Show that the following problem is a convex optimization problem.

$f(x,y,z)=2x^2-y+z^2 \rightarrow min! $



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.

  • 2
    $\begingroup$ It sounds good. $\endgroup$ – A.Γ. Jun 2 '19 at 20:55

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.


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