Gradient tabu search

Abstract

This paper presents a modification of the tabu search called gradient tabu search (GTS). It uses analytical gradients for a fast minimization to the next local minimum and analytical diagonal elements of the Hessian to escape local minima. For an efficient blocking of already visited areas tabu regions and tabu directions are introduced into the tabu list (TL). Trials with various well-known test functions indicate that the GTS is a very promising approach to determine local and global minima of differentiable functions. Possible application areas could be optimization routines for force field parameters or conformational searches for large molecules.

New Tabu Search based global optimization methods outline of algorithms and study of efficiency

“…A similar behavior is found for the parameter a (0.0 \ a 1.0), which determines the size of the cone of the TD. 35 Similar to large TR too large cones restrict the flexibility of the search. The search must take detours to the global minimum since more direct ways are set tabu by accident.…”

“…41 The eigenvector-following techniques, 42 which exploit the eigenvalues and associated eigenvectors of the Hessian are best suited to solve this problem but the necessary effort increases with NDIM((NDIM 11)/2 (NDIM being the dimensionality of the problem) and additionally an NDIM(NDIM matrix has to be diagonalized. Therefore to reduce the cost, the previously developed GTS 35 only exploits the diagonal elements of the Hessian and estimates step size and direction according to a linear ranking procedure. For the GOTS and the TSPA only functional evaluations should be used to estimate step size and directions.…”

“…In the present approaches this is no longer sufficient since the number of steps decreases while the step sizes increase. To overcome the resulting problems TR and TD are used that were already successfully employed in the GTS method 35 and which related to ideas introduced by F. Glover. 43 The TR is defined as follows:…”

“…Their effects are studied in Table 2 using the Rastrigin and the Ackley function as examples and illustrated in Figures 3-6. The values for the GTS, which were taken from our previous publication, 35 are given for an easier comparison. To estimate the influence on the convergence, the tables give how often the function, its gradient, and the diagonal elements of its Hessian had to be computed until the global minimum was reached.…”

“…35 It uses analytical gradients for a fast minimization to the next local minimum and the diagonal elements of the analytical Hessian to escape local minima. For the minimization a combination of the Steepest Descent and the Quasi-Newton method is used.…”