Pivot Methods for Global Optimizations

The pivot method for global optimization is based on pivot moves through phase space. We begin by assuming a continuous phase space S. Within phase space is defined a real function f(vec{x}). Our goal is to determine the global minimum value of this function within the defined phase space. 

Briefly, we start with N=2m initial probes, of which m probes will act as the pivot probes, and the remaining m probes will be relocated. A local selection of the m pivot probes begins with a search at each probe for its nearest neighbor, based on the distance of the probes. 

Once we have paired the probes, the probe with the lower value for the function f(vec{x}) is defined as the pivot probe, the other probe being the probe that will be relocated. For each pivot probe with parameter values vec{x}{B,i}, we explore phase space by placing the probe to be relocated near the pivot probe by changing its parameters according to a general q-distribution based on the Tsallis entropy for the placement of the probes near the pivot probes. 

The method quickly converges, does not require derivatives, and is resistant to becoming trapped in local minima. 

• "A New Approach to Global Minimization", A. Stanton, R. Bleil and S. Kais, J. Comp. Chem. 18, 594-599 (1997).

• "Pivot Method for Global Optimization", P. Serra. A.F. Stanton and S. Kais, Phys. Rev. E 55, 1162-1165 (1997).

• "Comparison Study of Pivot Methods for Global Optimization", P. Serra, A.F. Stanton, S. Kais and R. Bleil, J. Chem. Phys. 106, 7170-7177 (1997).

• "Symmetry Breaking and Stability of Binary Clusters", Pablo Serra and Sabre Kais, Chem. Phys. Letters 275, 211-214 (1997).

• "Pivot methods for global optimization: A study of water clusters (H₂0)_N with 2 ≤ N ≤ 33", Pablo Nigra and Sabre Kais, Chem. Phys. Letters, 305, 433-438 (1999).

• "Study of phase changes in water octamer using the parallel tempering and multihistogram methods", Pablo Nigra, Marcelo Carignano and Sabre Kais, J. Chem. Phys. 115, 2621-2628 (2001).