Opposition versus randomness in soft computing techniques Shahryar Rahnamayan, Hamid R. Tizhoosh * , Magdy M.A. Salama Faculty of Engineering, University of Waterloo, Waterloo, Canada Received 18 October 2006; received in revised form 30 March 2007; accepted 28 July 2007 Available online 6 August 2007 Abstract For many soft computing methods, we need to generate random numbers to use either as initial estimates or during the learning and search process. Recently, results for evolutionary algorithms, reinforcement learning and neural networks have been reported which indicate that the simultaneous consideration of randomness and opposition is more advantageous than pure randomness. This new scheme, called opposition-based learning, has the apparent effect of accelerating soft computing algorithms. This paper mathematically and also experimentally proves this advantage and, as an application, applies that to accelerate differential evolution (DE). By taking advantage of random numbers and their opposites, the optimization, search or learning process in many soft computing techniques can be accelerated when there is no a priori knowledge about the solution. The mathematical proofs and the results of conducted experiments confirm each other. # 2007 Elsevier B.V. All rights reserved. Keywords: Soft computing; Opposition-based learning; Random numbers; Opposite numbers; Differential evolution 1. Introduction The footprints of the opposition concept can be observed in many areas around us. This concept has sometimes been labeled by different names. Opposite particles in physics, antonyms in languages, complement of an event in probability, antithetic variables in simulation, opposite proverbs in culture, absolute or relative complement in set theory, subject and object in philosophy of science, good and evil in animism, opposition parties in politics, theses and antitheses in dialectic, opposition day in parliaments, and dualism in religions and philosophies are just some examples to mention. Table 1 contains more instances and corresponding details. It seems that without using the opposition concept, the explanation of different entities around us is hard and maybe even impossible. In order to explain an entity or a situation we sometimes explain its opposite instead. In fact, opposition often manifests itself in a balance between completely different entities. For instance, the east, west, south, and north can not be defined alone. The same is valid for cold and hot and many other examples. Extreme opposites constitute our upper and lower boundaries. Imagination of the infinity is vague, but when we consider the limited, it then becomes more imaginable because its opposite is definable. Many machine intelligence or soft computing algorithms are inspired by different natural systems. Genetic algorithms, neural nets, reinforcement agents, and ant colonies are, to mention some examples, well established methodologies motivated by evolution, human nervous system, psychology and animal intelligence, respectively. The learning in natural contexts such as these is generally sluggish. Genetic changes, for instance, take generations to introduce a new direction in the biological development. Behavior adjustment based on evaluative feedback, such as reward and punishment, needs prolonged learning time as well. In many cases the learning begins at a random point. We, so to speak, begin from scratch and move toward an existing solution. The weights of a neural network are initialized randomly, the population initialization in evolutionary algo- rithms (e.g., GA, DE, and PSO) is performed randomly, and the action policy of reinforcement agents is initially based on randomness, to mention some examples. Generally, we deal with complex control problems [1,2]. The random guess, if not far away from the optimal solution, can result in a fast convergence. However, it is natural to state that if we begin with a random guess, which is very far away www.elsevier.com/locate/asoc Applied Soft Computing 8 (2008) 906–918 * Corresponding author. E-mail addresses: shahryar@pami.uwaterloo.ca (S. Rahnamayan), tizhoosh@uwaterloo.ca (H.R. Tizhoosh), m.salama@ece.uwaterloo.ca (M.M.A. Salama). 1568-4946/$ – see front matter # 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.asoc.2007.07.010