# Can the simplex method be used for general monotonically increasing objective functions?

The simplex method relies on the fact that if you have a linear objective function and linear constraints, the optima must lie on one of the vertices of the simplex formed by the constraints. Let's say now that the objective function is not linear, but monotonically increasing in all the input variables. Let's take an example:

Minimize: $$e^x+e^y$$

s.t. $$A\vec{x} \leq \vec{b}$$

Where $$\vec{x} = (x,y)$$ and $$A$$ is a $$n \times 2$$ matrix.

Can we not say that the optima must still lie on one of the vertices?

No, we can't. Consider minimizing $$e^x + e^y$$ subject to $$x+y=1$$, $$x, y \ge 0$$. The minimum is at $$(1/2,1/2)$$ which is not a vertex.