ε Constrained CCPSO with Different Improvement Detection Techniques for Large-Scale Constrained Optimization

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1 216 49th Hawaii International Conference on System Sciences ε Constrained CCPSO with Different Improvement Detection Techniques for Large-Scale Constrained Optimization Chen Peng, Qing Hui Department of Electrical and Computer Engineering, University of Nebrasa-Lincoln Lincoln, NE , USA s: Abstract Although there are many studies on large-scale unconstrained optimization (e.g., with 1 to 1 variables) and small-scale constrained optimization (e.g., with 1 to 3 variables) using nature-inspired algorithms (e.g., evolutionary algorithms and swarm intelligence algorithms), no publicly available nature-inspired algorithm is developed for large-scale constrained optimization. In this paper, we combine a cooperative coevolutionary particle swarm optimization (CCPSO) algorithm with the ε constrained method to solve large-scale real-valued constrained optimization problems. The εccpso framewor is proposed, and three different algorithms based on the framewor, i.e.,, εccpsow and, are developed. The proposed algorithms compare favorably to the state-of-the-art constrained optimization algorithm εdeag on large-scale problems. The experimental results further suggest that with adaptive improvement detection technique is highly competitive compared with the other algorithms considered in this wor for solving large-scale realvalued constrained optimization problems. I. INTRODUCTION Nature-inspired meta-huristics such as evolutionary algorithms and swarm intelligence algorithms have been demonstrated to be effective optimization techniques [1], [2], especially for complicated problems such as nonconvex nonlinear optimizations with an unnown objective function. Hence, in recent years there have been many studies on nature-inspired algorithms for large-scale (e.g., 1 1D, D as dimensions) unconstrained optimization [3] [8], e.g., the MLCC method [8], CCPSO2 algorithm [6], and methods based on variable interactions [3], [7]. There also has been a lot of attention on algorithms for constrained optimization at smaller scale (e.g., 1 3D) [9] [11]. However, large-scale constrained optimization using nature-inspired algorithms is still a relatively new and under-explored area, and to the authors best nowledge, by far there is no nature-inspired algorithm nown to be capable of solving a broad spectrum of large-scale real-valued constrained optimization problems. Clearly, for solving general large-scale constrained optimization problems, we need to face both the difficulties in searching the minimum of the large-dimensional objective function and locating the feasible region defined by the large-dimensional constraint functions. Many practical applications, however, need such optimization techniques. For example, in power grid systems, identifying highly critical lines are crucial for system safety and security, and such vulnerability analysis usually involves solving optimization problems of a large number of variables [12]. A typical IEEE Three-Area RTS-96 system has a total of 73 buses and 185 transmission lines, and the standard IEEE 118-bus system has a total of 185 lines and 19 generation buses [12], [13]. Here we combine an algorithm in large-scale unconstrained optimization area, i.e., CCPSO2, with an effective constrained optimization technique, i.e., the ε constrained method, and propose a new framewor, i.e., εccpso, to solve large-scale constrained optimization problems. On one hand, CCPSO2 is a cooperative coevolutionary particle swarm optimization (CCPSO) algorithm using a random grouping technique, based on the cooperative coevolution (CC) strategy [14], [15], and has shown promising solution capability for large-scale unconstrained optimization. On the other hand, in [9] for 1 3D constrained optimization, the ε constrained method is adopted by a differential evolutionary (DE) algorithm, i.e., εdeag, which shows competitive results in the CEC 1 constrained real-parameter optimization competition [16], [17]. Experimental results in later sections will show that our proposed algorithms compare favorably to the εdeag algorithm on large-scale constrained optimization problems. In summary, the contributions of this paper are listed as follows. 1) This paper is, to the authors best nowledge, the first published wor on large-scale constrained optimization using nature-inspired algorithms (e.g., evolutionary algorithms and particle swarm optimization) tested by a comprehensive benchmar problem set. 2) A hybridized optimization algorithm framewor, i.e., εccpso, is proposed for large-scale constrained optimization, by combining the CCPSO2 algorithm and ε constrained method together. 3) Three different algorithms based on the εccpso framewor are proposed for large-scale constrained optimization, i.e., with ε-criteria based improvement detection, εccpsow with fixed-size improvement detection window, and with adaptive improvement detection window /16 $ IEEE DOI 1.119/HICSS

2 4) Eleven benchmar problems proposed for the CEC 1 special session on constrained real-parameter optimization (i.e., CEC 1CRPO) are extended from a maximum of 3D to a maximum of 1D for our computer experiments. 5) A comprehensive study is carried out comparing the proposed algorithms with the state-of-the-art constrained optimization algorithm εdeag on the eleven large-scale benchmar problems, with an emphasis on among the other three εccpso algorithms. This paper is organized as follows. Section II introduces and formulates the general real-valued constrained optimization problems. Section III introduces several relevant techniques, i.e., particle swarm optimization (PSO), cooperative coevolution, and ε constrained method, as bases for our algorithms. In Section IV, the εccpso framewor and the ε level controlling method in εccpso are described. Furthermore, three different algorithms based on the εccpso framewor, i.e.,, εccpsow, and, are proposed. Section V presents comprehensive and comparative results on the proposed algorithms and the εdeag algorithm for the eleven benchmar problems of large-scale constrained optimization. II. CONSTRAINED OPTIMIZATION PROBLEMS Many industrial applications are essentially constrained optimization problems, with the physical or environmental constraints formulated as equality or inequality relationships. For example, the balanced coordinated resource allocation design of a networ system [18], the vulnerability analysis of power grids [12], and the optimization of cascading failure protection in complex networs [19] can all be formulated as constrained nonlinear optimization problems. Moreover, the above mentioned applications are all potentially largescale problems, since a networ can have a large number of nodes. Generally speaing, a constrained optimization problem can be expressed in the following formulation minimize: subject to: f (x), g i (x), i = 1,...M g h j (x)=, j = 1,...M h x min x x max, = 1,...,D, where x =(x 1,x 2,...,x D ) R D is a D dimensional vector, and f (x) : R D R is called an objective function. The problem has M g inequality constraints g i (x) and M h equality constraints h j (x) =. Each element x R of the vector x is also called a variable, and has a lower bound x min and an upper bound x max. Usually the equality constraints are relaxed into inequality constraints of the form h j ( )= h j ( ) h δ, (1) where the tolerance level h δ specifies how much the equality constraints are relaxed. III. RELATED STUDIES To solve large-scale constrained optimization problems, our εccpso framewor combines a cooperative coevolutionary PSO algorithm [6] with the ε constrained method [9]. The update law of a classical PSO algorithm is given as follows v i,d (t + 1)= ωv i,d (t)+c 1 r (i) 1 (t)(p i,d(t) x i,d (t))+ c 2 r (i) 2 (t)(g d(t) x i,d (t)), (2) x i,d (t + 1)= x i,d (t)+v i,d (t + 1), where x i,d, v i,d,andp i,d denote the dth dimension of the position x i, velocity v i, and personal best p i of particle i, respectively; and g d denotes the dth dimension of the global best g of the swarm. The parameters ω, c 1, and c 2 are called inertia weight, cognitive attraction, and social attraction, respectively. In addition, r (i) 1 and r (i) 2 are random variables independently and uniformly sampled from [,1] for particle i in each iteration. In the terminology for many nature-inspired algorithms, an iteration t as in (2) is also called a generation. A. Cooperative coevolution The idea of cooperative coevolution (CC) is to partition the dimensions/variables space into certain groups and evolve the current group s variables using the best solutions found in the other groups [14], [15]. To maximize the performance of CC algorithms, correlated variables should be partitioned into the same group, and uncorrelated variables into different groups. To this end, different grouping strategies have been proposed and investigated [3], [4], [6] [8]. For example, in multilevel cooperative coevolution (MLCC) [8], at each generation a group size s is probabilistically chosen from a set S according to the recorded performance of each group size in S, then the variables are divided into K = n/s groups. In CCPSO2 [6], this procedure is simplified by choosing a new group size s randomly in uniform distribution from S only if there is no improvement to the global best g in the last generation. Competitive results have shown the effectiveness of CCPSO2 for large-scale unconstrained continuous optimization. Instead of using the classical update law (2), Ref. [6] shows that a Cauchy and Gaussian-based update law with ring topology wors better with the random grouping cooperative coevolutionary PSO. The update law is shown as follows { p i,d (t)+c i (t) p i,d (t) l i,d (t), if rand < σ, x i,d (t +1)= l i,d (t)+n i (t) p i,d (t) l i,d (t), otherwise, (3) where rand denotes a uniform random function on (,1), C i and N i are random variables following the standard Cauchy distribution and the standard Gaussian distribution 1712

3 for particle i, independently sampled at each generation t. Besides, l i,d is the dth dimension of the neighborhood s best l i of particle i. The probability of choosing Cauchy update is specified by a user-defined parameter σ [,1], andthe probability of choosing the Gaussian update is 1 σ. B. The ε constrained method There are many constraint handling methods for natureinspired-algorithms, as introduced in [11]. Some of the common ones include penalty method, stochastic raning, ε-constrained method, multi-objective approaches, ensemble of constraint-handling techniques, etc. In this paper, the ε-constrained method [1] is chosen to be the constraint handling component of our algorithm, mainly because of its simplicity and competitive performance, as already demonstrated in many wors [9], [1], [2], [21]. Besides, in the CEC 1 competition & special session on constrained realparameter optimization, an ε-constrained differential evolution algorithm with an archive and gradient-based mutation (edeag) demonstrated highly competitive results [9], and was the winner of the competition. In the ε-constrained method, for any ε, the ε level comparison < ε between two pairs of particles objective values and constraint violations ( f 1,φ 1 )and(f 2,φ 2 )isdefined as follows f 1 < f 2, if φ 1,φ 2 ε; ( f 1,φ 1 ) < ε ( f 2,φ 2 ) f 1 < f 2, if φ 1 = φ 2 ; φ 1 < φ 2, otherwise, where the constraint violation can be calculated in various ways. Here we use the following constraint violation function M g i=1 (max{,g i(x)}) 2 + M h j=1 (max{, h j (x)}) 2, φ(x)= if x [x min,x max ], = 1,...,D,, otherwise, where the inequality constraint h j ( )= h j ( ) h δ isa relaxation for the equality constraint h j ( ) with tolerance h δ. The definition of the ε level comparison < ε allows transforming a constrained optimization problem into an unconstrained problem. In case of ε =, < ε is equivalent to the comparison < of objective function values. In case of ε =, < ε is equivalent to the lexicographic order comparison where constraint violation precedes objective function values. Furthermore, it can be proved that for a certain ε value, the order of the particles is well-defined, i.e., if ( f 1,φ 1 ) < ε ( f 2,φ 2 ) and ( f 2,φ 2 ) < ε ( f 3,φ 3 ),then( f 1,φ 1 ) < ε ( f 3,φ 3 ). More properties of the ε level comparison can be found in [9], [1]. IV. ε CONSTRAINED CCPSO This section introduces the εccpso framewor and the novel ε level control method in this framewor. Furthermore, for different new group size selection methods, three different algorithms based on the εccpso framewor are proposed. A. εccpso framewor The framewor of εccpso is shown in Algorithm 1. As in CCPSO2, the dimensions of the solutions are divided into K groups, with each having size s (d = Ks). Here, the jth group part of the dimensions of a vector z is denoted as z.d j, j [1...K], andb( j,z) returns the vector (g.d 1,g.d 2,...,g.d j 1,z,g.d j+1,...,g.d K ).InεCCPSO, the comparisons between objective function values in CCPSO2 are replaced by the ε-comparisons between the pairs of objective function values and constraint violations. Furthermore, using ε-comparison, the notion of ε-minimizations, i.e., min ε and arg min ε, can be introduced. For example, min ε [ f (z i ),φ(z i )] denotes the minimal pair [ f (z i ),φ(z i )] by the criteria of ε-comparison. Let N (i) denote the neighborhood particles of particle i. Since we use a ring topology, we have N (1) ={n,1,2}, N (n) ={n 1,n,1} and N (i)={i 1,i,i + 1} for i = 2,...n 1. Furthermore, for the clarity of representation, let us define P(x) = [ f (x),φ(x)] as the pair of objective value and constraint violation corresponding to solution x. Thus, i = arg min ε i N (i) [ f (z i ),φ(z i )] = arg min ε P(z i ) i N (i) denotes the neighborhood best of particle z i by means of ε-comparison. Algorithm 1 Pseudocode for εccpso 1: Initialize each particle solution. Let initial personal best be the same as the initial solution. 2: Initialize ε = ε() for constraint handling; 3: Compute the initial global best g and neighborhood s best l based on ε-minimization; 4: Randomly choose a group size s from S and let K = n/s (the jth group part of the dimensions of vector z is denoted as z.d j, j [1...K]); 5: repeat 6: if global best has not been improved then 7: randomly choose a group size s from S and let K = D/s; 8: Ind randomly permutate indices from [1,...,D]; index vector 9: Construct K groups using Ind; 1: for each group j [1,...,K] do 11: for each particle i [i,...,n] do 12: Evaluate b( j,x i.d j ) and b( j,p i.d j ) 13: Update ε based on the current FES; 14: if P[b( j,x i.d j )] < ε P[b( j,p i.d j )] then 15: p i.d j x i.d j ; 16: if P[b( j,p i.d j )] < ε P(g) then 17: g.d j p i.d j ; 18: end for 19: for each particle i [i,...,n] do 2: l i.d j p i.d j,wherei = arg min ε P[b( j,p i.d j )]; i N (i) 21: end for 22: end for 23: Update positions of all particles using (3); 24: until Stop criteria met 1713

4 B. Controlling ε level in εccpso In an algorithm utilizing the ε constrained method, the ε level needs to be controlled such that it gradually reduces from a large value (corresponding to large tolerance of constraint violations) to zero (corresponding to zero tolerance of constraint violations). Here, based on the ε level control method used in [9], we adopt a modified version for largescale constrained optimization, which is shown as follows ( ) ε()= 1 N 1 2 N φ(x i ()) + min φ(x i()) i=1 i=1,...,n ( ) ε() 1 t cp ( t) ε( t)= T c, < t Tc,, t > T c, c p ( t)= { max { c min p, logε λ logε() log(1 T λ /T c ) }, t T λ, c min p cp (T c t), t > T λ, cp = cmin p c p (T λ ) T c T λ. where t is the current fitness evaluations, i.e., FES, and the initial ε level ε() is chosen to be the middle of the best value and average value of initial constraint violations of the particles. Besides, T c =.95Max FES is the fitness evaluations when ε is decreased to zero, i.e., ε(t c )=, where Max FES is the maximum number of fitness evaluations, and T λ is the fitness evaluations when the ε level is decreased to ε λ, i.e., ε λ = ε(t λ ). The control parameter c p is set fixed for t T λ, and is gradually reduced to its specified minimum value c min p when t > T λ, with c p (T c )=. C. εccpso with adaptive improvement detection In unconstrained CCPSO2, a new group size s is selected from S if the global best has not been improved in the last generation. Note that since in unconstrained optimization, the global best can only be changed to improve the objective function, we can consider any update of the global best as an improvement. However, when combining CCPSO2 with the ε constrained method, a problem of detecting the improvements arises. In εccpso for constrained optimization, new global best solution can also be chosen for the improvement of constraint violations. Now we consider several different ways of determining whether the global best has been improved or not in the last generation. First, let be the εccpso algorithm that detects the improvement of global best using ε- comparison, i.e., the global best is considered improved if P(g new ) < ε P(g old ),whereg new denotes the new global best and g old denotes the old global best. This is reasonable based on the new ε-criteria. Thus, from Algorithm 1, we now that any update of the global best is considered as an improvement in. This is directly derived from and is similar to the original CCPSO2 algorithm. Secondly, let εccpsow be the εccpso algorithm that detects the improvement of the global best using the ordinal comparison of objective function values (i.e., fitness values) with an improvement detection window of a specified size. In εccpsow, the new global best is considered to have been improved if the new global best objective value is smaller than the minimum of the global best fitness values in the last w generations, i.e., f (g new ) < min{ f (g) : g S g (w)}, where S g denotes the set of global best in the last w generations, and w is called the size of the improvement detection window. The rationality behind the design of εccpsow is that in a constrained optimization, both the global best fitness value and constraint violation should be reduced in order for an update of global best to be called an improvement. However, an update based on ε-comparison with a new ε level does not necessarily reduce the fitness value, thus it is reasonable to compare the current global best fitness value with a certain number of previous global best fitness values to decide whether the update is an improvement. Note that different improvement detection window size has different effect. For some problems, εccpsow with w = 1 yields better performance than using w = 1, while sometimes using w = 1 yields better performance. To automatically select an appropriate improvement detection window size, finally we propose an εccpso method which uses an adaptive detection window, i.e.,. We adopt the adaptive weighting scheme used in [8], where a performance record list R = {r 1,r 2 } is used and is updated according to r i = f (g old) f (g new ). f (g old ) The probability p 1 and p 2 of selecting window size 1 and 1 respectively in each generation is computed as e 7r i p i = e 7r, 1 + e 7r 2 where constant 7 and the natural exponential constant e are empirical values. V. COMPUTER EXPERIMENTS In this section, we show the experimental results of the proposed three different algorithms, i.e.,, εccpsow, and, compared with the stateof-the-art constrained optimization evolutionary algorithm εdeag. First, the eleven benchmar problems and the parameter settings for our algorithms are described. Second, the results of the εccpsow algorithm with w = 1andw = 1 are compared to with adaptive improvement detection technique. Then, and are compared to εdeag on the benchmars of 1D. Finally, we compare the results of and on the benchmars of higher order dimensions, i.e., 5/1D, and show that is more favorable for a general 1714

5 application. All results here are the average of 25 runs of the corresponding algorithms. Table I PARAMETER SETTINGS Name Value N 3 D 1/5/1 σ.5 Max FES 2*D S {2,5,1,5,1} for D<25; {2,5,1,5,1,25} otherwise h δ.1 ε λ.34 c min p 3 T c.95 Max FES T λ.8 Max FES Table II PROPERTIES OF THE ELEVEN CEC 1 PROBLEMS Problem Search range Objective Number of constraints Equalities Inequalities C1 [,1] D N 2N C2 [ 5.12,5.12] D S 1S 2S C3 [ 1,1] D N 1N C4 [ 5,5] D S 2N2S C5 [ 6,6] D S 2S C9 [ 5,5] D N 1S C12 [ 1,1] D S 1N 1S C14 [ 1,1] D N 3S C16 [ 1,1] D N 2S 1S1N C17 [ 1,1] D N 1S 2N C18 [ 5,5] D N 1S 1S The parameter settings for the different εccpso algorithms (i.e.,, εccpsow, and ) are shownintablei. Table III εdeag EXPERIMENTS AT DIFFERENT SCALES C4 C12 Avg. Fitness Max. Const. 7D D D D Unsolvable 35D D E+143 5D Unsolvable 1D Unsolvable A. Benchmars and experimental setup We adopt and extend eleven benchmar problems proposed for the CEC 1 special session on constrained real w = 1 w = (a) C2 of 1D w = 1 w = (c) C2 of 5D w = 1 w = (b) C5 of 1D w = 1 w = (d) C5 of 5D Figure 1. Fitness values of εccpsow with w = 1andw = 1 compared to solving two benchmar problems of 1/5 dimensions parameter optimization (i.e., CEC 1CRPO) [17], i.e., C1, C2, C3, C4, C5, C9, C12, C14, C16, C17, and C18. The properties of these problems are shown in Table II, where S is for separable and N for Non-separable. For example, C4 has a separable objective function, 2 non-separable and 2 separable equality constraints, and no inequality constraint. Detailed problem definitions can be found in [17]. Roughly speaing, these problems are defined in the formulation of (1). Moreover, all the variables of a problem have the same lower bounds and upper bounds, i.e., x min = x min and x max = x max. Besides, a random translation vector o [o min,o max ] D is used, where o min and o max are dependent on the problem. We extend the original CEC 1CRPO problems from a maximum of 3D to a maximum of 1D. For example, in C2, we have x min = 5.12 and x max = From the code of the original C2 problem, min(o) =.4966 and max(o)=.4934, thus we let o min =.5 and o max =.5, and generate o randomly in uniform distribution over the region [o min,o max ] 1. After the translation vectors o for the problems are generated, they are fixed for all experiments. B. Testing different improvement detection window strategies There are cases in which different improvement detection window sizes mae huge differences to the performance of the εccpsow algorithm. As shown in Fig. 1, when solving Problem C2 with εccpsow, using w = 1 results in better performance than using w = 1. However, when solving Problem C5, w = 1 yields better performance (note that 1715

6 Table IV 1D EXPERIMENTS εdeag Avg. Fitness Max. Const. Avg. Fitness Max. Const. Avg. Fitness Max. Const. C E E E E E-1.E+ C E+.E E+.E E-2.E+ C E E E E E E+5 C E E E E+ Unsolvable C E+2.E E E E E+2 C E+3.E E+5.E E+7.E+ C E E E E+3 Unsolvable C E+4.E E+11.E+ 7.77E+12.E+ C E-3.E E-2.E E+.E+ C E+1.E E+1.E E+2.E+ C E+.E E+1.E E+3.E+ in ε constrained optimization, the global best fitness values are not always decreasing, since the particles are compared using new ε-criteria continually). Because of the adaptive improvement detection window, for both C2 and C5, the performance of is in between εccpsow with w = 1 and with w = 1. For the other problems which are not shown in the figure, it is either that using w = 1and w = 1 in εccpsow do not mae much difference, or that the average fitness value of almost coincides to one of the two cases of εccpsow (i.e., w = 1orw = 1). Thus, the algorithm can roughly represent the class of εccpso algorithms with improvement detection window technique in terms of the average fitness value performance. For the rest of the experimental studies, we thus use the algorithm to compare with other types of algorithms on large-scale constrained optimization. C. Comparing and with εdeag Although there is no nature-inspired algorithm nown to be capable of solving a broad class of large-scale real-valued constrained optimization problems, constrained optimization algorithms of smaller scales are abundant, and many of them are well studied [11]. The εdeag algorithm is an ε constrained differential evolutionary algorithm which maes use of an archive and gradient-based mutation. It has been shown to be quite efficient for 1/3D real-valued constrained optimization [9]. The code of the algorithm can be downloaded from [22], which is also used in this research. However, as the number of dimensions increases, the efficiency decreases, as shown in Table III for Problems C4 and C12. Furthermore, when the number of dimensions is greater than a certain limit (i.e., 97D for C4 and 4D for C12), overflows occur during the execution of εdeag for these two problems. In Table IV, we compare the average fitness values (Avg. Fitness) and the maximum constraint violations (Max. Const.) of the final global best solution of the 25 runs of the and algorithms with those of the solutions of the εdeag algorithm. As shown in the table, both and surpass εdeag for all problems except C1, where and have not found a final feasible solution. Note that the global bests of and have better fitness values than that of εdeag for all problems; and except for C1, the maximum constraint violations of both and are all smaller than that of εdeag. D. Comparing with The final average global best fitness values (Avg. Fitness) and maximum constraint violation (Max. Const.) of and applied on the eleven benchmar problems of 1/5/1D are shown in Tables IV, V, and VI, respectively. Besides, the evolution process of the average fitness values of and solving six of the benchmars (i.e., C3, C4, C9, C14, C16, and C18) of 1/5D are shown in Fig. 2 and Fig. 3. From the results we can see that for six of the benchmars, i.e., C1, C2, C4, C12, C16, and C17, in the cases of 1/5D, the performance of the two algorithms are too close to tell which is better. However, for the set of problems that outperforms, i.e., C3, C5, C9, C14, and C18, the performance gaps are clear. We would thus conclude that although is better in some situations, is more favorable for more general optimization problems. VI. CONCLUSION Large-scale constrained optimization using natureinspired algorithms (e.g., PSO, DE) is still a relatively new and under-explored area, and by far there is no publicly available nature-inspired algorithm developed for such a purpose. In this paper, by combining two different techniques from two different areas (i.e., CCPSO2 and the ε constrained method), we propose the εccpso framewor for large-scale real-valued constrained optimization. Based on the εccpso framewor, we further develop three different algorithms with different global best improvement detection techniques. Experimental results show that the 1716

7 (a) C3 (b) C4 (c) C (d) C14 (e) C16 (f) C18 Figure 2. Fitness values of and solving six benchmar problems of 1 dimensions Table V 5D EXPERIMENTS Table VI 1D EXPERIMENTS Avg. Fitness Max. Const. Avg. Fitness Max. Const. C E-1.E E-1.E+ C E+.E E+.E+ C3 7.65E E E E+2 C E+ 7.66E E E+2 C E+2.E E+2.E+ C E+3.E+ 2.82E+7.E+ C E E E E+3 C E+2.E E+5.E+ C E+.E E+.E+ C E+4.E E+4.E+ C E+1.E E+1.E+ Avg. Fitness Max. Const. Avg. Fitness Max. Const. C E-1.E E-1.E+ C E+.E E+.E+ C E E E E+2 C E E E E+2 C E+2.E E+2.E+ C E+3.E E+9.E+ C E E E E+3 C E+2.E E+2.E+ C E+.E E+.E+ C E+4.E E+4.E+ C E+1.E E+1.E+ proposed algorithms compare favorably to the state-of-theart constrained optimization algorithm εdeag on large-scale problems. It is also shown that with adaptive improvement detection technique is more favorable than the other two algorithms for more general problems. REFERENCES [1] A. E. Eiben and J. E. Smith, Introduction to evolutionary computing. Springer Science & Business Media, 23. [2] J. Kennedy, J. F. Kennedy, and R. C. Eberhart, Swarm intelligence. Morgan Kaufmann, 21. [3] M. N. Omidvar, X. Li, Y. Mei, and X. Yao, Cooperative Co- Evolution With Differential Grouping for Large Scale Optimization, IEEE Trans. Evolutionary Computation, vol. 18, no. 3, pp , Jun [4] Y. Mei, X. Li, and X. Yao, Cooperative Coevolution With Route Distance Grouping for Large-Scale Capacitated Arc Routing Problems, IEEE Trans. Evolutionary Computation, vol. 18, no. 3, pp , Jun [5] B. Kazimipour, X. Li, and A. K. Qin, Initialization methods for large scale global optimization, in IEEE Congress on Evolutionary Computation. IEEE, 213, pp [6] X. Li and X. Yao, Cooperatively Coevolving Particle Swarms for Large Scale Optimization, IEEE Trans. Evolutionary Computation, vol. 16, no. 2, pp , Apr [7] W. Chen, T. Weise, Z. Yang, and K. Tang, Large-scale global optimization using cooperative coevolution with variable interaction learning, in Parallel Problem Solving from Nature, PPSN XI. Springer, 21, pp

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