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Tuning a Simulated Annealing Metaheuristic for Cross-Domain Search

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Tuning a Simulated Annealing Metaheuristic for Cross-Domain Search Tuning a Simulated Annealing Metaheuristic for Cross-domain Search Warren G. Jackson, Ender Ozcan,¨ Robert I. John ASAP Research Group School of Computer Science University of Nottingham Jubilee Campus, Wollaton Road, Nottingham, NG8 1BB, UK Email: fpsxwgj,Ender.Ozcan,[email protected] Abstract—Simulated Annealing is a well known local search well for cross-domain search, their studies emphasise on metaheuristic used for solving computationally hard optimisation high level heuristic selection methods with disregard or lack problems. Cross-domain search poses a higher level issue where a of reasoning in the choice of move acceptance strategies single solution method is used with minor, preferably no modifica- tion for solving characteristically different optimisation problems. used for accepting or rejecting neighbourhood moves. Single The performance of a metaheuristic is often dependant on its point based metaheuristics, also referred to as local search initial parameter settings, hence detecting the best configuration, metaheuristics [4] perform an iterative search based on a single i.e. parameter tuning is crucial, which becomes a further challenge solution. They are usually employed as the move acceptance for cross-domain search. In this paper, we investigate the cross- component within hyper-heuristics and have also been used domain search performance of Simulated Annealing via tuning for solving six problems, ranging from personnel scheduling to on their own for tackling the cross-domain search problem. It vehicle routing under a stochastic local search framework. The is this component that is the focus of this study. empirical results show that Simulated Annealing is extremely The performance of heuristic optimisation methods is af- sensitive to the initial parameter settings leading to sub-standard fected by their parameter settings. Parameter tuning to find performance when used as a single solution method for cross- the best settings is therefore required to achieve the best domain search. Moreover, we demonstrate that cross-domain parameter tuning is inferior to domain-level tuning highlighting performance of such methods. The parameter setting prob- the requirements for adaptive parameter configurations when lem is itself another example of an computationally difficult dealing with cross-domain search. problem requiring an exponential number of evaluations in the number of parameters and parameter levels to find the best I. INTRODUCTION configuration. Surveys on parameter tuning (algorithm config- Many real-world combinatorial optimisation problems are uration) methods can be found in [5], [6]. Cross-domain search NP-hard [1]. Heuristic search methods are often preferred over however entails solving characteristically different problems. exhaustive search methods, such as dynamic programming This means that the optimal parameter settings are unlikely or math modelling in practise, considering that they may to be the same when solving different problems. A steady- struggle in finding good quality solutions to these problems, state memetic algorithm (SSMA) was tuned for cross-domain and even an acceptable solution in some cases in a prescribed search in [7]. Surprisingly, they showed that the best parameter time frame. Cross-domain search [2] is a high level prob- configuration obtained on a training set of problems and lem which entails solving multiple characteristically different on an extended set of problems remained the same for the combinatorial optimisation problems using a single solution SSMA. Some key details were however not explored; whether method. The cross-domain search problem is an interesting cross-domain tuning is a viable strategy for improving the one as the features of the problems being solved may be cross-domain performance, that is that it does not perform very different however the methods used to solve them do worse than per domain tuning, and whether the Taguchi so without modification. A single solution method capable of orthogonal array design of experiments method [8], as used solving the cross-domain search problem well is highly sought in the aforementioned paper, is a suitable strategy for tuning after as it can not only solve existing problems, but also new a cross-domain search method. and unseen problems without modification. Moreover, if this Simulated Annealing [9], abbreviated herein to SA, is an method can perform well across existing problems, then it is extremely popular metaheuristic used for solving optimisation a good candidate for solving new and unknown problems well problems. In this paper, we use SA as the move acceptance also. component of a stochastic local search metaheuristic, thus Hyper-heuristics [3] have been explored as solution methods eliminating any influence of heuristic selection methods, and for cross-domain search. Selection hyper-heuristics contain investigate different tuning strategies for improving its cross- two key components, a heuristic selection method, and move domain performance across six problem domains. acceptance strategy. Whilst hyper-heuristics appear to perform The rest of this paper is structured as follows. Section II pro- 978-1-5090-4601-0/17/$31.00 c 2017 IEEE vides an introduction to Simulated Annealing as a stochastic and its time based equivalent in Equation 2 where t0 and tfinal local search metaheuristic, and the methodology for this study are the parameters of the cooling schedule and are the initial is given in Section III. The results are discussed in Section IV, and final temperatures respectively. α is a value such that t0 × n and conclusions are provided in Section V. α = tfinal, i is the current iteration, and E is the current elapsed time scaled linearly in time between 0.0 (start) and II. SIMULATED ANNEALING 1.0 (end). Simulated Annealing (SA) is a stochastic local search meta- i heuristic used for solving many combinatorial optimisation ti = t0 × α (1) problems. The pseudo-code of SA as a single point based search algorithm is provided in Algorithm 1. The actual t E SA move acceptance component is depicted by Lines 9 t = t × final (2) E 0 t to 14 which is used as a part of other metaheuristics/hyper- 0 heuristics [10]. B. Temperature Settings SA works by accepting a candidate solution if its cost The setting of the initial temperature is important because is better than or equal to that of the current solution, or it directly affects the number of initial worse moves that if a random number in the range [0; 1] is less than some are accepted. SA relies on the acceptance of many worse probability P determined by the metropolis criterion [11]. moves initially in order to escape local optima in the search The metropolis criterion has two parameters, one being the landscape. If this setting is too low, then SA will become stuck. signed difference between the current and candidate solu- In contrary, if this setting is too high, then it is possible that tion (δ), and the other being a temperature (t) determined SA performs random walk of the search space for a significant by an accompanying annealing schedule encapsulated in the proportion of the search. There are several methods used in getCurrentT emperature() method on Line 10. the literature for determining this parameter value; some set this to be a factor of the cost of the initial solution [13], others Algorithm 1: A stochastic local search metaheuristic using propose that the maximum expected change in cost is used [9], Simulated Annealing as its move acceptance method. [14] such that all worsening moves are accepted, whereas a 1 s generateInitialSolution(); value is calculated in [13], [15] such that a percentage of moves are accepted at the initial temperature. 2 sbest s; 3 while terminationCriterionNotMet do The setting of the final temperature is equally as important 4 h getHeuristicT oApply(); because as the search nears its end, SA should accept very 0 5 s h(s); few to no worse moves, allowing the search to converge. If 0 this setting is too high, then the search will not converge 6 if f(s ) < f(sbest) then 0 resulting in a relatively poor quality solution being found. 7 sbest s ; 8 end On the other hand, if this setting is too low, then the search 0 9 δ f(s ) − f(s); could prematurely converge as there will be no worse moves 10 t getCurrentT emperature(); accepted for a significant proportion of the search. The final − δ temperature is usually set to some small value close to zero. 11 P e t ; 0 12 if f(s ) ≤ f(s) jj random 2 [0; 1] < P then III. METHODOLOGY 0 13 s s ; In this paper, we compare three strategies for tuning SA 14 end as a cross-domain stochastic local search algorithm. As the 15 end stochastic local search framework, the HyFlex framework [16] 16 return s ; best is used, which is a software framework developed to enable the implementation and testing of cross-domain heuristic search methods such as hyper-heuristics and metaheuristics. HyFlex A. Cooling Schedules contains a total of 6 problem domains, each with 12 problem SA requires the use of a cooling schedule to decrease the instances. Each problem domain contains a set of move temperature parameter, t, over time such that the probability operators (heuristics) which define the solution neighbour- of accepting a worse move decreases over time. Such an- hood. In this paper, we choose a minimum subset of move nealing schedules include linear cooling, geometric cooling, operators performing single perturbations on the solution, and and Lundy and Mees cooling [12]. Linear cooling decreases where necessary modify the existing framework, to facilitate a the temperature linearly as the search progresses whereas true stochastic local search framework. The number of move both geometric cooling and Lundy and Mees cooling use an operators and problem instances used in this study for each exponentially decaying function to decrease the temperature. problem domain are shown in Table I.
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