Performance Assessment of the Hybrid Archive-based Micro Genetic Algorithm (AMGA) on the CEC09 Test Problems

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1 Perormance Assessment o the Hybrid Archive-based Micro Genetic Algorithm (AMGA) on the CEC9 Test Problems Santosh Tiwari, Georges Fadel, Patrick Koch, and Kalyanmoy Deb 3 Department o Mechanical Engineering, Clemson University, SC, USA Dassault Systemes, Simulia Corp., Cary, NC, USA 3 Indian Institute o Technology, Kanpur, UP, India Abstract In this paper, the perormance assessment o the hybrid Archive-based Micro Genetic Algorithm (AMGA) on a set o bound-constrained synthetic test problems is reported. The hybrid AMGA proposed in this paper is a combination o a classical gradient based single-objective optimization algorithm and an evolutionary multi-objective optimization algorithm. The gradient based optimizer is used or a ast local search and is a variant o the sequential quadratic programming method. The Matlab implementation o the SQP (provided by the mincon optimization unction) is used in this paper. The evolutionary multi-objective optimization algorithm AMGA is used as the global optimizer. A scalarization scheme based on the weighted objectives is proposed which is designed to acilitate the simultaneous improvement o all the objectives. The scalarization scheme proposed in this paper also utilizes reerence points as constraints to enable the algorithm to solve non-convex optimization problems. The gradient based optimizer is used as the mutation operator o the evolutionary algorithm and a suitable scheme to switch between the genetic mutation and the gradient based mutation is proposed. The hybrid AMGA is designed to balance local versus global search strategies so as to obtain a set o diverse non-dominated solutions as quickly as possible. The simulation results o the hybrid AMGA are reported on the bound-constrained test problems described in the CEC9 benchmark suite. I. INTRODUCTION Multi-objective optimization has become mainstream in recent years and many algorithms to solve multi-objective optimization problems have been suggested. The use o multi-objective optimization in the industry has been accelerated by the availability o aster processing units and computational analysis tools or various engineering problems and disciplines. The ever increasing popularity o multi-objective optimization in industry and the need or aster optimization algorithms has led to the development o several multi-objective optimization algorithms (MOEAs) in the recent past [], [], [3], [4], [5], [6], [7], [8]. Whenever, a new multi-objective optimization algorithm is proposed, a perormance comparison o the proposed algorithm with the current state-o-the-art optimization algorithms is perormed, and generally the new algorithm is shown to be aster than the existing algorithms on a set o careully chosen synthetic benchmark problems. A special session on the perormance assessment o dierent MOEAs thus provides an opportunity or a air and unbiased comparison o dierent multi-objective optimizers. In this report, the perormance o the hybrid Archive-based Micro Genetic Algorithm (AMGA) on the unconstrained test problems described in the CEC9 technical report [9] is reported. The AMGA [] is a constrained multi-objective evolutionary optimization algorithm. It is a generational genetic algorithm since during a particular iteration (generation), only solutions created beore that iteration take part in the selection process. AMGA uses genetic variation operators such as crossover and mutation to create new solutions. For the purpose o selection, AMGA uses a two tier itness assignment mechanism; the primary itness is the rank which is based on the domination level and the secondary itness is based on the diversity o the solutions in the entire population. This is in contrast to NSGA-II, where the diversity is computed only among the solutions belonging to the same rank. The AMGA generates a very small number o new solutions at every iteration and can thereore be classiied as a micro genetic algorithm. Generation o a very small number o solutions at every iteration helps in reducing the number o unction evaluations by minimizing exploration o less promising search regions and directions. The AMGA maintains an external archive o good solutions obtained. Use o the external archive helps AMGA in reporting a large number o non-dominated solutions at the end o the simulation and also provides inormation about its search history which is exploited by the algorithm during the selection operation. At every iteration, the parent population is created rom the archive and binary tournament selection is perormed on the parent population to create the mating population. The ospring population is created rom the mating pool, and is used to update the archive. The size o the archive determines the computational complexity o the AMGA, however or computationally expensive optimization problems, the actual time taken by the algorithm is negligible as compared to the time taken by the analysis routines. The design o the algorithm is independent o the encoding o the variables and thus the proposed algorithm can work with almost any kind o encoding (so long as suitable genetic variation operators are provided to the algorithm). The algorithm uses the concept o Pareto ranking borrowed rom NSGA-II [3] and includes improved diversity computation and preservation techniques. The diversity measure is based on eicient nearest neighbor search [] and modiied crowding distance ormulation []. A more detailed description o the AMGA can be ound in the original study []. In this paper, only the modiications

2 done to AMGA to couple a gradient based optimizer are discussed. The gradient-based local optimizer used with AMGA is the Sequential Quadratic Programming (SQP) algorithm []. SQP is one o the most popular and robust algorithms or constrained nonlinear single-objective optimization. Although, the SQP is capable o solving constrained test problems, we report the simulation results only on the unconstrained problems because the scalarization scheme proposed in this paper works only with the unconstrained problems. The SQP algorithm attempts to approximate the objective unction using a quadratic model and the constraint unctions using a linear model o the optimization variables. The SQP algorithm requires the computation o the Hessian o the objective vector which is approximated using the BFGS method []. SQP has excellent local convergence properties and is shown to be aster than most other gradient based optimizers on a large set o test problems [3]. The application o SQP or multi-objective optimization requires scalarization o the objective vector. In order to ensure that the SQP works with non-convex problems, artiicial constraints are added to ensure that improvement in all the objectives is observed. The additional constraints ensure that the solution obtained using SQP always dominates the initial (starting) solution. In the worst case (when the starting solutions happens to be the local optimum), no improvement is observed. It should be noted that the addition o SQP to the AMGA does not directly aect its global search capability. It however speeds up the local search process thereby allowing more unction evaluations to be used or the global search. It should be noted that hybridizing evolutionary algorithms with mathematical programming techniques has been attempted in the past [4], [5], [6], [7]. Hybrid evolutionary algorithms are also oten reerred to as memetic algorithms owing to their use o local search techniques which are traditionally aster than a typical evolutionary algorithm. The novel concept proposed in this paper is the use o a starting reerence point to enable the local optimizer to attempt to simultaneously improve all the objectives and also work with non-convex problems. The reerence point is used to ormulate the additional constraints such that only the region dominated by the starting point is easible. The scalarization scheme employed in this paper and the modiication done to the AMGA to incorporate the SQP algorithm are discussed in the next section. II. HYBRIDIZATION OF THE AMGA A. The Scalarization Scheme The scalarization scheme to convert the multi-objective optimization problem to a single-objective optimization problem is discussed irst. The scalarization scheme also includes the reerence-point method to ensure its applicability to nonconvex problems. The speciic way in which the reerencepoint method is used here limits its applicability to unconstrained problems or constrained problems or which the starting (initial) solution is easible. The general problem statement or the unconstrained (unconstrained in this case reers to bound-constrained) multi-objective optimization is given by Equation. Minimize ( (x), (x),..., M (x)), Subject to x (L) i x i x (U) () i, i =,,...,n. Let the starting (initial) solution or the optimization be x initial. Let the value o the objective vector at the starting point x initial be initial. We want to use SQP with the initial guess solution as x initial, and obtain a new solution x inal such that the corresponding objective vector inal dominates initial.weuse initial as the reerence point in the scalarization scheme. The corresponding single-objective optimization problem is given by Equation. Minimize M i= i Subject to j (x) ( initial ) j, j =,,...,M, x (L) i x i x (U) i, i =,,...,n. The easible search region or the scalarized test problem is shown in Figure. One inequality constraint is added or each objective unction. The additional o inequality constraints ensures that the inal obtained solution dominates the starting solution when using SQP. I the constraints are not introduced, and the problem is non-convex, application o SQP may result in a solution which is non-dominated with respect to the starting solution. This scenario is not desirable, since it would not speed-up the convergence towards the Pareto-optimal rontier. It should be noted that there exist several other scalarization schemes such as Epsilonconstraint method [8] and Normal-constraint method [8]. These scalarization schemes are designed to obtain a uniorm distribution o points on the Pareto-optimal ront. In this case however, the uniormity o distribution o the points on the Pareto-optimal ront is taken care-o by the global optimizer AMGA. The SQP is used to speed-up the search process and obtain an improvement in the objective unction value as quickly as possible. It is also possible to use other weighting schemes such as Tchebyche metric [9] or scalarizing the optimization problem. No noticeable change in perormance was observed by using dierent weighting schemes. For the scalarization scheme to work eiciently, it is important that the objectives be normalized beore computing the weighted objective vector. The normalization o the objectives is done by linearly scaling the objectives or every solution in the current population in the range zero to one. The minimum and maximum value o every objective unction in the current population is used or the scaling operation. B. Incorporating SQP into AMGA The application o SQP to the scalarized single-objective optimization problem described in the previous subsection is akin to starting rom an initial solution and obtaining a inal solution such that it is a local optimum. It should be noted that i the starting point does not correspond to a local optimum, then there always exists a easible search direction ()

3 Feasible search region Fig.. Feasible search space Reerence point SQP is also speciied. This modiication is applied or a single generation. SQP is then applied to every solution in the ospring population. Once the iteration o SQP has inished and new solutions are obtained, the archive is updated using the obtained solutions. The mutation operator is switched back to the polynomial mutation. Again, a ixed number o generations is perormed, and the process is repeated until the allowed number o unction evaluations is exhausted. I at any instant, the limit on number o unction evaluations is reached, the optimization process is terminated, and the nondominated solutions in the archive are reported as the inal obtained solutions. The total number o unction evaluations includes the unction evaluations perormed by AMGA and the SQP. The pseudo-code o the hybridized AMGA is as ollows. which will improve all the objectives simultaneously. This easible search direction will guide the optimization process towards the nearest local optimum. Further, SQP is one o the astest known methods [3] to ind the local optimum o constrained nonlinear single-objective optimization problems. Thus, i this step is incorporated as the mutation step o an evolutionary optimization algorithm, mutating the ospring solution will drive that solution towards to nearest local optimum. A potential drawback o this approach is that every time a solution is mutated, it will hit its nearest local optimum. Also, all the solutions in the ospring population will get accumulated at their respective local optima. For highly multi-modal problems, it is desired that a disruptive genetic mutation operator is also incorporated such that a robust search or the global optimum is acilitated. Further, once the SQP is applied to a ew selected solutions rom the archive, a set o good solutions are obtained. It is then desired to explore the search space around those solutions and improve the diversity near the obtained solutions. Based on empirical investigation and experimentation with dierent schemes, it was observed that switching between the genetic polynomial mutation [] operator and the SQP algorithm at regular intervals (ater every ew generations) resulted in a search strategy which balanced diversity o the obtained solutions, the convergence rate, and the global search capability o the optimization algorithm. The generation scheme o the AMGA was modiied to incorporate the SQP algorithm. The optimization process starts with an initial population generated randomly using Latin Hypercube [] sampling. Selection, simulated binary crossover [], and polynomial mutation [] are then perormed or a ixed number o generations. Selection operation creates the parent population rom the archive and the genetic variation operators create the ospring population rom the parent population. At every generation, the ospring population is used to update the archive. Ater a ixed number o generations has been completed, the genetic mutation operator is replaced by the SQP method and the multi-objective problem is scalarized. The maximum number o unction evaluations allowed or The hybrid AMGA pseudo-code: Begin Initialize optimization parameters. 3 Generate initial population. 4 Evaluate initial population. 5 Update the archive (using the initial population). 6 repeat 7 Create parent population rom the archive. 8 Create mating pool rom the parent population. 9 I generation count a multiple o switch requency Create ospring population rom the mating pool by crossover. Mutate the ospring population. Evaluate ospring population. 3 Else 4 Use SQP on every individual in the ospring population. 5 Update the number o unction evaluations. 6 Update the archive (using the o-spring population). 7 until (termination) 8 Report desired number o solutions rom the archive. 9 End From the above pseudo-code, it is evident that the basic lowchart o the AMGA remains unchanged except an IF statement at step 9. Hence, the addition o SQP does not change the generational scheme employed by the AMGA. From an algorithm design perspective, any local search strategy can be used in step 4 o AMGA so long as it acilitates relatively aster convergence towards to nearest local optimum.

4 III. SIMULATION RESULTS The simulation results o the hybrid AMGA applied to bound-constrained multi-objective test problems are presented in this section. The ollowing optimization tuning parameters are used to report the simulation results. Size o the initial population or objectives = Size o the initial population or 3 objectives = 5 Size o the parent population or objectives = 3 Size o the parent population or 3 objectives = 4 Size o the archive = size o the initial population Number o solutions reported at the end o the simulation = size o the initial population Number o unction evaluations = T = 3, Probability o crossover =. Probability o mutation = /N, wheren is the number o optimization variables Distribution index or crossover =.5 Distribution index or mutation =.5 Number o generations or switching the mutation operator = T / Number o unction evaluations allowed or each SQP iteration = T / The typical value o the crossover and mutation indices used with a genetic algorithm is in the range 5 to 5. The smaller the value o indices, the larger is the perturbation in the design variables. The indices are chosen so as to balance the disruptiveness (required or ast and robust search) o the genetic variation operators whilst attempting to ind a ine-grained value or the objective unctions. Since SQP is used with the AMGA, a ine-grained (accurate) value or the objective unctions is ensured. Hence, in the present case, a small value (.5) or the crossover and mutation index is used because it acilitates robust search and increases the resilience to premature convergence. It thus reduces the probability o getting stuck at a local optimum. The other simulation related parameters are as ollows. Operating system: Windows XP Proessional Programming language or AMGA: JAVA Runtime or SQP: Matlab JVM CPU: Core Quad.4 GHz RAM: 4 GB DDR 66 MHz Execution time or single simulation or -objective test problems: 3 minutes approx. Execution time or single simulation or 3-objective test problems: 6 minutes approx. The execution time or the problems with three objectives is higher because o the larger size o the archive. The CEC9 test problem suite also includes three 5-objective test problems. The simulation with 5-objective test problems could not be perormed because o sotware related issues encountered when linking Matlab, C++, and JAVA. The perormance indicator used to quantiy the quality o the obtained results is the IGD metric [9]. The IGD metric measures how well is the Pareto-optimal ront represented by the obtained solution set. To quantiy this inormation, a large set o evenly spaced points on the Pareto-optimal ront is generated. Let the size o this set be H. The minimum Euclidean distance o each point in this set rom the obtained solution set is computed. Let this distance be l i or the i th element o the Pareto-optimal set. Then the IGD metric is given by IGD metric = H l i (3) H i= The IGD metric or the case o two objectives is pictorially depicted in Figure. The IGD metric measures both the convergence and the spread o the obtained solutions. Smaller the value o the IGD metric, better is the obtained solution set. l l Fig.. 3 random simulations are perormed or each problem and the minimum (Min), maximum (Max), mean, and standard deviation (Std) o the IGD metric are reported in Table I. TABLE I THE IGD METRIC Problem Min Max Mean Std UF UF UF UF UF UF UF UF UF UF The plots o the Pareto-optimal ront or all the test problems are shown in Figures 3 to. The plots o the mean o the IGD metric with the number o unction evaluations or all the test problems are shown in Figures 3 to. IV. DISCUSSION, CONCLUSION, AND FUTURE WORK As is evident rom Table I, the hybrid AMGA is able to ind an approximate solution set near the the global Paretooptimal ront or most problems. In most test problems, l H

5 Fig. 3. Pareto ront or problem UF Fig. 5. Pareto ront or problem UF Fig. 4. Pareto ront or problem UF Fig. 6. Pareto ront or problem UF4 global convergence is obtained but the complete Paretooptimal rontier is not discovered by the hybrid AMGA. The primary cause o this behavior is the objective unction proile which is multi-modal near the global Pareto-optimal rontier, and a slight perturbation in the optimization variables causes the solutions to become dominated. Also, the phenomenon o genetic drit causes the population to ollow the good solutions which get discovered early in the search process. This genetic drit results in the clustering o the solutions around these points. With the hybrid AMGA proposed in this paper, it is not possible to get the IGD metric at every unction evaluation. It is thereore not possible to get the mean o the IGD metric at every unction evaluation. Further, due to the inclusion o the SQP algorithm in the AMGA, the number o unction evaluations exhausted at any generation is dierent, and thereore the IGD metric cannot be computed at the same number o unction evaluations or dierent simulations starting with dierent random seeds. Hence, cubic spline interpolation was used to generate the plots o the mean value o the IGD metric. The cubic spline interpolates all the data points, and does not have overshoots. It thus gives an accurate interpolation o the data points. It is also evident rom the convergence plots that the IGD metric does not always monotonically decrease with the increase in the number o unction evaluations. This is due to way the AMGA algorithm works. The parent population is created rom the archive using only the diversity in the variable space. Hence, instead o picking the best solutions, it picks

6 ..6. Fig. 7.. Pareto ront or problem UF5..6. Fig. 9. Pareto ront or problem UF Fig.. Pareto ront or problem UF8 Fig. 8. Pareto ront or problem UF6. the most diverse solutions. Such a strategy helps in the case o multi-modal problems, but does not always improve the convergence measure. The plot o the mean o the IGD metric or the case o UF5 (Figure 7) shows that the smallest (best) value o the mean o the IGD metric is., however in the Table I, the mean value o the IGD metric is.94. This apparent discrepancy is due to the act that, the smallest value o the IGD metric need not be obtained at the end o the simulation, nor is its behavior monotonic with the number o unction evaluations or the case o hybrid AMGA. For every simulation, the best value o the IGD metric obtained at any stage o the simulation is reported as the IGD metric or that simulation. The mean value in the Table I represents the mean o the reported IGD metrics or each simulation. 3.. Fig.. Pareto ront or problem UF9

7 Fig. 9. Mean o the or problem UF7 Fig.. Mean o the or problem UF8 Fig.. Pareto ront or problem UF Fig. 3. Mean o the or problem UF.. 3 Fig. 5. Mean o the or problem UF3. 3 Fig. 7. Mean o the or problem UF5.. 3 Fig. 4. Mean o the or problem UF.. 3 Fig. 6. Mean o the or problem UF4. 3 Fig. 8. Mean o the or problem UF6 Fig.. Mean o the or problem UF9 Fig.. Mean o the or problem UF For generating the plots o the mean o the IGD metric, the mean is computed by using the IGD metric at that unction evaluation. Hence, the apparent discrepancy is due to the act that dierent simulations starting with dierent random seeds do not achieve their respective best value o the IGD metric simultaneously. It was also observed in many simulations, that a reasonably good convergence was obtained in under, unction evaluations. And, the additional, unction evaluations did not result in any signiicant improvement in the obtained solution set. This behavior is evident in test problems UF, UF3, UF5, UF6, UF7, UF8, and UF9. Also, aster reduction in the value o the IGD metric can be observed at certain stages or dierent test problems. This behavior can be attributed to the SQP algorithm. The SQP in certain cases results in signiicantly improved objective unction values which causes a sudden decrease in the value o the IGD metric. Overall, the proposed hybrid AMGA perorms reasonably well on the test problems used or this study. The inclusion o SQP speeds-up the search process and also helps in obtaining a ine-grained value or the objective unctions. Some limitations o the proposed hybrid AMGA also became evident during the simulation process which are mentioned below. In most cases, the global Pareto-optimal ront was ound, and the extreme solutions were discovered, however the diversity o the obtained solutions was not good. Better diversity preservation operators are needed to approximate and represent the entire Pareto-optimal ront. The scalarization scheme proposed to convert the multiobjective problem to a single-objective problem cannot

8 handle constrained test problems. In highly multi-modal problems, the search process oten gets stuck in a locally optimal basin, and urther improvement is not observed. These limitations provide an opportunity or the urther improvement o the proposed hybrid AMGA which shall be the ocus o the uture research. The hybridized AMGA algorithm and related resources can be downloaded rom stiwari/amga.html REFERENCES [7] S. Z. Martinez and C. A. C. Coello, Hybridizing an evolutionary algorithm with mathematical programming techniques or multi-objective optimization, in Genetic and Evolutionary Computation Conerence. ACM New York, NY, USA, 8, pp [8] A. Messac and C. A. Mattson, Normal constraint method with guarantee o even representation o complete pareto rontier, in press: AIAA Journal. [9] K. Deb, Multi-objective Optimization Using Evolutionary Algorithms. Chichester, UK: Wiley,. [] K. Deb and M. Goyal, A combined genetic adaptive search (geneas) or engineering design, Computer Science and Inormatics, vol. 6, no. 4, pp. 3 45, 996. [] W. L. Loh, On latin hypercube sampling, Annals o Statistics, vol. 33, no. 6, pp. 58 8, 5. [] K. Deb and R. B. Agrawal, Simulated binary crossover or continuous search space, Complex Systems, vol. 9, no., pp. 5 48, 995. [] E. Zitzler, M. Laumanns, and L. Thiele, SPEA: Improving the strength pareto evolutionary algorithm or multiobjective optimization, in Proceedings o the EUROGEN Conerence,, pp. 95. [] D. W. Corne, J. D. Knowles, and M. J. Oates, The pareto envelopebased selection algorithm or multi-objective optimization, in Parallel Problem Solving rom Nature. Berlin: Springer,, pp [3] K. Deb, S. Agrawal, A. Pratap, and T. Meyarivan, A ast and elitist multi-objective genetic algorithm: NSGA-II, IEEE Transactions on Evolutionary Computation, vol. 6, no., pp. 8 97,. [4] S. Watanabe, T. Hiroyasu, and M. Miki, NCGA: Neighborhood cultivation genetic algorithm or multi-objective optimization problems, in Proceedings o the Genetic and Evolutionary Computation Conerence GECCO,, pp [5] K. Deb and S. Tiwari, Omni-optimizer, a generic evolutionary algorithm or single and multi-objective optimization, European Journal o Operational Research, vol. 85, no. 3, pp. 6 87, 8. [6] H. Eskandari, C. D. Geiger, and G. B. Lamont, Fastpga: A dynamic population sizing approach or solving expensive multiobjective optimization problems, in Evolutionary Multiobjective Optimization Conerence EMO-7. LNCS 443 Springer-Verlag Berlin Heidelberg, 7, pp [7] L. Marti, J. Garcia, A. Berlanga, and J. M. Molina, Introducing moneda: Scalable multiobjective optimization with a neural estimation o distribution algorithm, in Genetic and Evolutionary Computation Conerence. ACM New York, NY, USA, 8, pp [8] O. Schuetze, G. Sanchez, and C. A. C. Coello, A new memetic strategy or the numerical treatment o multi-objective optimization problems, in Genetic and Evolutionary Computation Conerence. ACM New York, NY, USA, 8, pp [9] Q. Zhang, A. Zhou, S. Zhao, P. N. Suganthan, W. Liu, and S. Tiwari, Multi-objective optimization test instances or the cec 9 special session and competition, IEEE Congress on Evolutionary Computation, Tech. Rep., 8. [] S. Tiwari, G. Fadel, P. Koch, and K. Deb, Amga: An archive-based micro genetic algorithm or multi-objective optimization, in Genetic and Evolutionary Computation Conerence. ACM New York, NY, USA, 8, pp [] M. Soleymani and S. Morgera, An eicient nearest neighbor search, IEEE Transaction on Communications, vol. 35, no. 6, pp , 987. [] K. Schittkowski, Nlpql: A ortran subroutine solving constrained nonlinear programming problems, Annals o Operations Research, vol. 5, no. 4, pp , 986. [3], Test problems or nonlinear programming - user s guide, University o Bayreuth, Tech. Rep.,. [4] P. A. N. Bosman and E. D. Jong, Exploiting gradient inormation in numerical multi-objective optimization, in Genetic and Evolutionary Computation Conerence. ACM New York, NY, USA, 5, pp [5] X. Hu, Z. Huang, and Z. Wang, Hybridization o the multi-objective evolutionary algorthms and the gradient-based algorithms, in IEEE Congress on Evolutionary Computation, 3, pp [6] D. Sharma, A. Kumar, K. Deb, and K. Sindhya, Hybridization o SBX based NSGA-II and sequential quadratic programming or solving multi-objective optimization problems, in IEEE Congress on Evolutionary Computation, 7, pp