I have a numerical function with 5 variables that takes about 2 minutes to evaluate each time. Trying to minimize the function with the "SimmulatedAnnealing" method takes several days of calculation.

I want Mathematica to print the best solution found so far, each time a new best solution is found while the algorithm is still running.

This code outputs the current values of the solution each step, but since the method doesn't always move towards a solution that is better than the previous one at each step, the last solution printed is not always the best solution so far

NMinimize[{StressError[p1, p2, p3, p4, p5], p1 > 0, p2 > 0, p3 > 0, p4 > 0, p5 > 0}, {{p1, 0.5, 2}, {p2, 0.5, 2}, {p3, 0.5, 2}, {p4, 
           0.5, 2}, {p5, 0.5, 2}}, Method -> "SimulatedAnnealing", 
         StepMonitor :> Print[{p1, p2, p3, p4, p5}]]

A way of doing this would be evaluating the function at each step, and only printing if it is the best solution found so far. Example:

BestSoFar = Infinity
NMinimize[{StressError[p1, p2, p3, p4, p5], p1 > 0, p2 > 0, p3 > 0, p4 > 0, p5 > 0}, {{p1, 0.5, 2}, {p2, 0.5, 2}, {p3, 0.5, 2}, {p4, 
           0.5, 2}, {p5, 0.5, 2}}, Method -> "SimulatedAnnealing", 
 StepMonitor :> (If[StressError[p1, p2, p3, p4, p5] < BestSoFar, 
    BestSoFar= StressError[p1, p2, p3, p4, p5]; Print[{p1, p2, p3, p4, p5}]])]

The problem with the previous code is that it would evaluate the function again at each step (which, considering that there are several hundreds of steps in the calculation, and evaluating the function takes about 2 minutes, would add a considerable amount of time to the calculation); the algorithm should already evaluate the function at each step, so evaluating it again shouldn't be needed.

Do you know a more efficient way to do the same?