I have a strategy f that takes parameters x,y (for x,y taking values in integer ranges). I get two grids (of returns and volatility values) from computing f(xi,yi) for integer ranges x1 <= xi <= x2 and y1 <= yi <= y2.

My question: what is the standard optimization techniques of determining areas in this discrete grid that are robust with respect to both returns and volatility values? The simplest way that comes to mind is given by this pseudo-code

threshold_ret = ...;   // minimum required threshold
threshold_vol = ...;   // maximum allowed volatility
threshold_rr = ...;    // minimum required returns stability radius 
threshold_rv = ...;    // minimum required volatility stability radius
candidates = empty list;
foreach (xi,yi) do
     ret = returns at xi,yi;
     vol = volatility at xi,yi;
     rr = maximum stability radius for returns grid at xi,yi;
     rv = maximum stability radius for volatility grid at xi,yi;
     if (ret > threshold_ret) and (vol < threshold_vol) and
        (rr > threshold_rr) and (rv > threshold_rv) then
            add (xi,yi,rr,rv) to candidates list;
     end if
end for

Also, assume I have two acceptable regions R1 and R2 from the candidates list. For R1, let stability radius for returns grid be rr1, and for volatility grid be vr1. similarly, let these quantities be rr2 and vr2 for region R2. Given that there are different cases such as

(rr1 > rr2) and (vr1 < vr2)
(rr1 < rr2) and (vr1 < vr2)
(rr1 > rr2) and (vr1 > vr2)
(rr1 > rr2) and (vr1 > vr2)

what is the reasonable metric to use to select from these regions? What if there are many such regions?



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