Quad Search and Hybrid Genetic Algorithms

A bit climber using a Gray encoding is guaranteed to converge to a global optimum in fewer than evaluations on unimodal 1-D functions and on multi-dimensional sphere functions, where bits are used to encode the function domain. Exploiting these ideas, we have constructed an algorithm we call Quad Search. Quad Search converges to a local optimum on unimodal 1-D functions in not more than function evaluations. For unimodal 1-D and separable multi-dimensional functions, the result is the global optimum. We empirically assess the performance of steepest ascent local search, next ascent local search, and Quad Search. These algorithms are also compared with Evolutionary Strategies. Because of its rapid convergence time, we also use Quad Search to construct a hybrid genetic algorithm. The resulting algorithm is more effective than hybrid genetic algorithms using steepest ascent local search or the RBC next ascent local search algorithm. 1 Background and Motivation There are several advantages to using a reflected Gray code binary representation in conjunction with genetic algorithms and local search bit climbers. For both unimodal 1-dimensional functions and separable multi-dimensional unimodal functions, proofs show that steepest ascent local search is guaranteed to converge to a global optimum after steps when executed on an -bit local search neighborhood. The proofs assume that the functions are bijections, so that search cannot become stuck on plateaus of equally good solutions. Furthermore, the proofs demonstrate that there are only 4 neighbors per step (or move) that are critical to global convergence [1]. Using a reflected Gray code representation, it can be proven that a reduction from to dimensions is always possible after at most 2 moves (even if the search repeatedly “steps across” the optimum). Hence the global optimum of any 1-dimensional unimodal real-valued function can be reached in at most steps. Under steepest ascent local search, each step requires evaluations. Thus, it follows that steepest ascent local search actually requires up to evaluations to converge to the global optimum. In practice, next ascent local search methods are often much faster than steepest ascent. However, the worst case order of complexity of next ascent algorithms is exponential. This is due to the fact that next ascent algorithms in the worst case can take very small steps at each iteration. The current paper exploits the ideas behind these convergence proofs to construct a new algorithm which we call Quad Search. Quad Search is a specialized form of steepest ascent that operates on a reduced neighborhood. The algorithm cuts the search space into four quadrant and then systematically eliminates quadrants from further consideration. On unimodal functions Quad Search converges to the global optimum in at most evaluations, as opposed to evaluations for regular steepest ascent. For multi-dimensional functions, Quad Search converges to a point that is locally optimal in each dimension. The new algorithm is tested on different types of unimodal functions. Quad Search is compared to steepest ascent and Davis’ Random Bit Climber, RBC, which is a next ascent local search method [2]. Quad Search using a Gray code representation converges after at most evaluations on classes of functions such as sphere functions where convergence proofs have also been developed for Evolution Strategies and Evolutionary Programming [1]. Given the convergence results on unimodal functions, we compare Quad Search with Evolution Strategies. The representations used by Evolution Strategies are continuous while Quad Search uses a discrete representation, but in practice, both encodings can be high-precision. Finally, one of the most exciting uses of Quad Search is in combination with genetic algorithms. We combine Quad Search with the “Genitor” steady-state genetic algorithm. We also look at hybrid forms of Genitor that use steepest ascent and RBC. Genitor in combination with Quad Search proved to be the most effective hybrid genetic algorithm. 2 The Gray code representation Gray codes have two important properties. First, for any 1-D function and in each dimension of any multi-dimensional problem, adjacent neighbors in the real space are also adjacent neighbors in the Gray code hypercube graph [3]. Second, the standard Binary reflected Gray code folds the search space in each dimension. For each reference point , there is exactly one neighbor in the opposite half of the search space. For example, in a 1-D search space of points indexed from to , the points and are neighbors. In effect these neighbors are found by folding or reflecting the 1-D search space about the mid-point between and . There are also reflections between each quartile of the search space. Thus, starting at some point in the first quartile of the search space we can define a set of four points that will be critical to Quad Search. These points are found at locations given by the integer indices and . An inspection of these integer values shows that each of the points is in a different quadrant of the search space. This follows from the following observations. Point is in the first quadrant and . Furthermore, is the last point in the second quadrant and therefore must also be in the second quadrant. Similarly is the first point in the third quadrant and therefore must be in the third quadrant. Finally is the last point in the fourth quadrant and therefore is in the fourth quadrant. If we interpret each of these integers as Gray encoded bit strings, we can find each of these points using exclusive-or (denoted by ) as follows: Fig. 1. The neighbor structure across and within quadrants. X N BB BB N BB N N BB