$$\beta$$ β -Hill climbing: an exploratory local search Mohammed Azmi Al-Betar Neural Computing and Applications ISSN 0941-0643 Neural Comput & Applic DOI 10.1007/s00521-016-2328-2 1 23 Your article is protected by copyright and all rights are held exclusively by The Natural Computing Applications Forum. This e-offprint is for personal use only and shall not be self- archived in electronic repositories. If you wish to self-archive your article, please use the accepted manuscript version for posting on your own website. You may further deposit the accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publication or later and provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's website. The link must be accompanied by the following text: "The final publication is available at link.springer.com”. 1 23 Author's personal copy Neural Comput & Applic DOI 10.1007/s00521-016-2328-2 ORIGINAL ARTICLE b-Hill climbing: an exploratory local search Mohammed Azmi Al-Betar1 Received: 11 December 2015 / Accepted: 13 April 2016 Ó The Natural Computing Applications Forum 2016 Abstract Hill climbing method is an optimization tech- due to the value range of their variables: continuous and nique that is able to build a search trajectory in the search discrete. In general, the continuous optimization problem is space until reaching the local optima. It only accepts the formulated as follows: uphill movement which leads it to easily get stuck in local minffðxÞjx 2 Xg; optima. Several extensions to hill climbing have been proposed to overcome such problem such as Simulated where f ðxÞ is the objective function; x ¼fxi j i ¼ 1; ...; Ng Annealing, Tabu Search. In this paper, an extension version is the set of decision variables. X ¼fXi j i ¼ 1; ...; Ng is of hill climbing method has been proposed and called b-hill the possible value range for each decision variable, where climbing. A stochastic operator called b-operator is utilized Xi 2½LBi; UBi, and LBi and UBi are the lower and upper in hill climbing to control the balance between the explo- bounds for the decision variable xi , respectively, and N is ration and exploitation during the search. The proposed the number of decision variables. method has been evaluated using IEEE-CEC2005 global Definition 1 The search space S¼fx ¼fðx ; v Þji 2 optimization functions. The results show that the proposed i i 1 ÁÁÁN ^ v 2 X gg is the set of all possible solutions for a method is a very efficient enhancement to the hill climbing i i given problem with a dimension N, where each member is providing powerful results when it compares with other a solution of a given problem. In continuous optimization, advanced methods using the same global optimization S RN . functions. The main purpose of solving optimization problems is to Keywords Hill climbing Á Metaheuristics Á Global find a solution xà 2S with a minimum value of the optimization Á Local search methods Á Operations research objective function such that f ðxÃÞ \ f ðxjÞ; 8xj 2S.This solution (e.g., xÃ) is called a global optimal solution. The emergence of metaheuristics for solving difficult 1 Introduction global optimization problems has been one of the most notable accomplishments over the last two decades [16, Optimization is a field of operations research concerned 18]. Metaheuristic-based method is an iterative improve- with finding a best configuration of problem variables to ment process that uses its operators and combines intelli- optimize its objective function f ðxÞ. The optimization gently the problem specific knowledge for exploring and problems are conventionally categorized into two groups exploiting the search space (S) in order to reach a good- quality solution [3, 22, 31]. The key research issue in applying metaheuristic-based & Mohammed Azmi Al-Betar methods to any global optimization problem is to make an [email protected] attempt to strike a balance between exploration and 1 Department of Information Technology, Al-Huson University exploitation during the search. It should be emphasized that College, Al-Balqa Applied University, during the exploration stage the search is encouraged to P.O. Box 50, Al-Huson, Irbid, Jordan 123 Author's personal copy Neural Comput & Applic explore the not-yet-visited search space regions if need be. 4. Side walk Explore NðxÞ and select the first solution x0 During the exploitation stage, however, the search con- that has the same value of objective function as x such centrates on the already-visited search space regions [29]. as f ðx0Þ¼f ðxÞ. Note that the side walk improvement It is widely accepted in the literature that the metaheuristic strategy is very useful to enable the local search-based methods can be classified into two classes: (1) local search- method in jumping from search space region to another based and (2) population-based methods. However, some in the same level with hope to empower it to escape researchers have classified the metaheuristic into nature-in- local optima efficiently. spired versus non-nature-inspired methods; single versus multiobjective algorithms; one versus various neighborhood Definition 3 The local optimal solution with respect to structures, and memory or memory-less [8]. the neighborhood structure N is a solution x such that Population-based methods normally begin with a pop- 8x 2 bs : f ðbsÞ Àf ðxÞ. The solution x is a local optimal ulation of random solutions. They iteratively recombine the solution if f ðxÞ\f ðxÞ; 8ðxÞ2bs. properties of current population to come up with a new Hill climbing (HC) is the simplest form of local search- population [6]. Besides, they normally use a random based methods where the search nature only accepts the method to diversify the search when necessary. The main ‘downhill’1 move. In this method, the exploitation concept advantage of population-based algorithms is that they are is activated where the search is improved using the accu- able to widely scan many search space regions at the same mulative search without any stochastic component. time. However, they are disadvantaged in their incapacity Therefore, HC is normally used to empower other EAs in to provide a precise local optimal solution for each visited terms of exploitation capability (see memetic algorithm region. That is why the problem of slow convergence might [3]). The main shortcoming with classical version of HC is occur. that it can easily get stuck in local optima [8]. With this Local search-based methods, the key issue of this paper, dilemma in mind, several extensions to HC have been initiated with a single provisional solution, x. That solution introduced to overcome such problem. The common idea iteratively undergoes changes using a neighborhood of these extensions lies in utilizing an intelligent stochastic structure (i.e., NðxÞ) until a locally optimized solution, operator that may help in avoiding the local optima trap. which is usually in the same region of the search space as The most popular extensions are Simulated Annealing (SA) the initial solution, is obtained. Therefore, the search space [17], Tabu Search (TS) [12], Greedy Randomize Adaptive S can be seen as a set of regions bs where bs S and Search Procedure (GRASP) [10], Variable Neighborhood [bs ¼ S. Local search-based methods are able to fine-tune Search (VNS) [14], and Iterated Local Search (ILS) [20]. the search space region to which they converge and find a The main objective of this paper is to produce an extension precise local optimal solution. However, they go through a variation of HC that utilizes an intelligent stochastic operator trajectory without doing a wider scan of the entire search able to avoid the local minima for global optimization. The space. proposed method is called b-hill climbing. In the improvement Definition 2 A neighborhood structure is a function N : loop of b-hill climbing, two consecutive operators have been S 2S which assigns to each x 2 S a set of neighbors triggered: neighborhood navigation (N operator) and b oper- NðxÞ S. Note that NðxÞ is called the neighborhood of ator. In N operator, a single move to the neighboring solution x. using a random walk acceptance rule is adopted. Thereafter, b operator constructs a new solution selecting the majority of the Function N has different operations to calculate the new values from the current solution while other values are neighboring solution for the solution x. Any local search- selected randomly from the range controlled by parameter b based method can carry out one or more operations to that tunes the balance between exploration and exploitation move from the current solution x to the neighboring solu- according to the nature of the search space. This idea stems 0 tion x 2NðxÞ as follows: from the uniform mutation operator of GA. 1. First improvement Explore the NðxÞ and select the The performance of b-hill climbing has been intensively first solution x0 better than x such as f ðx0Þ\f ðxÞ. evaluated using IEEE-CEC2005 [27] functions of total number 2. Best improvement Explore all solutions in NðxÞ and of 25 functions as used in [1, 2, 4, 5, 7, 11, 19, 21, 23–26, 30]. select the solution with the lowest objective function. The comparative results show the efficiency of the proposed Formally, select x0 2NðxÞ where f ðx0Þ\f ðx00Þ, method. 8x00 2NðxÞ. The rest of the paper is organized as follows: The related 3. Random walk Select randomly x0 2NðxÞ. Note that work of hill climbing and its extensions is overviewed in this strategy does not check whether the value of f ðx0Þ is lower or higher than f ðxÞ.