Questions tagged [coercive]

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

Filter by
Sorted by
Tagged with
0 votes
3 answers
70 views

How to show that a function $f(x,y) = x^2-3xy+5y^2$ is coercive?

Definition. A continuous function $f(x)$, defined on $\mathbb{R^n}$, is coercive if: $$\lim_{||x||\xrightarrow[]{}\infty}f(x)=+\infty$$ Problem. Is the following solution is correct: $f(x,y)=x^2-3xy+...
36n's user avatar
  • 11
0 votes
1 answer
41 views

Proof of Coerciveness of a Function [closed]

please may have a look . Help, Hints, $\textbf{Statement:}$ Let $f : \mathbb{R}^n \rightarrow \mathbb{R}$ be a $C^1$ function such that for every $x = (x_1, \ldots, x_n) \in \mathbb{R}^n$, the ...
E.K's user avatar
  • 1
0 votes
0 answers
13 views

Strongly monotone operator implies coercivity

I read from the book "Finite-Dimensional Variational Inequalities and Complementarity Problems, Volume I" (P. 156) and paper "Finite-dimensional variational inequality and nonlinear ...
zzgsam's user avatar
  • 107
4 votes
2 answers
250 views

Definitions of coercivity - functional analysis

In my PDE courses I've come across two different definitions or coercivity of a functional $\mathit{F}: \mathit{H} \rightarrow \mathbb{R}$ where $\mathit{H}$ is a Hilbert space. Definition 1: For the ...
Len's user avatar
  • 103
0 votes
0 answers
37 views

Why is th coerciveness of a bilinear form important for making sure a variational problem is well-posed?

Hello, I've come across the idea of the coerciveness of a bilinear form in the context of variational problems. Could you help me understand in simpler terms why this property is so important for ...
user134613's user avatar
1 vote
0 answers
28 views

Existence and uniqueness without ellipticity condition

Consider a function $a:[0,1]\to[0,\infty)$, e.g., $a(x)=x$. Are there existence theorems for equations of the form: \begin{align*} u-\big(a(x)u_x\big)_x=f(x) ? \end{align*} what are the boundary ...
Alberto's user avatar
  • 243
2 votes
1 answer
47 views

Negative definite forcing of an ODE ? (soft question)

Given an ODE in $\mathbb{R}^n$ $$\frac{d}{dt}x(t)=b(x(t)).$$ How can the assumption that $$\langle D b \xi,\xi\rangle \leq - c |\xi|,~~~~~\forall \xi \in \mathbb{R}^n$$ be interpreted? Does this stop ...
Monty's user avatar
  • 2,230
0 votes
1 answer
79 views

coerciveness of $\|Hx - y\|_2^2 + R(x)$ where $R$ convex

Let $x \in \mathbb{R}^d$, $y \in \mathbb{R}^m$, $H \in \mathbb{R}^{m,d}$ and $R$ proper, convex and lower-semicontinuous. What do we require $H$ to fulfill in order for $$x \mapsto \|Hx - y\|_2^2 + R(...
Pazu's user avatar
  • 1,077
2 votes
2 answers
186 views

Coercive functionals and minimizing sequences

My course notes provide just a brief mention to a common technique used in calculus of variations, but I can't understand it. Don't expect total rigor, since it's just a mention. Let $G:X\to \overline{...
Kandinskij's user avatar
  • 3,719
0 votes
1 answer
45 views

Coercivity of norm term with matrix

Let's start with $x \in \mathbb{R}^d$, $y \in \mathbb{R}^m$, $H \in \mathbb{R}^{m,d}$ and consider $$F:x \mapsto \|Hx - y\|_2^2 $$ I want to examine whether or not $F$ is coercive, i.e. if for all ...
Pazu's user avatar
  • 1,077
0 votes
1 answer
224 views

Prove that a strongly convex function is coercive

Let function $f : \mathbb{R}^n \to \mathbb{R}$ with $\|\cdot\|$ is Euclidean norm then we have the following definitions: A function $f$ is coercive if $\lim\limits_{\|x\|\rightarrow \infty}f(x) = \...
Trí Hoàng Minh's user avatar
4 votes
1 answer
179 views

Coercivity and spectral gap: understanding the equivalence

I am referring to this paper, p. 21. First, there is the following definition of coercivity: Let $L$ be an unbounded operator on a Hilbert space $\mathcal{H}$ with kernel $\mathcal{K}$ and let $\...
selector's user avatar
  • 447
0 votes
1 answer
67 views

Proof that bilinear form $(u,v)_{L^2(\Omega)} \mapsto (\nabla u, \nabla v)_{L^2(\Omega)}$ is not coercive in $H^1(\Omega)$

Let $\Omega \in \mathbb{R}^n$ be a bounded and regular domain. Take a look at the following bilinear form: \begin{align} (u,v)_{L^2(\Omega)} \mapsto (\nabla u, \nabla v)_{L^2(\Omega)} \end{align} In ...
milaking's user avatar
  • 183
5 votes
1 answer
158 views

Is the bilinear form $A(\cdot,\cdot)$ $l_{2}$-elliptic or coercive?

Consider the space $l_{2} = \{ (x_{n})_{n\in \mathbb{N}} \subset \mathbb{R} : \sum_{n=1}^{\infty} |x_{n}|^{2} < \infty \}$. Now, consider the bilinear form $A(\cdot,\cdot):l_{2}\times l_{2}\to \...
tnt235711's user avatar
  • 391
0 votes
2 answers
47 views

Find a not coercive function $f: \mathbb{R}^2 \to \mathbb{R}$ which satisfies: $\lim_{|x_1| \to \infty} {f(x_1, \alpha x_1)} = \infty$

I am self-studying "Introduction to nonlinear optimization" by Amir Beck, and after studying chapter two which is called "Optimality Conditions for Unconstrained Optimization" I ...
john's user avatar
  • 136
3 votes
1 answer
168 views

Functions that are coercive along every line

A function $f:\mathbb{R}^n\to\mathbb{R}$ is called coercive if $$ \lim_{\|x\|\to\infty}f(x)=\infty. $$ To show that $f$ is coercive, we need to prove that for every sequence $\{x_n\}$ with $\|x_n\|\to\...
Kittayo's user avatar
  • 699
3 votes
1 answer
102 views

Coercivity of an integral operator in $L^2$-norm

Let us consider the integral operator $T:L^2(0,1)\to [0,\infty)$ such that for all $k\in L^2(0,1)$, $$ T(k)=\int_0^1 k_t^2e^{-2 \int_0^t k_s d s} d t. $$ Is the operator $T$ coercive in the $L^2$ ...
John's user avatar
  • 13.2k
0 votes
1 answer
29 views

Prove a certain functional is unbounded from above

Let $\lambda\in\mathbb{R}$, $2<p\leq 2^*=(2N)/(N-2)$. For $u \in H^{1}_0({\Omega})$ where $\Omega$ is a domain of $R^N$. Define: $$ \varphi(u) = \displaystyle\int_{\Omega} \frac{1}{2}|\nabla u|^2 +...
Kimura Leo's user avatar
0 votes
1 answer
224 views

Proof about Garding's Inequality for a bilinear form

Considering the following bilinear form : $$a(\phi,\psi)=\intop_{\varOmega}\left(\sum_{g=1}^{G}D_{g}\nabla\phi_{g}\cdot\nabla\psi_{g}+\sum_{g,h=1}^{G}\varSigma_{gh}\phi_{g}\psi_{h}\right)\,\mathord{d}\...
maru0032's user avatar
0 votes
1 answer
146 views

Least square function is coercive if and only if matrix is injective

Let $\phi \in \mathcal{M}_{M,N}(\mathbb{R})$, $y \in \mathbb{R}^M$ and $f(x)=\frac{1}{2}||\phi x - y||^2$. I want to show that $f$ is coercive if and only if $\phi$ is injective. This is what I have ...
R0M2's user avatar
  • 3
1 vote
0 answers
91 views

Determine whether or not the function is coercive

Determine whether or not the function is coercive. $$f(x_1,x_2)=e^{x_1^2}+e^{x_2^2}-x_1^{200}-x_2^{200}$$ Intuitively, I know that this function is coercive because, for large enough $x_i$, we have ...
James Anderson's user avatar
0 votes
0 answers
65 views

Determine whether a function is coercive or not.

$f(x,y) = \frac{x^4}{x^2+1} + |y| $ I want to see if the given function is coercive or not. My questions is that I'm stuck on how to transform the function into $(x^2+y^2)$ form. I need to convert ...
jun's user avatar
  • 661
2 votes
2 answers
161 views

Conditions for Kernel coercivity on a bilinear form

Let $H = L^2([0,1])$, $K \in L^{\infty}([0,1]^2)$. I have the following bilinear form: $$ a(u,v)=\int_{[0,1]^2}{K(x,y) u(x) v(y) dx dy} \;\;\; \forall u,v \in H $$ What conditions do I have to set for ...
DMDemon's user avatar
  • 21
1 vote
0 answers
84 views

Proving that g is coercive and strictly convex

Hello can someone help me understand this? Suppose we have $g(a) =a^4+a^3+2a^2+a+1$. Prove that g is coercive and strictly convex on R. Therefore, g has a unique global minimizer a*.
Blake's user avatar
  • 107
0 votes
0 answers
47 views

Coercivity of functional vs Boundness of the set: how to prove this?

Referring to my previous question: Two theorems of existence of a minimizer in a Hilbert space. I am trying to prove that the coercivity of the functional allow us to skip the hypothesis of boundness ...
C. Bishop's user avatar
  • 3,418
-1 votes
1 answer
60 views

Coercity definitions in $\mathbb{R}$

Let $f:\mathbb{R}\to\mathbb{R}\cup\{-\infty, +\infty\}$ be a function. My definition of coervit is that $f$ is coercive if $$ (D)\qquad \lim_{|x|\to +\infty} f(x) = +\infty.$$ I am trying to justify ...
C. Bishop's user avatar
  • 3,418
1 vote
2 answers
375 views

Relation between coercivity and boundness from below

Let $f:\mathbb{R}\to\mathbb{R}$ be a function such that $$f(x)\ge \| x\|.$$ It is clear that, if $\|x\|\to +\infty$, it follows that $f(x)\to +\infty$, so $f$ is coercive. However, the inequality $f(x)...
C. Bishop's user avatar
  • 3,418
0 votes
1 answer
108 views

Coerzive bilinear forms in Sobolev Space $H^1(\Omega)$.

Consider $\Omega=[0,1]^3$ and the bilinear form $B(u,v)=\int_{\Omega} (\nabla u)^T A \nabla v +uv \,dx, \; \; u,v \in H^1(\Omega).$ The matrix $A \in \mathbb{R}^{n \times n}$ is symmetric and positive ...
Mia torett's user avatar
1 vote
0 answers
49 views

How to test coercivity for implicit functions?

Let $F(y), F:\mathbb{R}\to\mathbb{R}$ be a function which depends implicitly from the variable $y$. My question is: it is possible to check the coercivity of $F(y)$? If yes, how? Some examples or ...
C. Bishop's user avatar
  • 3,418
1 vote
2 answers
513 views

Determine whether $f$ is coercive or not.

I have difficulty in determine whether those functions are coercive or not. This is part of an exercise in Amir Beck's book "Introduction to Nonlinear Optimization". a) $f(x_1,x_2)=2x_1^2-...
FactorY's user avatar
  • 774
1 vote
0 answers
41 views

Prove the coercivity of a function

I have some difficulties with the following problem: $$ f:\mathbb{R}^n \to \mathbb{R} $$ $$ f(x) = \sum_{i=1}^{n} (e^{x_i} - b_ix_i) $$ $$ b_i > 0, i = 1,..., n. $$ I want to prove that : $f$ is ...
Quanticat's user avatar
0 votes
1 answer
154 views

The existence of minimum two-variable function

How can we say a function has a global minimum without calculating any critical points or major calculations? I want to know about the right theorem and how do we reduce the theorem to a problem like ...
Priya's user avatar
  • 189
1 vote
0 answers
65 views

Coercivity in the kernel of a bilinear form

Let $\Omega=(a,b)$ and $a:H^1(\Omega)\times H^1(\Omega)\to \mathbb{R}$ to be the bilinear form defined by, $$a(\sigma,\tau)=\int_a^b\sigma \tau.$$ Consider also the bilinear form $b:H^1(\Omega)\times ...
Gerschgorin's user avatar
1 vote
0 answers
135 views

Conditions for Coercivity Condition

Let $\ell^1(\mathbb{R}^d)$ be the Banach space of sequences in $\mathbb{R}^d$ for which the norm $\|(x_n)_n\|=\sum_{n=1}^{\infty} \|x_n\|_2$ is finite. Let $f:\mathbb{R}^n\rightarrow [0,\infty)$ be ...
ABIM's user avatar
  • 6,695
0 votes
1 answer
625 views

Does the function is coercive?

How to prove the following function is coercive? \begin{align*} f(x,y)= x^{2n}+y^{2n}-nx^2+2nxy-ny^2 \end{align*} I found the definition of a coercive function and tried some problems and figure out ...
Priya's user avatar
  • 189
1 vote
1 answer
82 views

For a fixed $f \in L^1(\Omega)$, find a coercive function $h$ such that $h(f) \in L^1(\Omega)$.

I am looking for a proof of the following statement : Let $(\Omega,\mathcal{A},\mu)$ a probability space and $f \in L^1(\Omega)$. Show that there exists a coercive function $h$ ($h$ is positive, non-...
Velobos's user avatar
  • 2,190
0 votes
0 answers
246 views

Reference for coercive property

Problem: Let $X$ be a Banach space and $H$ be a Hilbert space. Let $A$ be a linear continuous operator from $X$ to $H$. Here, we say that $A$ is coercive if there exists real number $c>0$ such that ...
Leonard Neon's user avatar
  • 1,354
0 votes
0 answers
98 views

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 ...
vidyarthi's user avatar
  • 7,028
3 votes
1 answer
229 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(...
M B's user avatar
  • 597
1 vote
0 answers
150 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 ...
Bernhard Listing's user avatar
1 vote
2 answers
320 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^...
M B's user avatar
  • 597
0 votes
2 answers
341 views

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 ...
M B's user avatar
  • 597
1 vote
0 answers
116 views

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 ...
Vefhug's user avatar
  • 294
5 votes
1 answer
892 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 ...
itpdg's user avatar
  • 243
0 votes
0 answers
63 views

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 [...
ABIM's user avatar
  • 6,695
0 votes
2 answers
98 views

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 ...
jmp2626's user avatar
0 votes
0 answers
73 views

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 ...
Ali Na's user avatar
  • 31
0 votes
2 answers
184 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+...
Phoenix_10's user avatar
0 votes
2 answers
251 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 $...
Guilherme's user avatar
  • 1,591
0 votes
2 answers
136 views

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} >...
praly's user avatar
  • 65