Gradient Tabu Search SVETLANA STEPANENKO, BERND ENGELS Institut fu¨r Organische Chemie, Universita¨t Wu¨rzburg, Am Hubland, D-97070 Wu¨rzburg, Germany Received 14 November 2005; Revised 22 May 2006; Accepted 23 May 2006 DOI 10.1002/jcc.20564 Published online 21 December 2006 in Wiley InterScience (www.interscience.wiley.com). Abstract: This paper presents a modification of the tabu search called gradient tabu search (GTS). It uses analyti- cal 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 intro- duced 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 opti- mization routines for force field parameters or conformational searches for large molecules. q 2006 Wiley Periodicals, Inc. J Comput Chem 28: 601–611, 2007 Key words: nonlinear global optimization; metaheuristic; tabu search; descent methods; mildest ascent strategy Introduction Formally, a general optimization problem can be stated as min Fðx i Þ subject to x i 2 D (1) where F(x i ) is continuous objective function with D ¼ {x i :lower bound  x i  upper bound}. Depending on the problem x i are binary numbers, integer, or continuous variables. Such optimiza- tion problems are very important for various fields from econo- metrics to bioinformatics and occurs also quite often in compu- tational chemistry. A well-known example from computational chemistry is the conformational search for a large and very flexi- ble molecule. This task comprises all ingredients of an optimiza- tion problem. It possesses a large number of possible solutions with similar quality and the handling of the problem necessitates the scan over a large space. Finally, as in many optimization problems of econometrics, it remains uncertain if the optimal so- lution was really found. In econometrics, there is recently a growing interest in solv- ing optimization problems using metaheuristics. 1–3 The tabu search (TS) is such a metaheuristic. To find the global mini- mum, the TS either selects the move leading to the largest decrease of F(x i ) or it follows the mildest ascent to escape a local minimum. This strategy is best described by the terms ‘‘steepest descent, mildest ascent’’. To select the right move, the whole neighborhood of the current trial solution is investigated. Reverse moves and cycles are avoided by the use of a tabu list (TL), where the moves previously done are memorized. In many applications in a wide variety of fields, the TS yielded solutions whose quality significantly surpasses that obtained by methods previously applied. 4–6 The TS is often applied in econometrics 3,7 but is less known in chemistry. A recent application stems from Baumann et al. who used it to identify good variable subsets within the frame- work of QSAR techniques. 8 Within such applications TS is extremely efficient. One reason for the efficiency is that the vari- ables x i are binary in this problem. Because of its high efficiency, the TS could be a method of choice for optimization problems in chemistry. But, in contrast to most existing applications of the TS, such problems represent continuous optimization problems for which the definition of the neighborhood is more complicated. In a simple discretization of the problem, for instance, the level of discretization determines the accuracy of the solution. This leads to problems if some pa- rameter x i must be much more accurate than others or if the nec- essary accuracy is not known in advance. Several attempts have been made to deal with continuous optimization problems. 9–20 Battiti and Tecchiolli introduced the continuous reactive TS, 10 which is a generalization of the reactive TS developed by the same authors. The continuous reactive TS method tries to locate most promising boxes. In a second step, refined solutions are Contract/grant sponsor: Deutsche Forschungsgesellschaft; contract/grant number: EN197/10-1 Contract/grant sponsor: SFB630; contract/grant number: TP C3 Correspondence to: B. Engels; e-mail: bernd@chemie.uni-wuerzburg.de q 2006 Wiley Periodicals, Inc.