Optimization without complex numbers

Question

We need to find a minimum of functions:

(1+$\sqrt x$)$^2$+$y^2$

Due to the fact that the function has a square root, the optimization algorithm goes into the area of complex numbers. How to make so that to receive decisions only in real numbers?

Answer 1

We have a sum of two squares $(1+\sqrt{x})^2+y^2$. This sum is minimal if both terms are minimal. Hence $x = 0$ and $y=0$ will minimize the expression. The function is unbounded from the top. This means that we will not be able to give a maximal value to this equation.

Answer 2

Since we have $$\frac{\partial f(x,y)}{\partial x}=2(1+\sqrt{x})\frac{1}{2\sqrt{x}}\neq 0$$ there are no extreme points.

Answer 3

You either use constrained optimization (optimize over $x\geq 0$) or if your solver does not support constrained optimization you can use some change of variables, such as defining $x=a^2$ and optimize over $a$ and $y$ instead.