If F is a very expensive scalar function, I would maybe try something like the following: make an interpolation of F (eg a spline? Or possibly even a Lagrange-type interpolation if your function is very smooth. Also if you have derivative information for F you might be able to use that with Hermite interpolation) at the points where you have evaluated it. At this point minimizing G under that approximation should be quick (if f1 and f2 are not too bad), so do that with any method you like. Then evaluate F at this minimum, improve your interpolation, and repeat until convergence.
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