An Empirical tudy of Control Parameters for Generalized Differential Evolution aku Kukkonen Kanpur Genetic Algorithms Laboratory (KanGAL) Indian Institute of Technology Kanpur Kanpur, PIN 8 6, India saku@iitk.ac.in, saku.kukkonen@lut.fi Jouni Lampinen Department of Information Technology Lappeenranta University of Technology P.O. Box, IN-585 Lappeenranta, inland jouni.lampinen@lut.fi KanGAL Report Number 54 Abstract In this paper the influence of control parameters to the search process of previously introduced Generalized Differential Evolution is empirically studied. Besides number of generations and size of the population, Generalized Differential Evolution has two control parameters, which have to be set before applying the method for solving a problem. The effect of these two control parameters is studied with a set of common bi-objective test problems and performance metrics. Objectives of this study are to investigate sensitivity of the control parameters according to different performance metrics, understand better the effect and tuning of the control parameters on the search process, and to find rules for the initial settings of the control parameters in the case of multi-objective optimization. It seems that in the case of multi-objective optimization, there exists same kind of relationship between the control parameters and the progress of population variance as in the case of single-objective optimization with Differential Evolution. Based on the results, recommendations choosing control parameter values are given.
Introduction Many practical problems have multiple objectives and several aspects cause constraints to problems. Generalized Differential Evolution (E) [ 4] is an extension of Differential Evolution (DE) [5, 6] for constrained multi-objective (Pareto-)optimization. DE is a relatively new real-coded Evolutionary Algorithm (EA), which has been demonstrated to be efficient for solving many real-world problems. In the earlier studies [4, 7, 8] E is compared with other multi-objective EAs (MOEAs). In these studies it has been noted that E performs better with different control parameter values from those usually used for the basic DE algorithm in singleobjective optimization. It has been also noted that the control parameter values should be drawn from a rather narrow range for some problems. However, more complete tests with different control parameter values has not been performed and reported before. Differential Evolution The DE algorithm [5, 6] was introduced by torn and Price in 995. Design principles of DE are simplicity, efficiency, and use of floating-point encoding instead of binary numbers. As a typical EA, DE has a random initial population that is then improved using selection, mutation, and crossover operations. everal ways exist to determine a stopping criterion for EAs but usually a predefined upper limit G max for the number of generations to be computed provides an appropriate stopping condition. Other control parameters for DE are crossover control parameter, mutation factor, and the population size NP. In each generation G, DE goes through each D dimensional decision vector x i,g of the population and creates corresponding trial vector u i,g as follows [9]: r, r, r {,,..., NP }, (randomly selected, except mutually different and different from i) j rand = floor (rand i [, ) D) + for(j = ; j D; j = j + ) { if(rand j [, ) < j = j rand ) u j,i,g = x j,r,g + (x j,r,g x j,r,g) else u j,i,g = x j,i,g } () Both and remain fixed during the whole execution of the algorithm. Parameter, controlling the crossover operation, represents the probability that an element for the trial vector is chosen from a linear combination of three randomly chosen vectors instead of from old vector x i,g. The condition j = j rand is to make sure that at least one element
is different compared to elements of the old vector. Parameter is a scaling factor for mutation and its value is typically (, +]. In practice, controls the rotational invariance of the search, and its small value (e.g..) is practicable with separable problems while larger values (e.g..9) are for non-separable problems. Control parameter controls the speed and robustness of the search, i.e., a lower value for increases the convergence rate but also the risk of stacking into a local optimum. Parameters and NP have the same kind of effect on the convergence rate as has. After the mutation and crossover operations trial vector u i,g is compared to old vector x i,g. If the trial vector has equal or better objective value, then it replaces the old vector in the next generation. DE is an elitist method since the best population member is always preserved and the average objective value of the population will never worsen. Generalized Differential Evolution everal extensions of DE for multi-objective optimization and for constrained optimization has been proposed [ 8]. Generalized Differential Evolution (E) [ 4, 7, 8] extends the basic DE algorithm for constrained multi-objective optimization by modifying the selection rule of the basic DE algorithm. Compared to the other DE extensions for multi-objective optimization, E makes DE suitable for constrained multi-objective optimization with noticeable fewer changes to the original DE algorithm. The modified selection operation for solving a constrained multi-objective optimization problem x i,g+ = u i,g minimize subject to {f ( x), f ( x),..., f M ( x)} (g ( x), g ( x),..., g K ( x)) T with M objectives f m and K constraint functions g k is presented formally as follows []: k {,..., K} : g k ( u i,g ) > x i,g if k {,..., K} : g k ( u i,g) g k ( x i,g) k {,..., K} : g k ( u i,g ) k {,..., K} : g k ( x i,g ) > k {,..., K} : g k ( u i,g ) g k ( x i,g ) m {,..., M} : f m ( u i,g ) f m ( x i,g ) otherwise (), () where g k( x i,g ) = max (g k ( x i,g ), ) and g k( u i,g ) = max (g k ( u i,g ), ) are representing the constraint violations.
The selection rule given in Eq. selects trial vector the next generation in the following cases: u i,g to replace old vector x i,g in. Both trial vector u i,g and old vector x i,g violate at least one constraint but the trial vector does not violate any constraint more than the old vector.. Old vector x i,g violates at least one constraint whereas trial vector u i,g is feasible.. Both vectors are feasible and trial vector u i,g has at least as good value for each objective than old vector x i,g has. Otherwise old vector x i,g is preserved. The basic idea in the selection rule given in Eq. is that trial vector u i,g is required to dominate compared old population member x i,g in a constraint violation space or in an objective function space, or at least provide an equally good solution as x i,g. After the selected number of generations the final population represents the solution for the optimization problem. Unique non-dominated vectors can be separated from the final population if desired. There is no sorting of non-dominated vectors during the optimization process or any mechanism for maintaining diversity of the solution. In the earlier studies [4,7,8] E was compared with other MOEAs. In these studies, it was observed that relatively small values for crossover control parameter and mutation factor should be used in the case of used multi-objective test problems. It was also noticed that suitable values for these control parameters should be drawn from a rather narrow range for some problems to obtain a converged solution along the Pareto front. However, performance was not thoroughly tested with commonly adopted metrics for MOEAs by applying different control parameter combinations. 4 Experiments In this investigation E was tested with a set of bi-objective benchmark problems commonly used in the MOEA literature. These problems are known as CH, CH, ON, KUR, POL, ZDT, ZDT, ZDT, ZDT4, and ZDT6 [9, pp. 8 6] and they all have two objectives to be minimized. With all the test problems the size of the population and the number of generations were kept fixed. rom the final population unique non-dominated solutions were extracted to form the final solution, an approximation of the Pareto front. Characteristics of the problems, the size of the population, and the number of generations are collected to Table. Relatively small number of generations was used for some problems to observe the effect of the control parameters. Different control parameter value combinations in the range [, ] and [, ] with a resolution of.5 were tested for all the test problems. Tests were repeated times with each control parameter combination and the results were evaluated using 4
Problem D eparability Pf, decision space Pf, objective space NP G max CH N/A line curve, convex CH N/A lines curves 5 ON separable line curve, non-convex 5 KUR separable points & 4 curves point & curves 5 POL non-separable curves curves ZDT separable line curve, convex 5 ZDT separable line curve, concave 5 ZDT separable 5 lines 5 curves 5 ZDT4 separable line curve, convex 5 ZDT6 separable line curve, concave 5 Table : Characteristics of the bi-objective optimization test problems (Pf = Pareto front) [9, pp. 8 6] [, pp. 6 8, 4 5], the size of the population, and the number of generations. convergence and diversity metrics. Closeness to the Pareto front was measured with a generational distance () [9, pp. 6 7], which measures the average distance of the solution vectors from the Pareto front. Diversity of the obtained solution was measured using a spacing () metric [9, pp. 7 8], which measures the standard deviation of the distances from each vector to the nearest vector in the obtained non-dominated set. maller values for both metrics are preferable, and the optimal values are zero. The number of unique non-dominated vectors () in the final solution was also registered. Results for the different problems are shown in igs. as surfaces in the control parameter space. The results for all the problems show that with a small value the final solution has greatest amount of unique non-dominated vectors and also values for performance metrics and are best. or the multidimensional (i.e. D > ) test problems, values for the performance metrics are best with low values, whereas for the one-dimensional test problems, does not seem to have notable impact on the solution. The value range for in the tests is larger than usually used for single-objective optimization. Based on the results, a larger value does not give any extra benefit in the case of multi-objective optimization and therefore its value can be chosen from the same value range as in the case of single-objective optimization. rom the results of the multidimensional test problems, one can observe a non-linear relationship between and values, i.e., a larger value can be used with a small value than with a large value, and this relationship is non-linear. An explanation for this can be found from a theory for the single-objective DE algorithm. A formula for the relationship between the control parameters of DE and the evolution of the population variance has been conducted in []. The change of the population variance between 5
CH, Cardinality CH, Generational Distance CH, pacing 8 6. 6 4.5.5.5.5 log ( ) 4.5.5.5.5.5..5..5.5.5 ء.5.5 ز س igure : Mean values of the number of unique non-dominated vectors in the final solution, generational distance (note logarithmic scale), and spacing for CH shown as surfaces in the trange shape of the spacing surface is due to only a few non-dominated vectors when is large. CH, Cardinality CH, Generational Distance CH, pacing 5.4.8.7 4..6.5...5.5.4..5.5.5.5.5.5.5.5.5.5 igure : Mean values of the number of unique non-dominated vectors in the final solution, generational distance, and spacing for CH shown as surfaces in the successive generations due to the crossover and mutation operations is denoted with c and its value is calculated as c = /NP + /NP +. When c <, the crossover and mutation operations decrease the population variance. When c =, the variance do not change, and when c >, the variance increases. ince a selection operation of an EA usually decreases the population variance, c > is recommended to prevent too high convergence rate with a poor search coverage, which typically results in a premature convergence. On the other hand, if c is too large, the search process proceeds reliably, but too slowly. When the size of the population is relatively large (e.g. NP > 5), the value of c depends mainly on the values of and. Curves c =. and c =.5 for NP = in the control parameter space are shown in ig.. These curves are also shown with 6
ON, Cardinality ON, Generational Distance ON, pacing 6 5 4.5.5..5..5..5.5.5..5..5.5.5.5.5.5.5.5.5.5.5 igure : Mean values of the number of unique non-dominated vectors in the final solution, generational distance, and spacing for ON shown as surfaces in the KUR, Cardinality KUR, Generational Distance KUR, pacing 6.5 5 4..5..5..5..5.5.5.5.5.5.5.5.5.5.5.5.5.5 igure 4: Mean values of the number of unique non-dominated vectors in the final solution, generational distance, and spacing for KUR shown as surfaces in the the and surfaces in igs.. As it can be observed, the surface for c in ig. corresponds well with the surfaces for and in most of the test problems proposing that the theory for the population variance between successive generations is applicable also in the case of multi-objective optimization with E. This is natural, since singleobjective optimization can be seen as a special case of multi-objective optimization, and a large c means slow but reliable search while a small c means opposite. In general, the results are good with small and values meaning also a low c value close to one. One reason for the good performance with small control parameter values is that the solution of a multi-objective problem is usually a set of non-dominated vectors instead of one vector, and conflicting objectives of the problem reduce overall selection pressure, maintain the diversity of the population, and prevent the premature convergence. Another reason is that all the multidimensional test problems here are separable or almost separable. With strongly non-separable objective functions, a larger 7
POL, Cardinality POL, Generational Distance POL, pacing 4.4. 5 5 5.5..5..5.8.6.4.5..5.5.5.5.5.5.5.5.5.5.5 igure 5: Mean values of the number of unique non-dominated vectors in the final solution, generational distance, and spacing for POL shown as surfaces in the ZDT, Cardinality ZDT, Generational Distance ZDT, pacing.8. 8 6 4.5.6.4..5.5..5.5.5.5.5.5.5.5.5.5.5 igure 6: Mean values of the number of unique non-dominated vectors in the final solution, generational distance, and spacing for ZDT shown as surfaces in the might work better. Also, most of the problems have relatively easy objective functions to solve leading faster convergence with a small, while a bigger might be needed for harder functions with several local optima. The results for most of the ZDT problems show that the value of control parameter must be kept low (e.g...) to obtain good results. This is due to the fact that in the ZDT problems, the first objective depends on one decision variable while the second objective depends on rest of the decision variables. This leads easily to a situation when the first objective gets optimized faster than the second one and the solution converges to a single point on the Pareto front since in E there is no mechanism for maintaining diversity of the solution. Therefore, if the optimization difficulty for different objectives varies, then the value for must be close to zero to prevent the solution converge to a single point of the Pareto-optimal front. Based on the results, our recommendations for the control parameter values are 8
ZDT, Cardinality ZDT, Generational Distance ZDT, pacing 6 5 4.5.4...5.5.5.5.5.5.5.5.5..5.5.5.5 igure 7: Mean values of the number of unique non-dominated vectors in the final solution, generational distance, and spacing for ZDT shown as surfaces in the ZDT, Cardinality ZDT, Generational Distance ZDT, pacing 8 6.8.6.4.. 4.5.4..5.8.6.4..5.5.5.5.5.5.5.5.5.5 igure 8: Mean values of the number of unique non-dominated vectors in the final solution, generational distance, and spacing for ZDT shown as surfaces in the [.5,.8] and [.,.6] for initial settings, and it is safer to use smaller values for. However, these recommendations are based on the results of the used test functions. In overall, it is wise to select values for control parameters and satisfying a condition. < c <.5. 5 Conclusions An extension of Differential Evolution (DE), Generalized Differential Evolution (E), is described. E has the same control parameters, crossover control parameter and mutation factor, as the basic DE has. uitable values for these control parameters has been reported earlier to be different compared to those usually used with the basic single-objective DE, but no extensive study was performed. 9
ZDT4, Cardinality ZDT4, Generational Distance ZDT4, pacing 8 6 5 4.5..5. 4.5.5.5..5.5.5.5.5.5.5.5.5.5.5 igure 9: Mean values of the number of unique non-dominated vectors in the final solution, generational distance, and spacing for ZDT4 shown as surfaces in the ZDT6, Cardinality ZDT6, Generational Distance ZDT6, pacing 4 8 6 4...5.5.5.5.5.5.5.5..5.5.5.5 igure : Mean values of the number of unique non-dominated vectors in the final solution, generational distance, and spacing for ZDT6 shown as surfaces in the Different control parameter values for E were tested using common bi-objective test problems and metrics measuring the convergence and diversity of the solutions. Based on the empirical results, suitable initial control parameter values are [.5,.8] and [.,.6]. The results also propose that a larger value than usually used in the case of single-objective optimization does not give any extra benefit in the case of multiobjective optimization. It also seems that in some cases it is better to use a smaller value to prevent the solution converge to a single point of the Pareto front. However, these results are based on mostly separable problems with conflicting objectives. Also in the case of multi-objective optimization, the non-linear relationship between and was observed according to the theory of the basic single-objective DE about the relationship between the control parameters and the evolution of the population variance. It is advisable to select values for and satisfying the condition. < c <.5.
.5 4.5.5.5 c =.5 c =...4.6.8 c.5.5.5.5.5.5..5 igure : Curves c =. and c =.5 for NP = in the In the future, the correlation between c and performance metrics should be studied also numerically. urthermore, the further investigation of the generalization of the theory for the evolution of the population variance to multi-objective problems is necessary. Also, extending studies for strongly non-separable multi-objective problems and problems with more than two objectives remains to be studied. pecial cases, as multi-objective problems without conflicting objectives and single-objective problems transformed into multi-objective forms, might be interesting to investigate. Acknowledgments. Kukkonen gratefully acknowledges support from the East inland Graduate chool in Computer cience and Engineering (ECE), the Academy of inland, Technological oundation of inland, and innish Cultural oundation. He also wishes to thank Dr. K. Deb and all the KanGAL people for help during his visit in pring 5. References [] Jouni Lampinen. DE s selection rule for multiobjective optimization. Technical report, Lappeenranta University of Technology, Department of Information Technology,. [Online] Available: http://www.it.lut.fi/kurssit/-4/778/mode.pdf, 5..6. [] Jouni Lampinen. Multi-constrained nonlinear optimization by the Differential Evolution algorithm. Technical report, Lappeenranta University of Technology, Department of Information Technology,. [Online] Available: http://www.it.lut.fi/kurssit/-4/778/decontr.pd, 5..6.
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