Pivot methods for global optimization
Abstract
A new algorithm is presented for the location of the global minimum of a multiple minima problem. It begins with a series of randomly placed probes in phase space, and then uses an iterative redistribution of the worst probes into better regions of phase space until a chosen convergence criterion is fulfilled. The method quickly converges, does not require derivatives, and is resistant to becoming trapped in local minima. Comparison of this algorithm with others using a standard test suite demonstrates that the number of function calls has been decreased conservatively by a factor of about three with the same degrees of accuracy. Two major variations of the method are presented, differing primarily in the method of choosing the probes that act as the basis for the new probes. The first variation, termed the lowest energy pivot method, ranks all probes by their energy and keeps the best probes. The probes being discarded select from those being kept as the basis for the new cycle. In the second variation, the nearest neighbor pivot method, all probes are paired with their nearest neighbor. The member of each pair with the higher energy is relocated in the vicinity of its neighbor. Both methods are tested against a standard test suite of functions to determine their relative efficiency, and the nearest neighbor pivot method is found to be the more efficient. A series of Lennard-Jones clusters is optimized with the nearest neighbor method, and a scaling law is found for cpu time versus the number of particles in the system. The two methods are then compared more explicitly, and finally a study in the use of the pivot method for solving the Schroedinger equation is presented. The nearest neighbor method is found to be able to solve the ground state of the quantum harmonic oscillator from a pure random initialization of the wavefunction.
Degree
Ph.D.
Advisors
Kais, Purdue University.
Subject Area
Chemistry|Mathematics|Physics
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