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.

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Lower Limits at infinity and growth properties

I am trying to prove the following theorem from "Variational Analysis"by Rockafellar and Wets: Following the proof guide I have come up with this proof: Setting $\bar{\gamma} = \liminf_{|x| ...
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23 views

Euler-Lagrange equations for sum of two actions/Lagrangians

Suppose we're given two action integrals : $$ S_1 = \int_{t_0}^{t_1} L_1(t,y,\dot{y}) dt \\ S_2 = \int_{t_1}^{t_2} L_2(t,y,\dot{y}) dt$$ The minimization of action integral $S = S_1 + S_2$ with ...
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Duality in deterministic stochastic control and convex conjugate

I am currently reading the book "Stochastic Multi-stage Optimization" and trying to solve the Stochastic Optimal Control problem given in this book in the framework of duality. The problem ...
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Moser Iteration for Laplacian with Hardy potential

I am reading the following proof of Cao-Yan's 2010 CVPDE Paper. It's a property for solutions of Laplace equation with Hardy potential $|x|^{-2}.$ The space dimension $N \geq 3.$ Their proof is in the ...
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Integration by parts with cross derivatives

I wish to solve the following simplified problem in the context of Weak Formulations $\large \iint(u \frac{\partial ^2 v}{\partial x ^2})dxdy + \iint(u \frac{\partial ^2 v}{\partial x \partial y})dxdy ...
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Expand this integral: $L(q) = \int \prod_{i=1}^{K} q_{i}(\theta_{i}) log\left( \frac{p(\theta,x)}{\prod_{i=1}^{K} q_{i}(\theta_{i})} \right) d \theta$

Suppose we have the following integral: $$ L(q) = \int q(\theta) log\left( \frac{p(\theta,x)}{q(\theta)} \right) d \theta$$ Letting $q(\theta) = \prod_{i=1}^{K} q_{i}(\theta_{i})$, we have: $$ L(q) = \...
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Euler-lagrange equation solution doesn't make sense

I have a real world based optimization problem, where the equation is $$ T=\int_{0}^{A}\frac{\sqrt{1+y'(x)^2}}{v(x)}\mathrm{d}x,\tag{1}$$ so that $$ L_{y'}=\frac{y'(x)}{\sqrt{1+y'(x)^2}\cdot v(x)}, \...
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62 views

What is $\max\langle x,Ax\rangle$ over subspaces non-invariant under $A$?

Let $A$ be an Hermitian matrix in a vector space $V$, and let $U\le V$ be a subspace of $V$. If $U$ is invariant under $A$, then the maximum of $\langle x,Ax\rangle$ over all unit vectors $x\in U$ ...
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Existence of solution of generalized variational inequality

I am studying the course Variational inequality and optimisation in analysis,we have the following definition: Definition:Let $K$ be a nonempty subset of $\mathbb{R^n}$, $F:K\rightarrow 2^\mathbb{...
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Nonexpansive mapping and resolvent

Suppose that $T$ is a maximal monotone mapping on Hilbert space, so that $$ \langle Tx - Ty, x - y \rangle \geq 0, $$ for all $x, y$. Then let $R = (I + T)^{-1}$ denote its resolvent. Suppose that $...
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Constructing a sequence of scalar to “merge” two sequences

I am currently trying to prove Proposition 1.4 in Variational Analysis by Prof. Boris Mordukhovich (Springer, 2018), namely Let $\Omega_1 \subset \mathbb{R}^n$, $\Omega_2 \subset \mathbb{R}^n$ and $...
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Infinitesimal variation of level sets of a function

Let $X$ be a complex manifold, and let $f:X\to \mathbb C$ be a holomorphic function. Fix a constant $C>0$. Assume $$ Y_t:=\{x\in X\mid |f(x)-t|^2=C\}$$ is smooth for all $t\in[0,\epsilon]$. Is ...
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Please Help: Utter confusion - First Variation of a function - Calculus of variations

So i've been at this for most of the night. i was originally asked to find the first and second variation of the problem $$\int_{0}^{1} \sqrt{\dot{x}^{2}+\dot{y}^2}~dt$$ but at this point i'll ...
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Calculate the first variation of $\int_{a}^{b} \sqrt{1+|\frac{dy}{dx}|^2}~dx$ by considering a parameterisation of y and x

So i've been given this job, i could use some help please. This is what i have so far: $$J(x,y,y') = \int_{a}^{b} \sqrt{1+\left|\frac{dy}{dx}\right|^{2}} dx \implies $$ $$J(t,x,y,\dot{x},\dot{y}) = \...
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Calculus of variation; Calculating First Variation

so from my understanding of the subject there seems to be a whole deluge of differing definitions for things such as the First variation for a functional. now i've been asked to calculate the first ...
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A quote from Euler and the relationship with Variational Analysis

Namely, because the shape of the whole universe is most perfect and, in fact, designed by the wisest creator, nothing in all of the world will occur in which no maximum or minimum rule is somehow ...
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Normal cone to the union of sets

For a set $A \subseteq \Bbb R^n$ and a point $\bar{x} \in A,$ the limiting (Mordukhovich) normal cone of $A$ at $\bar{x}$ is defined as $$N(\bar{x},A):= \limsup_{x\to \bar{x}} \widehat{N}(x,A),$$ ...
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Variational Calculas problem

For my Variational Calculus course we have been given the following problem: Given the following Euler Lagrange for the path P(t(s),s) $\frac{d}{dr}(\frac{((1-a/r)\frac{dt}{dr}}{\sqrt{(1-a/r)(dt/dr)^2-...
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Where is my error when setting up the differential equation to find the length of a path on a sphere

I have a sphere and I want to find the equation of the curves which give us the shortest path between two points laying on its surface. By using the Euler-Lagrange equation, I need to find the value ...
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Comparison theorem for variations inequalities

I have the following system $h(x,t) \geq g(x,t)$ $h_t(x,t) + \frac{1}{2}h_{xx}(x,t)\leq0$ $(h(x,t)-g(x,t))(h_t(x,t) + \frac{1}{2}h_{xx}(x,t)) = 0$ Is there anything we can say about the free ...
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Calculus of Variation: how to find extremas for the Functional?

I have the functional $I(t)=\int dt\ y^2(1-y')^2$ with the constraints $y(2)=1$ and $y(3) = \sqrt{3}$. I need to find the unique smooth extremal. I used the Euler-Lagrange equation $$\frac{dL}{dy}-\...
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Variational Inequalities and Convex functions

I am having troubles in understanding the proof of the following proposition. Proposition If $f(x)$ is a convex function and $x^*$ is a solution to VI($\nabla f$,$K$), then $x^*$ is a solution to the ...
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33 views

Extension of solution PDE

Let us consider a non-negative function $u \in C^{0,\alpha}(B_1)$ such that $\Delta u=1$ in the set $\{u>0\}\cap B_1$. Is it true that $$ \Delta u = \chi_{\{u>0\}} \quad\mbox{in }B_1? $$ In ...
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Jensen's inequality and convex Lagrangian

I was reading some lecture notes, and there was a following example that I didn't quite understand. If we have a following variational problem: $ \int_{a}^{b}f(u'(x))dx$ where the Lagrangian $f$ is a ...
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66 views

Using Lagrange multiplier in Euler-Lagrange Equation

I think I am doing something wrong when combining Lagrange multiplier and Euler-Lagrange equation. I need to maximize a functional of the form: $$ \int\!dx~{L(x, G, \dot{G})}~~~~~\text{where } L(a, b,...
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104 views

Generalization of minimal selection theorem

Consider a metric space $X$ and a set-valued map $F : X \to \mathbb{R}^{n}$. We define the minimal selection \begin{equation*} m(F(x)) := \arg\min \big\{ \lvert u \rvert : u \in F(x) \big\}, \end{...
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Approximating KL divergence between mixture of gaussian and standard normal

I am reading this paper, and I don't get the proposition given in the appendix of this paper. Proposition 1 Let $K,L \in \mathbb{N}$, a propbability vector $(p_1,...,p_L)$, and $\Sigma_i\in\...
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Why $p(x|z)$ is assumed to follow multivariate Gaussian distribution in Variational Autoencoder?

In VAE paper including Kingma's paper, they assumed $p(x|z)$ to be a Gaussian/Bernoulli distribution according to data type. $p(x|z)$ as a decoder functions to map the latent value $z$ to the ...
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Why both infima are in fact minima?

Let $L:\mathbb{T}^d\times\mathbb{R}^d\mapsto\mathbb{R}$ be a Lagrangian over the $d$-dimentional standard torus times $\mathbb{R}^d$. Let $\alpha >0$, the infinite horizon optimal control problem ...
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Laplace Equation on a circle with homogeneous Dirichlet boundrary

This is actually originally a Physics problem involving variational calculus, but I've come this far and beyond this point it's only mathematics thus I ask here! To begin with, I've derived a ...
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How inner product is equal to this expression

Basically i want to prove ($ \left\vert u\right\vert ^{q-2}u$ , for $q\geq2$) is strongly monotone. But i can't understand first expression. $\left\langle \left\vert u\right\vert ^{q-2}u-\left\vert v\...
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Why : $\int_{\Omega}\operatorname{div^{2}\vec{u}}dx=\int_{\Omega }(\nabla u)(\nabla u)^{t}dx$

Isee this steps in book PDE : I'm going to understand it : Steps : $$u\in\mathbb{R^{N}}$$ And $\Omega \subset\mathbb{R^{N}}$ $$K=\int_{\Omega}\operatorname{div^{2}\vec{u}}dx=\int_{\Omega }\...
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How to prove this identity for inner products?

$\newcommand\normt[1]{\left\lVert#1\right\rVert_{L^2}}$ We want to minimize the distance function or its square, $$f(\zeta_1,\zeta_2) = \normt{u(x)-\sum_{k=1,2}\sigma_kQ(x-\zeta_k)}^2 = \int \left[u(...
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Minimizing a functional with local constraint.

The problem that I am trying to solve is finding extremum of the integral: $\int|\vec{Q}(x,y)|^\alpha dxdy$, subject to a constraint: $\nabla\cdot\vec{Q}(x,y)=R(x,y)$. Now, if the constraint was ...
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Is the bijection of $3$ functions valid when you take them as a 3d vector?

Let $F_1,F_2,F_3$ be three functions from $\mathbb R\to\mathbb R$. $F_1,F_2,F_3$ are bijective (we can say that they are strictly increasing on $\mathbb R$ by the bijection theorem they are bijective)....
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Checking Euler-Lagrange equation in Sobolev spaces

I'm working on a problem in variational calculus. However, I don't feel completely sure of what I'm doing. Consider $Q:=(0,1)\times(0,1)\subseteq \mathbb{R}^2$ and define the following functional $$...
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Using these definitions how to prove $\gamma =\frac{\mu }{L^{2}}$ and $L=\frac{1}{\gamma }$

A mapping $\psi :H\rightarrow H$ is said to strongly monotone if $\exists $ $% \mu >0$ such that $$ \left\langle \psi \left( u\right) -\psi \left( v\right) ,u-v\right\rangle \geq \mu \left\Vert u-...
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Mean Field Assumption - Minimize KL Divergence

$Q_{1} = $ { $q(x): q(x) = \prod_{i=1}^{N} q_{i}(x_{i}) $ } where $q_{i}(x_{i})$ is a distribution which only depends on the $i^{th}$ variable $x_{i}$ of x. p(x) $\propto 1 - \prod_{i-1}^{N} x_{i} $, ...
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Problem with finite element method

Task picture Hey! I need some help with this task! Task: Derive variational formulation, finite element formulation, and time-marching scheme for problems (1a) - (1d). I know how to solve the ...
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Eulerian Variation of Four Velocity ??

Problem is Given in the Image, Iam unable to understand the way $\delta u^{\alpha}$ is calculated here, please help in this regard
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Finite Element Method

Hey! I need some help with this task! Task: Derive variational formulation, finite element formulation, and time-marching scheme for problems (1a) - (1d). Where should I start or how do I solve this?...
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25 views

Reducing trace minimization to generalized eigenvalue problem

I'm looking for insight in solving the following optimization over symmetric matrices A and positive-definite H. $$R=\max_{A}\frac{\text{tr}(HA)^2+2\text{tr}(HAHA)}{\text{tr}(AHA)}$$ The paper ...
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Prove that $g(x)=x^2$ is cocoercive map on [a,b].

plz use this definition $$\left\langle g\left( x\right) -g\left( y\right) ,x-y\right\rangle \geq \mu \left\Vert g\left( x\right) -g\left( y\right) \right\Vert ^{2}$$ and solve it.
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Calculus of variations with a change in parameter after some time.

$$\text{fun}:=\int_0^\infty e^{-rt}\left[u(c'(t))+w-p\,c'(t)\right]e^{-b\,c(t)}\,dt. $$ I am maximizing this objective function with some function c(t). As an extension, I want to get a solution when ...
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Generalized definition of the Green Formula (In Sobolev Spaces)

Given the classical version of Green's Formula: Let $\Omega$ be an open bounded subset of $\mathbb{R}^{n}$ and $u, v \in C^{2}(\overline{\Omega})$. \begin{equation} \int_{\Omega} Dv \cdot Du \ dx = -...
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Variational calculus with the Lagrangian function not explicitly dependent on $y'$

The standard problem in variational calculus is that given the functional $F[f]=\int_a^b \textrm{d}x f(x, y, y'),$ where $y\equiv y(x)$, find $y(x)$ that extremizes $F[f]$ under the boundary ...
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Calculus - variation iteration method

Something I don't understand The linear differential equation of first order $$u' + a(t)u = b(t) $$ Its correction functional can be written in the form $$u_{n+1}(t) = U_n(t) +\int_0^t \lambda \{\...
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How to compute some variations? From $\dot{f_\epsilon}=\dot{f}+\epsilon\dot{h}n+\epsilon h\dot{n}$ to $\delta\dot{f}=\dot{h}n+h\dot{n}$

I have a paper and on that paper I only can read: Let $f:\mathbb{S^{1}} \to \mathbb{R^2}$ be a function and $f_{\epsilon}=f+\epsilon hn$ where $\mathbb{S^1}$ is the unit circle and $\dot{f}^2=r^2$. ...
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How to find $\operatorname{argmin}_{\int_{\Omega}\Delta u=0, u(z_1)=u_1,…,u(z_m)=u_m}{\|\Delta u\|}$?

Let $d\in\mathbb{N}$ and let $\Omega$ be a non-empty bounded arcwise connected open subset of $\mathbb{R}^d$ with regular boundary. Denote by $C^1(\bar\Omega)$ the set of differentiable continuous ...
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24 views

Euler-Lagrange equations square integrable function

(A) I am trying to minimise the integral of the form $I=\int_{-\infty}^{\infty} F(x,y(x),y'(x)) dx $ (1), $y(-\infty)=y(\infty)=0$ (2) with the contrain that $y(x)$ is square integrable $\int y(x)^{...