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) $f(x_1,x_2)=2x_1^2-8x_1x_2+x_2^2$.
b) $f(x_1,x_2)=4x_1^2+2x_1x_2+2x^2_2$.
C) $f(x_1,x_2)=x_1^3+x_2^3+x_3^3$.
I'm trying to estimate $f(x,y)$ to the norm $\|x\|$.
For the definition of coercice function, the following is known
Let $f : \mathbb{R}^n \to \mathbb{R}$ is continuous function. The function f is called coercive if $\lim\limits_{\|x\| \to \infty} f(x) = \infty $.
Any help would be highly appreciated.
Best Answer
So coercice function are functions that go to infinity regardless of the direction. Therefore $(a)$ is not coercice function because along the line $(x_1,x_2)=(t,t(4+\sqrt{14}))$ is constantly zero. Although $||(x_1,x_2)||\to\infty$ as $t\to \infty$. Next $(b)$ function is coercice. Its graph is a paraboloid which grows in every direction. In point $(c)$ you can see that it is enough to consider function along the line $(x_1,x_2,x_3)=(t,-t,0)$.