigorous treatment of the functional derivative the way it's used by physicists.

A lot of theories in physics can be derived from a variational principle: some action functional S on a space of e.g. field configurations $\phi$ is given, and the equations of the theory follow from ...
3
votes
0answers
88 views

Convexity vs Quasi-convexity

It is well known that in optimization the concept of quasi-convex function is the following: $f: \Omega\subset \mathbb{R}^n \rightarrow \mathbb{R}$ is a quasi convex function if for every $\alpha\in\...
2
votes
0answers
32 views

variation method

Define $ F[\phi]=\int dr \{V(\phi)+\frac{1}{2} \kappa \vert \nabla \phi \vert^2 \}$, then how can we deduce that$ \frac{\delta F}{\delta \phi}= \frac{d V}{d \phi}-\kappa \nabla^2 \phi $? I feel ...
1
vote
0answers
24 views

Relation between two types of pseudomonotone function

As far as I know, there are two different definitions for pseudomonotone function. The first definition was introduced by H.Brezis in 1968: Pseudomonotone function in the sense of Brezis: Let ...
3
votes
1answer
107 views

Let $X$ be dense in $Y$. Looking for an example such that a bounded bilinear form fulfills an inf-sup condition on $X\times Y$ but not on $X\times X$

Let $X$ and $Y$ be two separable Hilbert spaces such that $X \subsetneq Y$ densely, continuously and the norms are related through $\|x\|_X^2=|x|_X^2+\|x\|_Y^2$ for all $x \in X$; e.g. take $X=H^1$, $...
5
votes
0answers
121 views

All the symmetries of the Dirichlet energy are conformal

It seems to be "folklore" knowledge that all the (source) symmetries of the $d$-Dirichlet energy are conformal maps. Specifically, I have found this nice proof for the following claim: Proposition:...
3
votes
1answer
112 views

A mistake on the Borwein - Preiss's Variational Principle statement??

I know the following generalization of Borwein - Preiss's Variational Principle(BPVP), known as Loewen-Wang's Variational principle (LWVP) $\textbf{Loewen- Wang Variational Principle}$ Let $...
0
votes
1answer
66 views

Understanding the proof for the existence of solutions of a variational inequality (by F. E. Browder)

I am trying to understand the proof of Theorem 1 in this paper (see page 784 for the proof). I understand that by the finite intersection property there exists an element $u_0$ in the weak closure of $...
2
votes
1answer
99 views

The integral under the variation $\delta$ sign

In physics books on classical field theory, the authors usually define the action as $$ S = \int\mathcal{L(\phi,\partial_\mu\phi)d^4x} $$ where $\mathcal{L}$ is the lagrangian density. Then, ...
2
votes
3answers
137 views

Euler-Lagrange formula

Let $y:[-1,1]\to [2-1,2+1]$ be a $C^1$-smooth function, and $F(y,y'):=y\sqrt{1+y'^2}$. Suppose $y(x)$ satisfy the Euler-Lagrange equation, i.e. $$\frac{\partial F}{\partial y}-\frac{d}{dx}\frac{\...
0
votes
0answers
21 views

About variational calculus conmutative

I need some help with the development of an expression, it is: $$\delta\left\{\frac{\partial^3F(q^1,q^2,t)}{\partial t\partial q^1 \partial q^2}\right\}$$ with each of $q^1,q^2$ are functions of $t$. ...
0
votes
1answer
98 views

An $\mathcal{L}^1$ optimization problem in function space

Let $f: (-\infty,\infty)\to [0,\infty)$ be a smooth map. Then, how to solve the following optimizatin problem w.r.t. $f$? \begin{align} \mathrm{minimize}_f&~~\int_{-\infty}^{\infty} |f(x) - \exp(-...
0
votes
1answer
63 views

Compute a epsilon normal set

Could you please help to me to compute the following $\epsilon$-normal set: Given $\epsilon>0$, how to compute the $\epsilon$-normal set of $C:=[2,\infty)\times \Bbb{R}$ at the point $(2,0)$. Thank ...
2
votes
1answer
233 views

$\epsilon$-normals to convex sets

I am reading the book by B. Mordukhovich, Variational analysis and generalized differentiation I. On page 6 it is stated the following inclusion: $$ \hat{N}_{\varepsilon }\left( \bar{x};\Omega \right)...