Show that the following problem is a convex optimization problem.
$f(x,y,z)=2x^2-y+z^2 \rightarrow min! $
$g_1(x,y,z)=y+x\le1$,
$g_2(x,y,z)=z-y\le1$
Convex optimization problem if:
(1) $f(x)\rightarrow min!$
(2) $f(x)$ is convex
(3) all constraints $g_i$ are convex, $ i=1,..,m$
My idea is to calculate the Hessian matrix of the objective function and constraints and check if the matrix is positive (semi) definite, which would imply (strictly) convex function.
$H_f(x,y,z)=\begin{pmatrix} 4&0&0\\0&0&0\\0&0&2\end{pmatrix}$
This is a positvie semidefinite matrix (Eigenvalues $\geq0$)
$\Rightarrow f(x)$ is convex
The Hessian matrix of $g_1$ and $g_2$ is a zero matrix which is both convex and concave.
So the problem is a convex optimization problem.
Is my computation/conclusion correct?
Thank you in advance.
Your idea is good, since checking via Hessian is a simple technique in case it's easy to calculate it. However, with more complex functions the Hessian can be hard to compute, so it's a good practice to prove convexity using "non derivative" methods.
In your case, ask yourself the following (works for the target function as well as the constraints): What are the atoms ("building blocks") of the function? Do you see a simple decomposition to the function? For example, is the function a summation of convex functions? Recall that summing convex functions maintains convexity.