Robust Fitness Landscape based Multi-Objective Optimisation

Transcription

1 Preprints of the 8th IFAC World Congress Milano (Italy) August 28 - September 2, 2 Robust Fitness Landscape based Multi-Objective Optimisation Shen Wang, Mahdi Mahfouf and Guangrui Zhang Department of Automatic Control and Systems Engineering, The University of Sheffield Sheffield, S 3JD, UK ( shen.wang@shef.ac.uk; m.mahfouf@shef.ac.uk; guangrui.zhang@shef.ac.uk). Abstract: Multi-objective optimisation is a challenging and essential topic of both engineering and computation fields. Population-based approaches are able to achieve a set of solutions as an approximation to the Pareto front and have therefore become popular. However, these algorithms have to use a significant number of evaluations to reach convergence, which can be time-consuming and/or economically expensive as far as real-world applications are concerned. Recently, an efficient hybrid framework, named EMO-FL, which combines a state-of-the-art multi-objective optimiser with an embedded approximate model of the objective functions, has been proposed to handle such problems. In this paper, a new multiobjective optimiser (DBSGA) is developed and successfully exploited to enhance the performance of this EMO-FL framework. Successful applications of the newly proposed algorithm include well-known challenging benchmark functions and the optimal design of alloy steels via the manipulation of chemical equations and processing conditions. Keywords: Multiobjective optimisation, Fitness Landscape, Evolutionary algorithms, Robustness, Search methods, Steel industry.. INTRODUCTION In a variety of real-world applications one should face the scenario where several objectives have to be optimised simultaneously. More often than not, objectives may conflict with each other leading to Multi-Objective Optimisation Problems (MOOPs). Solving MOOPs consists of finding a set of well distributed Pareto-optimal solutions along the Pareto front(s), which can represent the behaviour of all possible trade-off solutions. Evolutionary Algorithms (EAs) that maintain a population of solutions in each generation, instead of a single solution, are therefore able to address MOOPs in a single run. However, most of Multi-Objective Evolutionary Algorithms (MOEAs) require a number of evaluations to locate the Pareto-optimal solutions, which can be time-consuming or economically not viable. Hence, desired outcomes have to be reached using as a few evaluations as possible. The expensive optimisation has hitherto attracted increasingly more attention. One approach to handle such problems is to utilise a model to estimate the fitness of some or all of offsprings, known as fitness landscape in evolutionary optimisation. Some successful implementations can be found in the literature (Knowles, 25), (Nain and Deb, 23), and (Regis and Shoemaker, 24). A hybrid framework that combines a global model of the search space with the evolutionary optimisation paradigm was proposed and named EMO-FL (Efficient Multi-objective Optimisation with Fitness Landscape) (Wang and Mahfouf, 2). EMO-FL makes use of a global Multi-Layer Perceptron (MLP) model associated with Dynamic Selection of training data to accelerate NSGA-II (Deb et al., 22). The goal of this paper is to extend the framework of EMO-FL and enhance the efficiency of solving the problems that have many local Pareto-optimal fronts and the accuracy of the embedded MLP model. Firstly, a newly proposed MOEA is introduced to replace NSGA-II in the original EMO-FL framework. Secondly, the performance of the enhanced version is evaluated via a series of experiments carried-out on chosen test functions. This new version of EMO-FL is finally applied to the optimal design of alloy steels with respect to pre-specified mechanical properties, such as tensile strength and reduction of area. This paper is organised as follows. Section 2 presents a brief review of the original version of EMO-FL, including its achievements and shortcomings where improvements can be made accordingly. Section 3 outlines the methodology of the enhanced EMO-FL framework associated with a new sorting and selection scheme. Section 4 describes the simulation results conducted on some well-known benchmark test suites. Section 5 introduces the application of the proposed algorithm in the metal-making industry. Finally, Section 6 draws some pertinent conclusions and outlines potential future work. 2. REVIEW AND MOTIVATION The EMO-FL framework was originally implemented via a combination of NSGA-II and an MLP model. Compared with previous research (Deb et al., 2), this approach is able to locate the Pareto-optimal solutions of ZDT (Zitzler et al., 2) and DTLZ (Deb et al., 2) problems using fewer evaluations. EMO-FL also outperforms NSGA-II in the global optimisation due to the inclusion of the Controlled Reseeding operator. However, such benefits come with a price of a relatively complicated structure. Despite these achievements, some disadvantages of the employed optimiser may seriously influence the performance of the resulting EMO-FL. One crucial problem is the accuracy of the global MLP model, which affects the total performance significantly. Provided that the MLP structure is defined appropriately and an effective method for selecting training data is available, Copyright by the International Federation of Automatic Control (IFAC) 29

2 Preprints of the 8th IFAC World Congress Milano (Italy) August 28 - September 2, 2 the accuracy of the model largely depends on the distribution of training data over the entire search space. Within the framework of EMO-FL, all data used for training are the points that have been visited by the optimiser in past generations. However, the sorting and selection schemes of NSGA- II are inherently fitness-preferred, which is likely to lead to a biased search and non-uniformly distributed data records. Two cases are graphically presented in Fig.. Compared with the non-dominated solutions (black stars in the left plot), the dominated solutions (circles in the right plot) can prove helpful in updating the global approximate model although they are not the best candidates in the sense of Nondominated sorting. f (x) f 2 (x) f (x) Fig.. The non-dominated sorting leads to a biased search. The original version of EMO-FL suffers from a lack of selection pressure which is also caused by NSGA-II itself. Some domination-based MOEAs have been recognised for their in lack of selection pressure in latter runs. In the case of that all the current sub-optimal solutions, both parents and offsprings, are assigned as rank or belong to the first front (front-wise), no precise guidelines are available for the selection and reproduction operators to follow although some of them are closer to the Pareto fronts than others. This would make the algorithm itself fail to handle some particular problems where the global Pareto-optimal solutions weakly dominate the nearby sub-optimal ones. Moreover, the absence of an adequate selection pressure affects the entire system not to perform as efficiently as expected for solving problems that have many local Pareto-optimal fronts. It is worth noting that EMO-FL shows a similar search property with the embedded multi-objective optimiser. A diversity-preferred sorting and selection approach, which are able not only to provide appropriate selection pressure but also to maintain a reasonably good diversity of solutions during the entire optimisation process, is proposed here to improve the existing framework without any increase in complexity. 3. METHODOLOGY 3. Distance-Based Sorting and Selection f 2 (x) The proposed sorting method, called Distance-Based Sorting (DBS), aims at providing an important guideline to reproduction operators throughout the optimisation so that the associated MOEA can perform more efficiently than current domination-based approaches. The basic idea of DBS is to separate the search space into several identical subspaces where selection is mandatorily restricted. It should not be confused with the algorithms which make use of weighted aggregation and decomposition techniques; DBS attempts to only rank solutions. The detailed associated methodology is as follows: Step. Divide the insofar explored objective space into N identical subspaces; Step 2. In each subspace, solutions are ranked with respect to the Euclidean distance between the corresponding individuals and the pre-defined reference point. For instance, the solution that has the shortest Euclidean distance from reference point in a subpopulation is assigned as Rank. The second nearest one is such ranked 2 and so on. Because solutions in each subpopulation are separately sorted and isolated from those in other subpopulations, traditional selection schemes, such as tournament selection which is usually performed over all involved solutions, cannot be implemented with DBS. A simple approach for selecting parents for the next generation is hereby proposed to handle the associated problems. Two options are respectively provided for constructing DBSGA with a variable and a fixed population size, provided that the population size is larger than the number of subspaces. Adaptive Population: The best individual solutions which have the shortest Euclidean distance away from the reference point in each subspace are selected as parents for the next generation. Fixed Population: Firstly, all solutions that have a rank of are selected. An estimation of population density is then carried-out by counting the number of individuals located in each subspace. The solutions with rank 2 are sequentially selected from the sparser subspaces to the denser ones. DBS and the associated selection methods are inherently able to maintain a better diversity of solutions than the domination-based sorting approaches since elitist solutions are uniformly chosen over the entire search space explored so far. Furthermore, the solutions lying in a sparser subspace are given more priority than those in denser areas even if they are closer to the reference point. This behaviour can assist in improving the MLP model. A DBS-based MOEA can therefore be formulated for eliciting the enhanced version of EMO-FL. The Distance-Based Sorting Genetic Algorithm (DBSGA) is represented via the following pseudocode, using the operators and parameters of Table. Table. The operator and parameter Settings of DBSGA Operator/Parameter Details/Symbol Initialization Random Seeding Crossover Simulated Binary Crossover (P x, μ x ) Mutation Polynomial Mutation (P m, μ m ) Sorting Selection Population Size Generations Reference Point Distance-Based Sorting Best Individual of Sub-population N (Variable) λ Origin 29

3 Preprints of the 8th IFAC World Congress Milano (Italy) August 28 - September 2, 2 Procedure DBSGA Begin Initialization Randomly generate N seeds as parents; Recombination Implement crossover and mutation to reproduce 2N offsprings; Combine parents and offsprings to compose the overall population; Sorting and Selection Perform Distance-Based Sorting to rank all current solutions; Select N elitist solutions among the population; If the number of generations < λ; return to the step of Reproduction; else reported as the Pareto-optimal solutions; end end 3.2 The Enhanced EMO-FL Framework The overall structure of the enhanced EMO-FL framework is shown in Fig.2. Compared with the original version, NSGA- II is replaced by DBSGA as the main optimiser in Initialization, Controlled, and Uncontrolled processes. To initialise the entire mechanism, DBSGA is executed λ generations, defined as a cycle-length, to optimise the problem under investigation. All generated solutions and their objective values are stored in a database for creating and updating the MLP-based model. The MLP, widely known as a universal approximator, is able to represent high dimensional non-linear real-world problems in adequate predictive accuracy. Besides, a variety of mature training methods have already been developed. In this paper, the MLP model is initialised with two structures in order to provide adequate flexibility. It is reasonable to assume that one can always have an adequate level of information to estimate the maximal complexity of the problem at hand. A criterion, determining which structure is about to be used in the next cycle with respect to the current model predictive error, is also included. Provided that an adequately trained model is available, the optimiser can alternatively use the true objective functions or the MLP model to evaluate the offsprings under the Generation-based control (Jin et al., 22) in the controlled optimisation process. The control rule is defined as follows: where η represents the number of generations in which offsprings are evaluated using objective functions. η max and η min are the maximal and minimal values of η respectively,. E max is maximum value of the acceptable error. E(k) is the error of the MLP model in k-th cycle. The error, E(k), is required to describe the performance of the approximate model for all involved objectives. Moreover, the objectives may differ from each other in scale. A Root-Mean- Square of Combining Normalized Error (RMSCNE) is therefore derived to compute the overall model predictive error for MOOPs, i. e., where N is the number of objectives, P is the population size, y i (j) and y NN,i (j) are the values of the objective function and predictive output of the MLP model for the i-th solution and j-th objective respectively. In order to increase the usage of the trained model, an uncontrolled optimisation operation, where DBSGA works with the MLP model only, is subsequently carried-out. It is worth noting that neither the number of generations nor the population size affects the total evaluations since the objective values of all offsprings are estimated via the MLP model. The related parameters in this part are therefore adaptive, which enables EMO-FL to work faster than the optimiser by itself when the objectives are well fitted in a relatively large region. Solutions from both processes are sorted and selected so as to lead to the current elitists as long as the uncontrolled process finishes. The iteration will be repeated until desired outcomes are obtained or the pre-defined number of generations is reached. Fig. 2. The structure of the enhanced EMO-FL framework. 3.3 Training Data Selection and Controlled Reseeding An effective approach for selecting the training data for the MLP model, called Dynamic Selection, is included in this framework. The basic idea of this method tends to circumvent the challenges associated with real-world expensive optimisation problems, where no adequate data record is? 2 292

4 Preprints of the 8th IFAC World Congress Milano (Italy) August 28 - September 2, 2 available in advance or instantly collected data are not well distributed. The detailed implementation is described as follows. It is assumed that a set of data visited in past generations has been stored in the database. A total of N training data is required and d is the number of dimensions of the search space. n training data are necessarily selected from each dimension, n training = N training /d Step. Group all data records into p clusters via Fuzzy-C- Means algorithm. For each dimension, the search space is divided into p+ intervals, with width W(i) for i =, 2,... p+, (the width is the distance between every two closed clustering centres. For the first and last intervals, it can be computed with estimated upper and lower limits if not known exactly). Step 2. A Gaussian-type function can then be assigned to each of the p+ intervals. The height of each Gaussian function, φ(i), refers to the normalised width W(i)/W max. Step 3. Γ is a random number between and, By comparing Γ and φ(i), one of the intervals with φ(i) Γ is randomly chosen. γ (a reference mark for selecting training data in each dimension) can then be obtained as follows: (3) where σ is the width of the Gaussian distribution. Step 4. An ideal location of the training data in the current dimension can be calculated as follows. /2 (4) where C(i) is the centre of the chosen interval. Step 5. If no available data are present at the ideal position, the closest data point will be chosen and is then deleted from the database to avoid being selected more than once. Repeat Step 3 to Step 5 until n training is reached. Thereafter, Shift to another dimension until all dimensions are covered. Dynamic Selection allows the data that are located in sparser regions to be selected in priority. It also provides a possible way to extract the information about already explored regions in the search space, which can be shared to perform a Controlled Reseeding in the unexplored areas. This will assist the optimiser to find the global set of solutions. This process synergizes only with the MLP model, where no objective evaluation is required. 4. EXPERIMENTAL RESULTS In this section, a series of experiments were carried-out in order to check whether the enhanced framework can perform more efficiently for solving relatively difficult problems, compared with the original version. Furthermore, the new version should be able to represent a reasonably good MLPbased model as DBSGA can distribute the solutions more uniformly than NSGA-II. A. Parameter Settings The problems applied in this part were chosen from the wellknown benchmark test suites (Deb et al., 2) and given in Table 2. For a fairer comparative study, all parameter settings of the MLP-based model and genetic operators are the same as those used in the original version, as shown in Tables 3 and 4. Each experiment is executed via 2 independent runs. The results represent the average values of 2 runs. Table 2. The parameter settings of test functions Test Function Dimensions Number of Objectives ZDT4 2 DTLZ 7 3 DTLZ3 2 3 DTLZ6 2 3 DTLZ Table 3. The parameter settings of genetic operators Operator Parameter Value Crossover Probability Mutation Probability /dimension Crossover Index 5 Mutation Index 2 Population Size (Max.) Table 4. The parameter settings of the MLP-based model Name of Parameters Value Hidden Neurons in Structure Hidden Neurons in Structure 2 5 Size of Training Data/λ(η max ) 8/2(8) E max.5 Proportion of Selected Training Data B. Simulation Results EXPERIMENT I 6% (Training) 2% (Testing) 2% (Validation) In the first experiment, the enhanced EMO-FL was utilised to handle the problems that have either a weakly non-dominated Pareto front, such as DTLZ6, or many local Pareto-optimal fronts. The experimental results are graphically presented in Fig. 3. A comparison of simulation results conducted using both versions is described in Table 5. The difficulty of DTLZ6 makes most domination-based algorithms fail to approach its Pareto front. For DTLZ, 3 and ZDT4 where there are numerical local Pareto-optimal fronts, most MOEAs have to use tens of thousands of evaluations to converge to their global Pareto fronts. According to the simulation results presented in Table 5, the number of evaluations that are required to address these three problems is reduced by 8%, 25%, and 8% respectively. Furthermore, the proposed algorithm can successfully find the Pareto-optimal solutions of DTLZ6 in all 2 runs. 293

5 Preprints of the 8th IFAC World Congress Milano (Italy) August 28 - September 2, 2 f(3) f(3) f(2).5 The Pareto front of DTLZ Fig. 3. The experimental results of the chosen problems. Table 5. The comparison of simulation results conducted by both the enhanced and original EMO-FL Test Function New Version Evaluation Times Original Version ZDT DTLZ DTLZ DTLZ6 47 N/A EXPERIMENT II f() The Pareto front of DTLZ6.5 f(2).2.5 f() The goal of this experiment is to distinguish the effect, which is caused by the introduction of DBSGA, on the quality of the MLP-based model. Both versions of EMO-FL were applied on the chosen test functions, including DTLZ, DTLZ3, DTLZ7, and ZDT4. The means and variations of the Root- Mean-Square Combined Error (RMSCE), which is defined in (5), of the initial models over 2 runs are also computed. The results shown in Tables 6 and 7 illustrate a comparison of the accuracy of the initial model elicited in both versions.,.6 where N is the number of objective functions, P is the population size, y i (j) and y NN,i (j) are respectively the value of objective function and predictive output of the MLP model for the i-th solution and j-th objective. It is also worth noting that the initial approximate models of both versions should be derived based on the similar sample size (as shown in Table 6) for meaningful comparisons since the enhanced version has an adaptive population size. It can be seen that the accuracy of the initial MLP-based models in the enhanced version has been improved by approximate f(3) f(2) The Pareto front of DTLZ3 f(2) f() The Pareto front of ZDT4.5 f() 5 8% for DTLZ, 3 and ZDT4. The variations of the RMSCE are also significantly reduced for all test functions. Table 6. The RMSCE of the initial model of the enhanced EMO-FL framework Test Function Sample Size Average RMSCE Variation ZDT (e-4) DTLZ (e-4) DTLZ (e-4) DTLZ (e-3) 6.99 (e-9) Table 7. The RMSCE of the initial model of the original EMO-FL framework Test Function Sample Size Average RMSCE Variation ZDT (e-4) DTLZ (e-4) DTLZ (e-4) DTLZ (e-3).522 (e-7) 5. APPLICATIONS TO THE OPTIMAL DESIGN OF ALLOY STEELS In real-world applications, the optimisation problem usually involves multiple incommensurable and competing objectives. This is no exception for the steel industry. Some mechanical properties of alloy steels, which are usually considered as design targets, may conflict with each other, such as Ultimate Tensile Strength (UTS) and Reduction of Area (ROA), ROA reflecting the ductility of alloy-steel. The proposed approach is applied to the engineering problems relating to the simultaneous optimisation of UTS and ROA for a particular steel grade. Heat treatments are commonly used to develop the required mechanical properties in the steel industry. However, one can rarely compute the mechanical properties due to the available physical knowledge of heat treatments being inadequate. The data-driven models are therefore elicited for the prediction of the mechanical properties of alloy steels. In this research, all experiments are based on an artificial immune systems based multi-objective fuzzy model (Chen and Mahfouf, 29). There exist 4 and 28 embedded structures for the UTS and ROA predictions respectively in these models, which provide independent outputs. The average value of predictions of all these structures is used as the unique output. Moreover, the standard deviations (STDs) of the predictions of all structures in both models are also considered as constraints, in order to guarantee consistency of the model predictions. Many factors can influence the mechanical properties, including the weight percentages of the chemical composites, the tempering temperature, and other parameters. In this paper, the decision vector consists of the four most important chemical composites, Carbon (C), Molybdenum (Mo), Man- 294

6 Preprints of the 8th IFAC World Congress Milano (Italy) August 28 - September 2, 2 ganese (Mn), Chromium (Cr), as well as the tempering temperature. The parameters of the enhanced EMO-FL framework are same as those shown in Tables 2 and 3. EXPERIMENT The experiment aims to find the feasible space of the predictive model, where consistent predictions for both UTS and ROA can be achieved with the constraints of the pre-defined STDs. Two multi-objective problems were defined as follows: ). Maximizing UTS and ROA simultaneously, subject to the corresponding STDs, i.e. Maximize: UTS and ROA Subject to: STD UTS 2 and STD ROA 2 2). Minimizing UTS and ROA simultaneously, subject to the same STDs, i.e. Minimize: UTS and ROA Subject to: STD UTS 2 and STD ROA 2 The results are graphically shown in Fig. 4. The upper and lower figures present the simulation results obtained by the original and new versions respectively. With this comparison, it can be seen that the disconnected Pareto fronts (marked by stars) were successfully located via both versions. However, this is not enough for one to carry-out a reliable design since the information about the confidence field, where STDs of both objectives are strictly constrained, is not complete. Therefore, a two-step experiment was proposed in order to address such a problem. Firstly, the globally maximal and minimal Pareto fronts can be found as normal. Secondly, the ancillary front (marked by circles) can be achieved by moving the position of the reference point to the interested area. The entire feasible space of the predictive model is sketched by the updated boundary, as shown in the lower plot of Fig. 4. ROA (%) ROA (%) Pareto-front obtained by the Original EMO-FL UTS (N/mm 2 ) Pareto-front obtained by the Enhanced EMO-FL UTS (N/mm 2 ) Fig. 4. The Pareto front and ancillary boundary of the feasible space of the predictive model. 6. CONCLUSIONS AND FUTURE WORK New sorting and selection approaches have been reported in this research work, based upon which the distance-based sorting genetic algorithm was developed to enhance the framework of EMO-FL in several aspects, including the accuracy of the MLP-based model and the speed of convergence. Moreover, the benchmark problem of DTLZ6 that suffers from the fact that its Pareto front weakly dominates the nearby sub-optimal solutions was successfully addressed. Other simulations using well-known challenging benchmark test functions were carried-out. A comparative study between the enhanced and original EMO-FL proved that the improved version is much more efficient in solving relatively difficult MOOPs. However, it is worth noting that the distance-based sorting can only provide a uniformly distributed set of solutions since it is not domination-based. A domination-based archive operator is therefore required to extract the Paretooptimal solutions. Furthermore, an approach for adaptively defining the reference point will be investigated for the DBS to address even more complex real-world problems. REFERENCES Chen, J. and Mahfouf, M. (29) An Artificial Immune Systems based Predictive Modelling Approach for the Multi-Objective Elicitation of Mamdani Fuzzy Rules, in Proc. of The 29 Int. Conf. on SMC., pp Deb, K., Pratap, A., Agarwal, S., and Meyarivan, T. (22) A Fast and Elitist Multiobjective Genetic Algorithm: NSGA-II, IEEE Trans. Evol. Comput., vol. 6, no. 2, pp Deb, K., Thiele, L., Laumanns, M., and Zitzler, E. (2) Scalable Test Problems for Evolutionary Multi- Objective Optimization, Computer Engineering and Networks Laboratory (TIK), Swiss Federal Institute of Technology (ETH), Zurich, Switzerland, Tech. Rep. 2. Jin, Y., Olhofer, M., and Sendhoff, B. (22) A Framework for Evolutionary Optimization With Approximate Fitness Functions, IEEE Trans. Evol. Comput., vol. 6, no. 5, pp Knowles, J. (25) ParEGO: A Hybrid Algorithm With On- Line Landscape Approximation for Expensive Multiobjective Optimization Problems, IEEE Trans. Evol. Comput., vol., no., pp Nain, P. and Deb, K. (23) Computationally Effective Search and Optimization Procedure Using Coarse-to- Fine Approximations, in Proc. of The 23 Congr. Evol. Comput. (CEC), vol. 3, pp Regis, R. and Shoemaker, C. (24) Local Function Approximation in Evolutionary Algorithms for the Optimization of Costly Functions, IEEE Trans. Evol. Comput., vol. 8, no. 5, pp Wang, S. and Mahfouf, M. (2) Efficient Multi-Objective Optimization with Fitness Landscape - A Special Application to the Optimal Design of Alloy-Steels, in Proc. of The 2 Congr. Evol. Comput. (CEC), pp Zitzler, E., Deb, K., and Thiele, L. (2) Comparison of Multiobjective Evolutionary Algorithms: Empirical Results, Evolutionary Computation, vol. 8(2), pp