[donkeybeer]

Its saying a very simple thing with somewhat convoluted language. First of all for full generality they are treating functions of Rn -> R (if the output is also a vector then it becomes a different type of problem, I believe its called multimodal optimisation).

And the f1, f2, ... fm are simply the (inequality) constraints that all candidate solutions must satisfy.

For example, you might want to maximize the volume of something you want to build from sheet metal, then f0 could be the expression for the volume of body and the constraint could be one inequality ie area(x) <= your_maximum_budget_for_sheet_metal etc