Duelist Algorithm: An Algorithm in Stochastic Optimization Method

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1 Duelist Algorithm: An Algorithm in Stochastic Optimization Method Totok Ruki Biyanto Department of Engineering Physics Insititut Teknologi Sepuluh Nopember Surabaya, Indoneisa Henokh Yernias Fibrianto Department of Engineering Physics Institut Teknologi Sepuluh Nopember Surabaya, Indonesia Hari H. Santoso Research Center for Metrology - The Indonesian Institute of Sciences Lembaga Ilmu Pengetahuan Indonesia (LIPI) Jakarta, Indonesia juara10@yahoo.com Abstract This paper proposes an algorithm for optimization based on how human fight and learn from each duel. Since this algorithm is based on population, the proposed algorithm starts with an initial set of duelists. Duels among these duelists forms the core of proposed evolutionary algorithm. The purpose of each duel is to determine the winner and loser. Then, each loser learns from the winner that beats him while the winner keep innovating new skill or technique that may upgrade their fighting capabilities. A few duelists with highest fighting capabilities are called as champion, champion s job is to train a new duelists that as good as himself to join the tournament as a representative of each champion. Champion is assured to be still the same as always with no change at all on their fighting capabilities. And then all the duelist are re-evaluated, and duelists with worst fighting capabilities is eliminated to maintain the amount of duelists. Applying and comparing the proposed algorithm to solve some optimization problems, shows its ability and in dealing with different type of problems compared to any other optimization algorithms. Keywords optimization; algorithm; duelist; fighting; capabilites; I. INTRODUCTION This paper proposes a new evolutionary algorithm for optimization based on genetic algorithm which is inspired by basic fighting competition. Optimization itself is a process to achieve something better. For example, let there be a problem f(x) then optimization is a process to find optimum value of x which is can be maximum, minimum or at specific point in between. Different methods have been proposed for solving an optimization problem. One of this methods is genetic algorithm (GA) which is based on natural selection by evolving a population of candidate solution for defined objective function [1]. On the other hand, a different method for optimization called ant colony optimization is inspired by foraging behavior of real ants [2]. Another type of method is inspired by social behavior of animals which is called as particle swarm optimization (PSO) [3]. There s also a method for optimization which inspired by imperialistic competition called imperialist competitive algorithm (ICA) [4]. Almost all of this example are population based algorithm which is mean that there s a set of population and keep improving itself each iterations [5]. Nowadays, all this optimization methods are very useful for solving multiple problems starting from process industry, energy management, scheduling, resource allocation or even pattern recognition and machine learning [6-13]. In this paper, a new algorithm based on genetic algorithm is proposed which is inspired by human fighting and learning capabilities. As an overview, in genetic algorithm there are two ways to evolve an individual into a new one. First is crossover where an individual mate with another individual to produce a new offspring, this new offspring s genotype are based on their parents. The second one is mutation where an individual mutate into a new one. In duelist algorithm (DA), all the individual in population are called as duelist, all those duelists would fight one by one to determine the champions, winners and losers. The fight itself just like real life fight where the stronger isn t always win, there s a probability that the weak one would be lucky enough to win. In order to improve each duelist, there are also two ways to evolve. One of them is innovation which is similar to mutation in genetic algorithm. The difference is only winners would possibly innovate. The other one is called as learning, losers would learn from winners. In genetic algorithm, both mutation and crossover are seem to be blind in producing any solution to find the best solution. Blind means that each solution produced in genetic algorithm aren t always a better solution, in fact it may turn out to be the worst one. Duelist algorithm tries to minimize this blind effect by giving different treatment on each duelist based on their classification. This paper described how duelist algorithm is designed and implemented.

2 II. REVIEW OF A DUEL Duel can be interpreted as a fight between one or more person(s) with other person(s). Fight itself isn t always using physical strength but also intellectual capability and reasoning, for example chess and bridge. Common type of duel which include physical strength is boxing, boxing is one of world s most popular sport where two persons need to knock down each of them under certain rules [14]. Soccer is also categorized as a duel where two teams must score goals to win the match, soccer is much more complicated than boxing knowing that teamwork plays important role [15]. Within every duel, there re must be a winner and a loser and rules that defined both of them. Take soccer for example, winner in soccer match is defined as team which score more goal(s) than their opponent. Not only score goals as much as possible, each team mustn t break the rules. In a match which include human, winning isn t always determined by each participant s skill or strength. Luck also take a part, it seems to be not scientific enough but luck just a word to describe all unknowns factor that may affect the match. After the match, knowing who s the winner and loser can be useful too. Loser can learn from how the winner wins, and winner can upgrade their ability and skill by training or trying something new. In the proposed algorithm, each duelist do the same to be unbeatable, by upgrading themself whether by learning from their opponent or developing a new technique or skill. III. DUELIST ALGORITHM The flowchart of proposed algorithm is shown in Figure 1. First, population of duelist is registered. Each duelist has their properties which is encoded into binary array of one and zero. After that, each duelist is evaluated to determine their fighting capabilities. And then a duel schedule is set to each duelist that contain a set of duel participants, in this duel, each duelist would fight one on one with other duelist. This one on one fight is used rather than gladiator battle to avoid local optimum. Each duel would produce a winner and a loser based on their fighting capabilities and their luck. After the match, a champion is also determined. This champions are those who has the best fighting capabilities of all duelists. Then, each winner and loser may upgrade their fighting capabilities while each champion train a new duelist that as good as themself and will be joining the next match that. Each loser would learn from their opponents how to be a better duelist by replacing a specific part of their binary array with winner s binary array value. On the other hand, winner would try to innovate a new technique or skill by changing their binary array value into something new. After that, each duelist fighting capabilities is re-evaluated for the next match. All the duelists then reevaluated through post-qualification and sorted to determine who deserve to be champions. Since there are new duelists that was trained by champions, all the worst duelists is eliminated to maintain the amount duelists in the tournament. Classification of each duelist is shown in Figure 2. This process will keep looping until tournament is finished. A. Registration of Duelist Candidate Each duelist in a duelist set is registered using binary array which contain only zero and one. This kind of set is also used in genetic algorithm called as population, binary array in genetic algorithm is called as chromosome but here the term is skillset. In and N var-dimensional optimization problem, the duelist would be binary length x N var length array. B. Pre-Qualification Pre-qualification is a test given to each duelists to measure or evaluate their fighting capabilities based on their skillset. C. Determine Board of Champions Board of champions is determined, this board is created to keep the best at the game without any possibilities of being eliminated from the game. Each champion should trains a new duelist to be as similar as himself with same dueling capabilities as himself, this new duelists would then join the duel while all the champions are out of duel but still in game. D. Define A Duel Schedule Between Each Duelist Duel between each duelist is set randomly. Each duelist that join the match then fight based on their fighting capabilities and luck to determine who is the winner and the loser. This duel is using a simple logic. If duelist A s fighting capabilities plus his luck is higher than duelist B s, then duelist A is the winner and vice versa. Duelist s luck is purely random function which equal to determined range of random number. Using luck in match may avoid local optimum. START REGISTRATION PRE-QUALIFICATION DETERMINE BOARD OF CHAMPIONS, CHAMPIONS TRAINS A NEW DUELIST THAT AS SIMILAR AS HIMSELF DUEL BETWEEN EACH DUELIST (CHAMPIONS EXCLUDED) DETERMINE WINNERS AND LOSERS WINNER TRAINS HIMSELF TO BE MORE ADVANCE POST-QUALIFICATION ELIMINATE SOME OF WORST DUELISTS IS TOURNAMENT FINISHED? YES END Fig. 1. Duelist Algorithm flowchart LOSER LEARNS FROM WINNER WHO BEATS HIM NO

Champion Winner Loser Fig. 2. Duelist s Classification. Worst Duelist E. Duelist s Enhancement After the match, each duelist are categorized into champion, winner and loser.

3 Champion Winner Loser Fig. 2. Duelist s Classification. Worst Duelist E. Duelist s Enhancement After the match, each duelist are categorized into champion, winner and loser. To improve each duelist fighting capabilities there re three kind of treatment given based on their match result. First treatment is for losers, each loser is trained by learning from winner. Learning means that loser may copy a fraction part of winner s skillset or binary array. The second treatment is for winners, each winner would trains on their own to be a new kind of duelist which hopefully to be a better one. This treatment consist of winner s binary array random manipulation. F. Elimination Since there are some new duelists joining the game, there must be an elimination to keep duelists quantity still the same as defined before. Elimination is based on each duelist s dueling capabilities which was determined and sorted before at post-qualification step. Duelist with worst dueling capabilities is eliminated and the amount of duelist that being eliminated is equal to amount of champions in game. IV. EXPERIMENTAL STUDIES This section discuss about Duelist Algorithm performance using 2 benchmarks. All these problems are maximization problems. The detail of these functions are listed below. Problem M 1 : f = (x. sin(4. x) + 1,1. y. sin (2. y)) Fig. 3. 3D plot of function in problem M 1 The initial population of 100 duelists, max generations of 200, luck coefficient of 0 and mutation probability of 0.5 is set. Result of first test is shown in Figure 4. Fig. 4. Duelist Algorithm s maximum cost versus iteration in problem M 1. Maximum value is found at which is supposed to be For comparison, a continuous GA is applied with population of 100 individuals, max generations 200, mutation probability of 0.5 and crossover probability of 1. To eliminate any unfair factor such as random population, predetermined population is used for both algorithm. Since both of them have a pseudo-random function (e.g. mutation), both of them always give different result every time we put it on a test. Figure 5 shows GA and DA comparison for the first test using M 1 as optimization problem. Problem M 2 : f = ( x 2 + y 2. cos(x y). e (cos(x.(y+5)) 7 ) Figure 3 shows a 3D plot of function of problem M 1 in interval 0 < x,y < 10. Fig. 5. Duelist Algorithm and Genetic Algorithm s maximum cost versus iteration in problem M 1. Another type of problem called M 2 is also given to both GA and duelist algorithm. Figure 6 shows 3D plot function of problem M 2 in interval 0 < x,y < 10 with maximum value

Particle Swarm Optimization (PSO) and Imperialist Competitive Algorithm (ICA) is also used make another comparison. Figure 10 shows the result of ICA using 100 countries, revolution rate of 0.

And Figure 10 shows the result of PSO using 100 swarms, 200 iterations, speed constants of 0.4 and 0.6, and theta within range of 0.5 to 0.9.

4 Particle Swarm Optimization (PSO) and Imperialist Competitive Algorithm (ICA) is also used make another comparison. Figure 10 shows the result of ICA using 100 countries, revolution rate of 0.3, assimilation coefficient of 2, assimilation angle coefficient of 0.5, zeta of 0.02, damp ration of 0.99, 8 initial imperialists and 200 decades. And Figure 10 shows the result of PSO using 100 swarms, 200 iterations, speed constants of 0.4 and 0.6, and theta within range of 0.5 to 0.9. This experiment shows that DA is faster than PSO and GA in reaching global optimum. Fig. 6. 3D plot of function in problem M 2 Test is taken using same setting as before and result is obtained as shown in Figure 7. Figure 7 shows that Duelist Algorithm achieve faster and better solution which is in 143 iterations and maximum value at While GA need 166 iterations and find maximum value at To make a valid comparison, we take 10 tests of both algorithm for problem M2 and the result is shown at Figure 8 and Figure 9. Fig. 10. ICA s maximum costs versus iteration in problem M 2. Fig. 7. Duelist Algorithm and Genetic Algorithm s maximum cost versus iteration in problem M 2. Fig. 11. PSO s maximum costs versus iteration in problem M 2. Fig. 8. Duelist Algorithm s maximum costs versus iteration in problem M 2. V. CONCLUSION In this paper, an optimization algorithm based on how duelist enhance himself to win a fight is proposed. In general, duelist algorithm refers to genetic algorithm. Each individual of the population is called duelist. Each duelist then fight with other duelist to determine who is the winner and the loser. Winner and loser have their own way of enhancing themself. Winner enhance himself by mutating itself. In the other hand, loser enhance himself by learning from the winner. After several enhancements and duels, some duelists will become the best solution for given problem. The algorithm is tested by 2 optimization problems or benchmarks, results shows that this algorithm is able to find the global optimum of these functions. The algorithm is also compared with genetic algorithm, particle swarm optimization and imperialist competitive algorithm for one of the optimization problems. REFERENCES Fig. 9. Genetic Algorithm s maximum costs versus iteration in problem M 2. [1] M. Mitchell, An introduction to genetic algorithms: MIT press, [2] M. Dorigo and C. Blum, "Ant colony optimization theory: A survey," Theoretical computer science, vol. 344, pp , 2005.

5 [3] R. Poli, J. Kennedy, and T. Blackwell, "Particle swarm optimization," Swarm intelligence, vol. 1, pp , [4] E. Atashpaz-Gargari and C. Lucas, "Imperialist competitive algorithm: an algorithm for optimization inspired by imperialistic competition," in Evolutionary computation, CEC IEEE Congress on, 2007, pp [5] J. E. Beasley and P. C. Chu, "A genetic algorithm for the set covering problem," European Journal of Operational Research, vol. 94, pp , [6] S. Hartmann, "A competitive genetic algorithm for resourceconstrained project scheduling," Naval Research Logistics (NRL), vol. 45, pp , [7] G. C. Dandy, A. R. Simpson, and L. J. Murphy, "An improved genetic algorithm for pipe network optimization," Water Resources Research, vol. 32, pp , [8] G. Jones, P. Willett, and R. C. Glen, "Molecular recognition of receptor sites using a genetic algorithm with a description of desolvation," Journal of molecular biology, vol. 245, pp , [9] R. L. Johnston and H. M. Cartwright, Applications of evolutionary computation in chemistry vol. 110: Springer Science & Business Media, [10] K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan, "A fast and elitist multiobjective genetic algorithm: NSGA-II," Evolutionary Computation, IEEE Transactions on, vol. 6, pp , [11] H. H. Balci and J. F. Valenzuela, "Scheduling electric power generators using particle swarm optimization combined with the Lagrangian relaxation method," International Journal of Applied Mathematics and Computer Science, vol. 14, pp , [12] M. Colombetti and M. Dorigo, "Evolutionary computation in behavior engineering," Evolutionary computation: theory and applications, pp , [13] D. B. Fogel, "An evolutionary approach to the traveling salesman problem," Biological Cybernetics, vol. 60, pp , [14] K. Woodward, Boxing, Masculinity and Identity. The" I" of the Tiger: Routledge, [15] T. Reilly, Science and soccer: Routledge, 2003.

Duelist Algorithm: An Algorithm Inspired by How Duelist Improve Their Capabilities in a Duel

Duelist Algorithm: An Algorithm Inspired by How Duelist Improve Their Capabilities in a Duel Totok Ruki Biyanto* a, Henokh Yernias Fibrianto a, Gunawan Nugroho a, Erny Listijorini b, Titik Budiati c, Hairul