Can I trust NMinimize

I have a function $$f(x,y)$$ for which Minimum[{f[ x, y], x > 0, y > 0}, {x, y}] does not find a solution. However, NMinimize[{f[ x, y], x > 0, y > 0}, {x, y}] reveals that the minimum is $$0$$ at $$(x_1,y_1)$$.

The first function tries to find a analytical solution, while the second only identifies numerical solutions.

The documentation says "NMinimize returns a machine-number solution". Does this mean that a numerical $$(x_1,y_1)$$ is found which minimizes $$f(x,y)$$. Does this (even anayltically) imply that $$f(x,y)\geq 0$$ for all $$x > 0, y > 0$$?

• The documentation of FindMinimum states that it only searches for a local minimum, so no: you cannot be sure that the global minimum is > 0. I recommend checking out NMinimize and the tutorial about optimization: reference.wolfram.com/language/tutorial/… – Sjoerd Smit Sep 23 at 9:22
• Thank you, I used NMinimize now and updated the question. – Paul Sep 23 at 9:45
• Without any further details about f I would say that in general you can never have any true guarantees about the minimum found by numerical methods. If f is a black-box function, there is always a possibility that it has a difficult-to-find global minimum that NMinimize didn't detect. The remark about machine-numbers simply means that the result is a double-precision floating point number. – Sjoerd Smit Sep 23 at 10:35
• You may take a look at Resolve and ForAll. If your function is not too complicated, you may be able to use those to prove that your function is always positive. For example: Resolve[ForAll[x, x > 0, x^2 > 0]]. – Sjoerd Smit Sep 23 at 10:52