# 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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### 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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### 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 ...
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 $\... 0answers 44 views ### Showing that invertibility implies coercivity I'm trying to proof the following: Let$f$be a continous sesquilinear form on a Hilbertspace$H$and let$A: H \to H$be its dedicated Operator such that $$f(u,v) = \langle u, Av \rangle_H \;\;\; \... 2answers 94 views ### Coercivity - Weak Poisson's equation Given the weak formulation of the Poisson equation, i.e. For given source function f\in H^{-1}(\Omega) find u \in H_0^1(\Omega) such that$$\int_{\Omega}\nabla u \cdot \nabla v \, dx= \int_{\Omega}... 1answer 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: ... 0answers 22 views ### 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? 2answers 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(... 1answer 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 \... 0answers 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. 1answer 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\... 0answers 39 views ### 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{... 1answer 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 ... 0answers 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. ... 0answers 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 ... 1answer 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 ... 0answers 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.$$How do I show ... 2answers 691 views ### Check if function is coercive I am trying to check if the function U(x,y)=6\ln x+\ln y is coercive. I know, that i need to check if \lim_{\vert (x,y) \vert \rightarrow \infty} U(x,y) = +\infty, and so far I have \lim_{(x,y) \... 2answers 273 views ### Linear function can be coercive? we know that \,f:\mathbb R^n \to \mathbb R\, is coercive if for any \, x\in \,\mathbb R^n\, %there exists a constant \,c\in \mathbb R\, such that %\big\langle \,f\left(x\right),\,x\,\big\... 1answer 446 views ### Why do we require coercivity for showing uniqueness of PDEs instead of injectivity? From what I gather, when working with PDEs we can show the uniqueness of a solution to certain PDE's by requiring that a bilinear form is continuous and coercive. The coercivity of the bilinear form ... 1answer 218 views ### Showing Coercive Condition in Lax-Milgram As a part of a Lax-Milgram problem, I need to show a particular operator is coercive, but I am having trouble finding the appropriate bounds. I need to show that there is some \alpha >0 such ... 0answers 49 views ### How can we show that (u,v)\mapsto\int u\cdot v+(u^0\cdot u)\cdot v+\nabla u:\nabla v is coercive on H_0^1? Let d\in\left\{1,\ldots,4\right\} \lambda denote the Lebesgue measure on \mathbb R^d \Lambda\subseteq\mathbb R^d be bounded and open \Delta t>0 \nu>0 Note that the norm induced by ... 0answers 194 views ### In Calculus of Variation: Problem applying variational principle theorem Let f:\mathbb R^m \rightarrow [0,+\infty)\; be a smooth function that vanishes on a finite set A\; where \vert A \vert\; \ge 2 and the maps v:(l^{-},l^{+}) \rightarrow \mathbb R^m\; ... 1answer 307 views ### Range of gradient map for coercive function I am given a differentiable (and therefore continuous) function f: \mathbf{E} \to \mathbb{R} which satisfies the following growth condition:$$ \lim_{||x|| \to \infty} \frac{f(x)}{||x||} \to +\... 1answer 485 views ### A question from definition of coercive function. I have one doubt about the definition of Coercive function: Definition: if for a functionf: \mathbb{R}^n \rightarrow \mathbb{R}$: $$\lim_{\|x\| \to +\infty} f(x) = +\infty,$$ then$f$is a ... 1answer 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$is ... 1answer 805 views ### coercive implies bounded, variational principle I have a question about the proof of the variational principle, see below. Any help is much appreciated! How does it follow from coercivity that$(x_k)_k$is bounded? Why is$\alpha_0 > - \infty$? ... 2answers 511 views ### How to show that$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}...
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 $\iff \forall \alpha\in\mathbb{R}.\{x|f(x)\leq\alpha\}$ ...