Questions tagged [coercive]

For specific questions related to properties of coercive functions. In particular, these are commonly used in the optimization community.

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The local minimizer of a function on a discrete domain

Consider a corecive function $h$ on $\mathbb{R}^n$ and a minimization problem on $h$: Minimize $h(x)$, with $x\in \{(x_1,x_2\ldots,x_n):x_i\in\{-1,0,1\}\}$. How can we determine the number of local ...
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62 views

Show that $f(x_1,x_2)=2x_1+(x_2-x_1^2)^2+(1-x_1)^2$ is coercive

I am trying to show that the function $$f(x_1,x_2)=2x_1+(x_2-x_1^2)^2+(1-x_1)^2$$ is coercive on $\mathbb{R}^2$. To show the function is coercive, we require $\|(x_1,x_2)\|\rightarrow+\infty\implies f(...
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26 views

Coercivity of a bilinear form and the associated differential equation

I have to prove that the following bilinear form is not coercive in $H_0^1(0,1)$. $$a(u,v)=\int_0^1x^2u'(x)v'(x)dx$$ Is it as simple as saying since there is no lower bound given on $x^2$ that is ...
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61 views

How to show that $f(x_1,x_2)=e^{x_1^2}+e^{x_2^2}-x_1^2-x_2^2$ is coercive?

I am trying to show $$f(x_1,x_2)=e^{x_1^2}+e^{x_2^2}-x_1^2-x_2^2$$ is a coercive function. I considered the inequality $e^a\geq 1+a \ \ \forall a\in\mathbb{R}$, so that \begin{align} e^{x_1^2}+e^{x_2^...
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Understanding the Definition of Coercive Functions

A function $f$ defined on $\mathbb{R}^n$ is said to be coercive if $$\lim_{\|\vec{x}\|\rightarrow \infty}f(\vec{x})=+\infty.$$ I do not understand the idea of taking the limit as the norm approaches ...
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Weak formulation and Lax-Milgram lemma for $\partial_t u = \partial_x (u^2 \partial_x u)$

Consider the PDE $$\partial_t u = \partial_x (u^2 \partial_x u)$$ with some initial condition $u_0(x)$ and homogeneous Dirichlet boundary conditions for $x \in (0,1)$ I want to study existence and ...
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43 views

Coercive/(weakly) semicontinuous function: extreme values

Consider functionals of the form $$\phi : X \rightarrow \mathbb{R} \cup\{+\infty\},$$ where $X$ is an arbitrary, normed vector space. In particular, $X$ may be of infinite dimension. I would be fine ...
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Coercivity definition

In Textbook: M. G. Larson, F. Bengzon, The Finite Element Method:  Theory, Implementation, and Applications, Springer, 2014, the  coercivity definition is given as; $ a(\cdot , \cdot)$ is coercive. ...
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Coervive Map on a Banach Space

If $B$ is a finite-dimensional Banach space then norm-coercivity and coercivity coincide since the weak and strong topologies coincide. However, if $B$ is infinite-dimensional, say $B=L^p$ for $p\in [...
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Is this function coercive

I am trying to figure out whether $f(x_1,x_2)=(x_2-2x_1)^4+64x_1x_2$ is coercive. I.e.: as $\Vert x\Vert \to \infty$, does $f\to \infty$? $x\in \mathbb{R}^2$ My instinct is yes, but I've been unable ...
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Is this function coercive? Is my method reasonable?

Is this function coercive? $$f(x) = (x_1+2x_2)^2$$ I thought that because, for $x=(t, -t/2)'$ (whose norm goes to infinity when t does), $f(x)=0$, that this would mean that $f$ is not coercive, but ...
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44 views

How to show $f(x_1,x_2) = (x_1-x_2)^4 - 5 x_1 x_2$ is not coercive?

How can you show that $f(x_1,x_2) = (x_1 - x_2)^4 - 5 x_1 x_2$ is not coercive? I somehow have to show that $\lim_{||x||\to\infty} \neq \infty$. I tried expressing the function as a function of $x_1^2+...
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72 views

If a operator $L$ in a Hilbert space is self-adjoint, then $L$ is coercive?

Let $H=(H,(\cdot, \cdot))$ be a Hilbert space and $L:D(L) \subset H \longrightarrow H$ a linear operator densely defined. If $L$ is self-adjoint operator, then $L$ is coercive, that is, there exists $...
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Strong convexity inequality implied by a second derivative which is positive definite outside a convex set

I am interested in having a proof of the following result: Let $f: \mathbb{R}^n \to \mathbb{R}$ be a $C^2$ function satisfying $$ \frac{\partial ^2 f}{\partial x\partial x}(x) \geq \underline{f} >...
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117 views

Proof that bilinear form is coercive

Hi everyone I'm stuck with a proof and would be happy if anyone could help me out. Let V be a Hilbert Space and $A:V\times V \rightarrow \mathbb{R}$ symmetric, elliptic (coercive with constant $\...
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52 views

Coercivity of a functional on $H^1([0,T]^2)$

Let $I: X=H^1(\mathbb{T}^2,\mathbb{C})\rightarrow\mathbb{R}$ be the functional $$ I(u)=\int_{[0,T]^2}\vert \nabla u\vert^2 \ dx + \frac{1}{2} \int_{[0,T]^2}(1-\vert u \vert^2)^2 \ dx - \lambda \int_{[...
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116 views

Variational formulation, coercivity of: $ a(v,v) = \int_{\Omega} (\Delta v)^2 + \int_{\partial\Omega} v^2$

i'm trying to solve the Poisson equation: $$ \begin{split} -\Delta u &= f \quad \text{in } \Omega\\ u &= g \quad \text{on } \partial \Omega, \end{split} $$ where $ \Omega$ is bounded a ...
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74 views

Coercive Function Proof General Inequality

Let $f(x_1, x_2) = 3 {{{x_1}}^{2}}+{{{x_2}}^{4}}-2 {{{x_2}}^{2}}-2{x_1} {x_2}+2 {x_1}-1 $ The above function is a coercive function, now prove that: Show that there exist constants $\lambda$ and $\...
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Showing that invertibility implies coercivity

I'm trying to proof the following: Let $f$ be a continous sesquilinear form on a Hilbertspace $H$ and let $A: H \to H$ be its dedicated Operator such that $$f(u,v) = \langle u, Av \rangle_H \;\;\; \...
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Coercivity - Weak Poisson's equation

Given the weak formulation of the Poisson equation, i.e. For given source function $f\in H^{-1}(\Omega)$ find $u \in H_0^1(\Omega)$ such that $$\int_{\Omega}\nabla u \cdot \nabla v \, dx= \int_{\Omega}...
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37 views

Boundedness of a function implies coercivity of certain functional

I am reading a paper, which states Let $\Omega=(x_0,y_0)$ be an open interval in $\mathbb{R}$ and let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a Borel function. Consider the functional $$ F_\Omega: ...
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Is the Gaussian function a coercive function?

Is the Gaussian function a coercive function? If so, is it possible to use coerciveness to show that the Gaussian function has a bounded support?
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166 views

Relationship between convex functions and coercivity

I was struggling with this problem: Let $f : \mathbb{R}^n \rightarrow \mathbb{R}$ be continuously differentiable convex function. Show that for any $\epsilon > 0$ the function $g_\epsilon (x) = f(...
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45 views

For which values of $\alpha , \beta \in \mathbb{R}$ is $f(x,y) = x^2 + y^2 + 2\alpha xy + \beta$ coercive?

I know that a function is coercive if $lim_{||x|| \to \infty}f(x) = \infty$, but I don't know how to find the values of $\alpha$ and $\beta$ that would make this function coercive. I figured that $\...
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58 views

Coercivity and PS condition implies global minimizer

I am unable to get any intution to prove the following statement. Any $f:X\to\mathbb{R}$ which is coercive and satisfies PS condition has a global minimizer, provided $X$ is a reflexive Banach space.
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66 views

Equivalent definition of coercivity of a bilinear form on a normed vector space

I am searching for the reason why the coercivity of a bilinear form on a normed vector space $a\colon V\times V\to\mathbb{R}$ is defined like this: "$\exists\, \alpha>0\,$ s. t. $a(\xi,\xi)\geq\...
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On definition of coercivity over an open domain

By far I have seen, in convex analysis and optimization literature, a coercive function $f: \mathbb{R}^n \to \mathbb R$ is defined as \begin{align*} f(x) \to +\infty \text{ as } \|x\| \to \infty. \end{...
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227 views

$f:R^n→R$ is a continuously differentiable convex function then $f$ is coercive.

$f:\Bbb R^n\to \Bbb R$ is a continuously differentiable convex function. I want to show that for any $\varepsilon \gt 0$ the function $g(x)=f(x)+\varepsilon \vert |x|\vert ^2$ ...
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35 views

If $V(x_1,x_2) = V_1(x_1)+V(x_2)$ then $V$ is coercive iff $V_1,V_2$ are coercive.

According to my notes $V$ is coercive iff $\lim_{\|x\| \to \infty} V(x) = +\infty$. I have been translating this as $\forall M \in \mathbb{R}. \exists R > 0.\|x\| > R \implies f(x) > M$. ...
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137 views

Unique stationary point for a coercive function must be a global minimum?

Suppose we have a $C^{1}(D)$ function $f : D \to \mathbb{R}$ where $D$ is an open subset of $\mathbb {R}^n$. I would like to claim $f$ has a unique global minimum. I have shown that $f$ is coercive on ...
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237 views

Convex function with unique critical point is coercive

Let $V:\mathbb{R}^d \to \mathbb{R} \in \mathcal{C}^1$ such that $V$ is convex and $V$ has a unique critical point. Then $V$ is coercive. This was an example given in one my lectures. But it ...
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103 views

Coercivity of this function.

Let the following hold: Let $H$ be a Hilbert space and $F:H \rightarrow \mathbb{R}$ be continuous and convex with $$\lim_{\|x\| \rightarrow \infty} \frac{|F(x)|}{\|x\|} =\infty.$$ How do I show ...
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691 views

Check if function is coercive

I am trying to check if the function $U(x,y)=6\ln x+\ln y$ is coercive. I know, that i need to check if $\lim_{\vert (x,y) \vert \rightarrow \infty} U(x,y) = +\infty$, and so far I have $\lim_{(x,y) \...
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273 views

Linear function can be coercive?

we know that $\,f:\mathbb R^n \to \mathbb R\,$ is coercive if for any $\, x\in \,\mathbb R^n\,$ $%there exists a constant \,c\in \mathbb R\, such that$ $ %\big\langle \,f\left(x\right),\,x\,\big\...
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446 views

Why do we require coercivity for showing uniqueness of PDEs instead of injectivity?

From what I gather, when working with PDEs we can show the uniqueness of a solution to certain PDE's by requiring that a bilinear form is continuous and coercive. The coercivity of the bilinear form ...
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218 views

Showing Coercive Condition in Lax-Milgram

As a part of a Lax-Milgram problem, I need to show a particular operator is coercive, but I am having trouble finding the appropriate bounds. I need to show that there is some $\alpha >0$ such ...
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How can we show that $(u,v)\mapsto\int u\cdot v+(u^0\cdot u)\cdot v+\nabla u:\nabla v$ is coercive on $H_0^1$?

Let $d\in\left\{1,\ldots,4\right\}$ $\lambda$ denote the Lebesgue measure on $\mathbb R^d$ $\Lambda\subseteq\mathbb R^d$ be bounded and open $\Delta t>0$ $\nu>0$ Note that the norm induced by $...
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In Calculus of Variation: Problem applying variational principle theorem

Let $f:\mathbb R^m \rightarrow [0,+\infty)\;$ be a smooth function that vanishes on a finite set $A\;$ where $\vert A \vert\; \ge 2$ and the maps $v:(l^{-},l^{+}) \rightarrow \mathbb R^m\;$ ...
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1answer
307 views

Range of gradient map for coercive function

I am given a differentiable (and therefore continuous) function $f: \mathbf{E} \to \mathbb{R}$ which satisfies the following growth condition: $$ \lim_{||x|| \to \infty} \frac{f(x)}{||x||} \to +\...
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485 views

A question from definition of coercive function.

I have one doubt about the definition of Coercive function: Definition: if for a function $f: \mathbb{R}^n \rightarrow \mathbb{R}$: $$ \lim_{\|x\| \to +\infty} f(x) = +\infty, $$ then $f$ is a ...
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3k views

Proving that a strongly convex function is coercive

I am having trouble with this proof. I am given the following 2 definitions: 1) A function $f$ is coercive if $\lim_{||x|| \rightarrow \infty} f(x) = \infty$ 2) A $C^2$ function $f$ is ...
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805 views

coercive implies bounded, variational principle

I have a question about the proof of the variational principle, see below. Any help is much appreciated! How does it follow from coercivity that $(x_k)_k$ is bounded? Why is $\alpha_0 > - \infty$? ...
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511 views

How to show that $f(x,y)=x^4+y^4-3xy$ is coercive?

How to show that $f(x,y)=x^4+y^4-3xy$ is coercive ? This is my attempt : $$f(x,y)=x^4+y^4-3xy$$ $$f(x,y)=x^4+y^4\left(1-\frac{3xy}{x^4+y^4}\right)$$ As $||(x,y)|| \to \infty $ , $\frac{3xy}{x^4+y^4}...
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745 views

Coercivity and Compactness

I need to prove the following: Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$ be continuous on all of $\mathbb{R}^n$. $f$ is coercive $\iff \forall \alpha\in\mathbb{R}.\{x|f(x)\leq\alpha\}$ ...