# Convex Optimization Problem (example)

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.

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

## 1 Answer

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.