One of the historically interesting advantages of Minuit is that it was probably the first minimization program to offer the user a choice of several minimization algorithms. This could be taken as a reflection of the fact that none of the algorithms known at that time were good enough to be universal, so users were encouraged to find the one that worked best for them. Since then, algorithms have improved considerably, but Minuit still offers several, mostly so that old users will not feel cheated, but also to help the occasional user who does manage to defeat the best algorithms. Minuit currently offers five commands which can be used to find a smaller function value, in addition to a few others, like MINOS and IMPROVE, which will retain a smaller function value if they stumble on one unexpectedly (or, in the case of IMPROVE, hopefully). The commands which can be used to minimize are:
This is the best minimizer for nearly all functions. It is a variable-metric method with inexact line search, a stable metric updating scheme, and checks for positive-definiteness. It will run faster if you SET STRATEGY 0 and will be more reliable if you SET STRATEGY 2 (although the latter option may not help much). Its main weakness is that it depends heavily on knowledge of the first derivatives, and fails miserably if they are very inaccurate. If first derivatives are a problem, they can be calculated analytically inside FCN (see elsewhere in this writeup) or if this is not feasible, the user can try to improve the accuracy of Minuit's numerical approximation by adjusting values using the SET EPS and/or SET STRATEGY commands (see Floating Point Precision and SET STRATEGY).
This is equivalent to MIGRAD, except that if MIGRAD fails, it reverts to SIMPLEX and then calls MIGRAD again. This is what the old MIGRAD command used to do, but it was removed from the MIGRAD command so that users would have a choice, and because it is seldom of any use to call SIMPLEX when MIGRAD has failed (there are of course exceptions).
This is not intended to minimize, and just scans the function, one parameter at a time. It does however retain the best value after each scan, so it does some sort of highly primitive minimization.
We have retained this Monte Carlo search mainly for sentimental reasons, even though the limited experience with it is less than spectacular. The method now incorporates a Metropolis algorithm which always moves the search region to be centred at a new minimum, and has probability e(- F/Fmin) of moving the search region to a higher point with function value F. This gives it the theoretical ability to jump through function barriers like a multidimensional quantum mechanical tunneler in search of isolated minima, but it is widely believed by at least half of the authors of Minuit that this is unlikely to work in practice (counterexamples are welcome) since it seems to depend critically on choosing the right average step size for the random jumps, and if you knew that, you wouldn't need Minuit.
This genuine multidimensional minimization routine is usually much slower than MIGRAD, but it does not use first derivatives, so it should not be so sensitive to the precision of the FCN calculations, and is even rather robust with respect to gross fluctuations in the function value. However, it gives no reliable information about parameter errors, no information whatsoever about parameter correlations, and worst of all cannot be expected to converge accurately to the minimum in a finite time. Its estimate of EDM is largely fantasy, so it would not even know if it did converge.