coulomb excitation data analysis codes; gosia 2007 - Physics and ...

10 MINIMIZATION BY SIMULATED ANNEALINGR.W. IbbotsonThe problem of minimizing an arbitrary function can be approached computationally using a wide varietyif techniques; some approaches are better suited to certain problems than others. A common feature of mostof these techniques is that steps are taken iteratively in the variable space which reduce the “cost” function(whatever function is being minimized). For a one-dimensional minimization problem with cost functionC(x), a step ∆x is chosen so that C(x + ∆x) < C(x) and repeated steps are taken in x until the value ofx for which C(x) is a minimum is found. This approach will be referred to here as “strict minimization”,since the cost function is never allowed to increase as the minimization procedure is executed. This is alogical and efficient approach, but certain problems will not be solved properly by this method. Specifically,if the function being minimized has several local minima, only one of which is the true global minimum ofthe function, strict minimization will only converge to the global minimum if the starting point is chosensomewhere in the “well” of the global minimum. Any algorithm which only takes steps “downhill” runs therisk of not converging to the global minimum, unless it is started at a point sufficiently close to this globalminimum. For example, if a strict minimization procedure is used to find the minimum of the functionshown in Figure 12 and the starting point is chosen in region II, the procedure will converge to a value ofx ≈ +1.4 (C(x) ≈ 8.8), rather than the global minimum value at x ≈−2.0 (C(x) ≈ 2.2). The generallocation of the global minimum is often known in physics-related problems from a consideration of thephysical properties of the system in question. In large Coulomb-excitation problems, however, the fit qualitymay be sufficiently dependent on certain unknown quantities such as the sign of a static E2 matrix elementthat an alternative to strict minimization should be considered. The method of Simulated Annealing (SA)is an attempt to overcome this shortcoming of standard minimization techniques by allowing some stepsuphill, in a controlled fashion.Figure 12: An example of a function which can cause difficulties with strict minimization procedures. Aprocedure using a starting value of x in region “II” (x >0) will not converge to the global minimum atx ≈−2.0.10.1 The Development of Simulated AnnealingIn 1953, Metropolis et al. showed that it is possible to reproduce the thermodynamic properties of a systemof particles by what amounts to a Monte-Carlo integration over configuration space [Me53]. The methodproposed by Metropolis involves beginning with these particles (or molecules) in a lattice, and choosing new148