Proving convexity of a bivariate function

Question

Let

\begin{align} f(x,y)=\left(\frac{x}{y}+\sqrt{2-\frac{1-x^2}{y^2}}\right)^2, \end{align}

where $0 \le x\le 1$, $-1 \le y \le0$, $x^2 + y^2 \le 1$ and $x^2 + 2y^2 \ge 1$. Is $f(x,y)$ convex with respect to $x$ and $y$ on its domain?

Answer 1

The feasible region of all the constraints in the model is the following red-shaded region:

In order for a model to be convex, both the objective function and the feasible region of the model must be convex. A set is said to be convex if the line segment connecting any two points $x_1$ and $x_2$ in the set also belongs in the set:

$$\lambda x_1 + (1-\lambda)x_2\qquad \forall\lambda\in[0,1]$$

Thus, we can test a model’s convexity for any two points $x_1$ and $x_2$ that satisfies the mentioned property like so:

Thus, since the feasible region is not convex, then the entire model is not convex.