Simple Optimization (SOPT) for Nonlinear Constrained Optimization Problem

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1 (ISSN 4-6) Journal of Science & Enineerin Education (ISSN 4-6) Vol.,, Pae-3-39, Year-7 Simple Optimization (SOPT) for Nonlinear Constrained Optimization Vivek Kumar Chouhan *, Joji Thomas **, S. S. Mahapatra *** * Student, Department of Mechanical Enineerin, Christian Collee of Enineerin and Technoloy, Bhilai, Chhattisarh, India, vivekchouhan@mail.com ** Associate Professor, Department of Mechanical Enineerin, Christian Collee of Enineerin and Technoloy, Bhilai, Chhattisarh, India, joji.siji@mail.com *** Professor, Department of Mechanical Enineerin, National Institute Technoloy, Rourkela, Orissa, India, mahapatrass3@mail.com Abstract In recent years, Meta-heuristic alorithm becomes popular for solvin optimization problems. There are problems which are difficult to solve usin traditional methods. SOPT is an efficient meta-heuristic technique to solve such nonlinear problems. In this paper, SOPT alorithm is coded usin MATLAB and three test functions for constrained optimization havin different characteristic are run. The example considered for this alorithm were attempted previously by various researchers usin different optimization techniques such as Leaue championship alorithm(lca), Particle swarm optimization(pso), Artificial bee colony(abc). However, Comparison between results obtained by the proposed alorithm and those obtained by different optimization alorithm shows the better performance of proposed alorithm. Index Terms: SOPT, Meta-heuristics, constrained optimization.. INTRODUCTION In an Optimization technique a real function is maximized or minimized by selectin input values from a iven set of values[9]. A eneral optimization problem selects n decision variables x..., x xn from a feasible reion in order to optimize (either maximize or minimize) a iven objective function f x, x,... x ) of n decision ( n variable. These types of problems may either be linear or nonlinear. Most of the real world problems are nonlinear in nature. In a Nonlinear prorammin problem the objective function is nonlinear and feasible reion may be formed by nonlinear constraints. Thus eneral equation for nonlinear proram for maximization can be written as: Maximizef(x, x,. x n ) subjected to: j x j =,,. J k x = k =,, K x i (L) x x i (U) i =, N Decision variables (: These are the quantities which are needed to be determined. Objective function f(: These are expressions which are to be maximized or minimized. Upper and lower bound(x (U), x (L) ): These are the maximum and minimum limit assined for the decision variable. Constraints: These are the conditions for the variables that are needed to be satisfied. There are basically two approaches for solvin the optimization problems: ) Traditional ) Meta-heuristic. Traditional approach For linear optimization type of problems traditional methods like simplex methods can be used to solve the problems exactly. For nonlinear problems radient based methods like olden section search, Fibonacci search etc. are used. These methods require the problems to be differentiable. Traditional methods sometimes fail to solve complex nonlinear problems. To overcome the difficulties of traditional methods researchers are now-adays developin new Meta-heuristic alorithms. These types of alorithms are population based alorithms where a set of solutions are randomly enerated and then 3

2 Journal of Science & Enineerin Education (ISSN 4-6) Vol.,, Pae-3-39, Year-7 they are moved systematically towards the optimum. These problems do not uarantee an optimum solution but it has been seen that for most of the problems they ive near optimum solutions.. Meta-heuristic approach Meta-heuristic alorithm is an iterative eneration process it utilizes the concept of explorin and exploitin the search space efficiently in order to find near optimal solution. Here startin with a feasible solution. In each iteration, search the neihbourhood of the current solution and terminate when a better solution is obtained. This technique may be Trajectory based method: Here search process is identified by a trajectory in the search space. It focuses on modifyin and improvin a sinle solution. Tabu search, simulated annealin Population based method: Every step of search process has a population-a set of solution. It maintains and improves a set of solutions. Genetic alorithm, bee colony optimization, ant colony optimization.. LEAGUE CHAMPIONSHIP ALGORITHM, PARTICLE SWARM OPTIMIZATION, ARTIFICIAL BEE COLONY. Leaue championship alorithm [3] It is a population based alorithm. Idea came from competition of sport teams in a sport leaue, team plays for several weeks (Iterations). Leaue schedule is iven, a number of individual as a team plays in pairs and the outcome of the ame is decided as team wins or loses the match. Playin strenth (fitness value) with teams intended formations are iven. Modellin a match analysis, each team makes its own playin stratey (New solution) for the next week contest and the process is repeated for several number of seasons.. Particle swarm optimization [] It optimizes the problem havin the candidate solutions, dubbed particles are moved around in the search space accordin to the simple mathematical equations over the particles position and velocity. Its local best known position called pbest and the best known position of the entire swarm called best which influences each particle movement. This way swarm moves toward best solution. The possible solution of the function to be optimized is iven by position of the particle which is updated in each iteration usin equation (). Velocity of a particle in each iteration is iven by equation (). If R is rane of vector x the velocity is normally initialized in the rane ( R, +R). v i = ωv i + φ p r p pbest i x i + φ r (Best x i ) () x i = x i + v i () The parameter ω is called the inertia weiht and controls the manitude of the old velocity in the calculation of the new velocity, whereas φ p and φ determine the sinificance of pbest and best respectively r p and r are the random numbers enerated in the rane [,]. v i at any iteration is constrained by the parameter v max which is normally taken about % of the rane of v. If in any iteration position of the particle crosses the boundary then velocity is adjusted so that particles position reaches to the boundary which is called clampin of velocity..3 Artificial bee colony [] In ABC alorithm, the position of a food source represents a possible solution to the optimization problem. This alorithm simulates the intellient forain behaviour of a honeybee swarm. The colony consist of three types of bees: employed bees, onlooker bees, scouts. Employed bee search for food around food source in memory and shares this information with the onlooker bee waitin in the hive. This information contains distance, direction and location. The nectar amount of food source decides the quality (fitness) of the associated solution. The onlooker bees checks its nectar amount and its closeness to the hive, if new food source is havin hiher nectar (better fitness) then it forets the previous food source and remembers only new one. Then the nectar is loaded from food source and unloaded to the food store after that it miht dance and other nest mate recruited before returnin to the same food source. A scout may an onlooker who tries to find out the food source with the information iven by employed bees. Searchin of nearby food source by employed and onlooker bees are done accordin to followin equation. v ij = x ij + ij (x ij x kj ) Where, i, k,.. NS NS is number of food sources and j, D D is dimension of te problem. k and j are randomly chosen indexes. ij is a random number between [,-] 36

3 Journal of Science & Enineerin Education (ISSN 4-6) Vol.,, Pae-3-39, Year-7 3. SOPT [4] It is a population based alorithm and iterations are performed basically in two staes named exploration and exploitation steps. Which are performed one by one. In Exploration stae, where best solution is enerated usin equation (), if obtained solution is better than the worst solution in population then worst solution is replaced by new enerated solution. X i,new = X i,best + C R i () Similarly Exploitation step is performed usin equation () X i,new = X i,best + C R i () and operation continues repeatedly until a final condition is met. Where, X i,new is the i t parameter of the new solution for any iteration performed. X i,best is the i t parameter of the best solution in the same iteration.c, C are the positive constant and C is half of C.R i is the normally distributed random variable with a mean zero. σ i is the standard deviation of i t parameter of all the members in the population. Value of control variable C can be taken in between to. Value of control variable C is automatically decided as half of value of C []. There are alorithms which do not contain any control parameters e.. TLBO, Jaya alorithm [8] and other alorithms are havin more numbers of control parameters. SOPT is an alorithm which is havin only one control parameter effectively. It also make only two function evaluations in each iteration and therefore computationally less expensive if problem is solved in same number of iterations. The main advantae of SOPT is its simplicity with equivalent efficiency. 4. TEST PROBLEMS Minimize x ) ( x x 7) f ( ( x ( 4.84 ( x.) ( x.) ( x ( x.) 4.84 x x, ) Minimize f ( x ( x h( x x x, x x x, 3 Maximize xx x 4x 6 f ( x x ( x x ( x x x x,. RESULT AND DISCUSSION SOPT alorithm is coded in MATLAB and in this alorithm C is taken as. and C is.7 that is half of C. Three test problems are selected to check the performance of SOPT and the results are compared with the results obtained by LCA, PSO [7], ABC [6]. Each time SOPT ives fairly ood result. The Values obtained by solvin the objective functions for are x=.468, x=.3866, function value obtained= 3.98and for x=.776, x=.7, function value obtained =.7 and for 3 x=.6887, x=.868, function value obtained= For solvin problem and problem 3 less than iterations are required as shown in fiure 4 and fiure 6, while solution for problem is obtained in less than 3 iterations as shown in fiure. Fiures to 3 shows the surface plot of the objective function. 3 is a maximization type of problem which is first converted to minimization type by multiplyin it by - [9]. Table-: function values obtained for constrained optimization problem 3 LCA PSO ABC SOPT

4 Function Values f(x, Function Values f(x, f(x, Function Value Journal of Science & Enineerin Education (ISSN 4-6) Vol.,, Pae-3-39, Year x x Fi-: Surface plot of function Fi-: converence curve for function x - Fi-: Surface plot of function - -. x Fi-6: Converence curve for function x Fi-3: Surface plot of function 3 x 6. CONCLUSION For the optimization purpose a number of Meta-heuristic alorithms are used. SOPT is a population based alorithm consists of two staes: exploration and exploitation. Iterations are to be done for each stae and for each iteration there are two function evaluations. Here only one parameter is required to be set so that number of problems with different parameters can be solved. But in other alorithms parameters are more, to et a best set of parameter value number of experiments are to be performed. A result of experiment shows that SOPT performs satisfactory to et optimum value. It can also be used for multi-objective constraints problems Fi- 4: Converence curve for function REFERENCES []. D. Karaboa, B. Akay, 9 A comparative study of Artificial Bee Colony alorithm Appl. Math.comput. 4(), 8-3. []. J. Kennedy and R.C.Eberhart, Particle Swarm Optimization, IEEE International Conference on Neural Networks, Vol. 4, pp94 948,99. [3]. Ali HusseinzadehKashan, An efficient alorithm for constrained lobal optimization and application 38

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