I have a function f:R2 -> R2. The variables are x,y, the coordinates of a point on a plane, and the function assigns 2 "properties" to each point that can be calculated through more or less complex algorithms so that to each pair (x,y) of coordinates corresponds a unique pair (v,w) where I have called v=f1 and w = f2.

I'm trying to find a point (x,y) where the values of both u and v are as low as possibly achievable with the minimization algorithm, probably not absolute minimums. The way I do is I select a large number of randomly selected points and at each one of them test for the v and w values. What I haven't clearly decided is the most convenient approach in order to minimize both values. For only one value, clearly you accept a point if it has v (or w) less than the previous value, otherwise reject the point. For the 2 values, you may find e.g. a point with lower v than the previous point but maybe higher w. Therefore, should I try to minimize something like u*v? Or maybe Sqrt(u2 + v2)? Or some other quantity? The range of u is the interval [0,1] but w may take values from 0 to wmax where wmax is also a function of the position (x,y).