A Surrogate-Assisted Memetic Co-evolutionary Algorithm for Expensive Constrained Optimization Problems

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1 A Surrogate-Assisted Memetic Co-evolutionary Algorithm for Expensive Constrained Optimization Problems C. K. Goh,D.Lim,L.Ma,Y.S.Ong and P. S. Dutta Advanced Technology Centre Rolls-Royce Singapore chi.keong.goh, Centre for Computational Intelligence Nanyang Technological University dlim, mali, Abstract Stochastic optimization of computationally expensive problems is a relatively new field of research in evolutionary computation (EC). At present, few EC works have been published to handle problems plagued with constraints that are expensive to compute. This paper presents a surrogate-assisted memetic co-evolutionary framework to tackle both facets of practical problems, i.e. the optimization problems having computationally expensive objectives and constraints. In contrast to existing works, the cooperative co-evolutionary mechanism is adopted as the backbone of the framework to improve the efficiency of surrogate-assisted evolutionary techniques. The idea of randomproblem decomposition is introduced to handle interdepencies between variables, eliminating the need to determine the decomposition in an ad-hoc manner. Further, a novel multi-objective ranking strategy of constraints is also proposed. Empirical results are presented for a series of commonly used benchmark problems to validate the proposed algorithm. I. INTRODUCTION Evolutionary Algorithm (EA) are powerful optimization tools that have been applied to complex constrained optimization problems with a great degree of success. Apart from tight constraints, researchers also have to deal with the increasing computational cost of today s applications. The computational cost typically increases with size, complexity, and fidelity of the considered problem model. In design analysis and optimization processes where high-fidelity analysis codes are used, the evaluation of each design quality requiring the simulation of the high-fidelity analysis codes, such as Computational Structural Mechanics (CSM), Computational Fluid Dynamics (CFD) or Computational Electro Magnetics (CEM), costs minutes to hours of supercomputer time. Therefore, the optimization goal in Expensive Constrained Optimization Problems (ECOPs) is not only to obtain a feasible and satisfactory solution but to achieve it within a small number of function evaluations. However, there is a surprising lack of efforts to tackle both constraints and high computational cost simultaneously. Within the field of EC, few publications [1][2][3] explicitly consider constraints among the existing evolutionary works designed for solving expensive optimization problems. Most techniques for constrained optimization problems [4][5] are typically built upon basic EAs that require thousands of function evaluations to find a satisfactory feasible solution. The computational efforts involved often exceed the maximum budget allocated. In this work, a surrogate-assisted cooperative coevolutionary Memetic algorithm (SCCMA) is developed for ECOPs. Instead of building a Memetic algorithm [6], [7] that seeds from canonical EA, the cooperative co-evolutionary paradigm [8], [9] forms the backbone of our proposed approach for handling expensive constrained optimization problems. Preliminary studies on surrogateassisted co-evolutionary search has been conducted in Ong et al. [10]. The SCCMA exts the idea of randomproblem decomposition [11] to handle interdepencies between variables. The new decomposition scheme eliminates any need for determining suitable subpopulation size and its corresponding members of the co-evolution. Further, constraints are handled by means of a novel multi-objective ranking strategy to besiege the co-evolutionary search towards regions containing feasible solutions. The SCCMA is also hybridized with a sequential quadratic programming (SQP) solver to exploit the well established efficient convergence capabilities of the SQP towards feasible local optimum. The organization of the paper is as follows: Section II introduces the general framework of the proposed SCCMA. Extensive empirical studies are conducted to compare and analyze the performance of SCCMA based on benchmark problems in Section III. Conclusions are drawn in Section IV. II. SURROGATE-ASSISTED MEMETIC CO-EVOLUTIONARY FRAMEWORK Over the years, numerous efforts have been gearing towards improving the efficiency of existing optimization algorithms. In evolutionary computation, one such research direction is on the use of surrogate models that are less computationally intensive [12], i.e., the Surrogate-Assisted Evolutionary Algorithms (SAEAs). Using surrogate models, the computational burden /11/$ IEEE 744

2 can be greatly reduced since the efforts involved in building the models and optimization using them are much lower than the standard approach of directly coupling the simulation codes with the optimizer. Surrogate-Assisted Memetic Algorithms (SAMAs) represents one of the most sucessful implementations of SAEAs, capable of balancing the exploration quality and exploitation efficiency demands of the optimization problem. In SAMAs, the numerous extra fitness evaluations in the exploitation phase to balance exploration is tolerated by the use of approximated fitness function. The use of exact models for the objective and constraint functions are interspersed with computationally cheap surrogate models during local search stage. Extensions to enhance search efficiency and approximation accuracy using improved training methods [13], gradient information [14] and multi-level surrogates [15][16] were also considered recently. This section describes the implementation details of the proposed SCCMA, which integrates the concepts of co-evolution and constraint-handling techniqes within SAMAs. A. Algorithm Overview The outline of the surrogate-assisted co-evolutionary memetic algorithm is given in pseudocode 1. The algorithm starts with the co-evolutionary optimization phase, decomposing the problem by partitioning the decision space and solving for the different partitions. In this phase, the original exact objective function is used to build up a database of design points for building the surrogates. The fitness of each individual is determined with a new hybrid MO ranking scheme that guides the search towards the promising feasible regions. The best solution(s) will be then updated into the archive. Details of the co-evolutionary optimization phase and the new MO ranking scheme is described in the rest of this subsection. Surrogate models are used only during the local search to estimate the objective and constraint values of each individual in the subpopulation using previously encountered neighbourhood design points. In this paper, simple local polynimial regression models are adopted and are fairly accurate at the point of interest where local search can be conducted accurately at low computational cost. The output solution of the local search is then evaluated with the exact model. In this work, the local search operator used is the SQP solver to enhance algorithmic efficiency in finding feasible local optimum. B. Multi-objective Ranking Scheme In the Multi-objective (MO) approach to COP, constraints are treated as objectives to maintain a balance between constraint satisfaction and optimization. In [17] and [18], constraints are treated as many objectives in an optimization framework adopting the Vector Evaluated Genetic Algorithm (VEGA), an existing Multi-Objective Evolutionary Algorithm (MOEA). The major disadvantage of using each constraint as an indepent objective is that the evolutionary pressure is very low and the search process often stagnates rapidly in Algorithm 1: Surrogate-assisted Memetic co-evolutionary algorithm Problem decomposition; Generate subpopulation for each partition of decision variables; while computational budget is not exhausted do *Co-evolutionary Optimization Phase* Select representatives for each subpopulation; for each subpopulation do Generate new subpopulation using evolutionary operators; for each individual in subpopulation do Collaborate with representatives from other subpopulations to form complete solution; Evaluate using exact model and assign Fitness; Update the database with new design points; Update Archive with best solution(s); *Local Search Phase* for each subpopulation do for each individual in subpopulation do Apply local search using local surrogate models of the objective and constraint functions; Evaluate using exact model and assign Fitness; Update the database with new design points; Update Archive with best solution(s); Repartitioning of decision variables; Assign decision variables to respective subpopulation; high-dimensional objective space [19], [20]. Without the preference for feasibility, MO approaches have very low selection pressure towards constraint satisfaction. This is particularly undesirable for problems with very few feasible solutions. The main idea of the proposed MO scheme is to provide sufficient bias (which is lacking in MO approaches) towards feasible regions with the deterministic ranking method while maintaining diversity (which is lacking in ranking approaches) with the MO method. The rules of the MO ranking scheme are given by: a feasible solution is preferred compared to the infeasible counterpart, between any two feasible solutions, the solution having better objective value is preferred, and between any two infeasible solutions, the solution that dominates in a Pareto optimal sense is preferred. 745

3 Contrary to existing approaches, the objective value is not considered as part of the MO criteria, further alleviating the selection pressure problem associated with the MO approaches. Three different constraint measures are used to drive the optimization process towards the regions containing feasible solutions. Minimize maximum constraint violation: Minimize: max { } G1 i,h1 j i =1,...,n g,j =1,...,n h where: G1 i (x) =max{0,g i (x)} H1 j (x) =abs ( h j (x) ε ) (1) Minimize count for violated constraints: ng Minimize: i=1 G2 i + n h j=1 H2 i { where: 1, if g i (x) > 0 G2 i = 0, otherwise. { H2 i = 1, if h i (x) >ε 0, otherwise. (2) Algorithm 2: Random Decomposition COEA (rdcoea) Random problem decomposition; Generate subpopulation for each partition of decision variables; while Computational budget is not exhausted do Select representatives for each subpopulation; for Each subpopulation do Generate new subpopulation using evolutionary operators; for Each individual in subpopulation do Collaborate with representatives from other subpopulations to form complete solution; Evaluate and Assign Fitness; Random repartitioning of decision variables; Assign decision variables to respective subpopulation by tournament selection; maximize ratio of rarely satisfied constraints Maximize: ng i=1 G2 i 1+G3 i + n h j=1 H2 j 1+H3 j (3) where: G3 i and H3 j are the ratio of the total number of solutions satisfying g i and h j to population size, respectively. All the three measures bias the search towards unique solutions, providing the diversity necessary to facilitating improved exploration of the search space. C. Cooperative Coevolution In conventional cooperative co-evolutionary algorithms, decomposition of the problem space is typically pre-determined before the search begins. For example, a common approach is to assign the i-th subpopulation to represent the i-th variable. Since each subpopulation is optimizing only a subset of the design space, it is necessary to concatenate the individuals in the i-th subpopulation with representatives from the other subpopulations to form a valid candidate solution for subsequent evaluation. When assessing the fitness of an individual from the i-th subpopulation, the individual will combine with representatives from j-th and k-th subpopulation to compose a valid candidate solution. The representative can either be the elite or some randomly chosen individual from the subpopulations [9]. To date, few has considered an intelligent grouping of variables. Yang et al [11] considered the random assignment on equal number of design variables in each of the subpopulations to encourage the emergence of best decomposition along the evolutionary process. Here we ext the idea by randomizing both the subpopulation size and design variables assigned to each subpopulation. The pseudocode of our COEA with random decomposition is depicted in Algorithm 2. III. EMPIRICAL STUDY This section begins with a brief description on the benchmark problems considered in the present study. Subsequently, the comparative study between SCCMA and various algorithms are presented. A. Constraint-Optimization Benchmark Test Functions Constrained problems are characterized by the type of objective functions, the type of constraints, and the number of active constraints. Active constraints are inequality constraints that take the value of 0 at the global optimum to the problem. The types of constraints are broadly categorized as linear inequalities (LI), nonlinear inequalities (NI), linear equalities (LE), and nonlinear equalities (NE). The type of constraints has a direct implication on the difficulty of the problem. The linearity of the constraints determines the level of difficulty in finding a feasible solution. The feasible design space for a problem with only linear constraints is convex, i.e., feasible solutions can be found easily by minimizing the constraint violations directly. On the other hand, in the case of nonlinear constraints, the differences in the constraint violation between slightly infeasible solutions and feasible solutions may not be indicative of the difference between their locations in the decision space. Therefore, solutions with similar degree of violation may be far apart in the decision space. The difficulty of a constrained optimization problem can be measured in terms of the ratio between the size of the feasible search space and that of the entire search space. This ratio can be estimated by the following equation ρ = Ω S where S is the number of solutions randomly generated from the design space and Ω is the number of feasible solutions (4) 746

4 TABLE I SUMMARY OF BENCHMARK CHARACTERISTICS. Problem d Type of f ρ LI NI LE NE a g01 13 Quadratic g03 10 Nonlinear g05 4 Nonlinear g07 10 Quadratic g09 7 Nonlinear g10 8 Linear g11 2 Quadratic g13 3 Nonlinear out of these solutions. Intuitively, it is harder to find feasible solutions for problems with small ρ values as compared to larger ρ values. In this paper, eight representative constrained benchmark functions of diverse characteristics are used to evaluate the performance of SCCMA. These test functions have been used in many previous works [21], [22], [23], [24], [25]. This test suite is selected based on their small ρ values and for the range of different constraint types to challenge the algorithm s ability in locating feasible solutions. Test function characteristics, based on the properties described above, are summarized in Table V. The ρ values are taken from [22]. B. Impact of Random Decomposition In this section, experiments are conducted to analyse the effects of random decomposition. The best results obtained, MFE and the ratio of feasible runs (FR) of the co-evolutionary algorithm (COEA) and rdcoea implementations for g01, g03, and g07 are summarized in Table III-B. By comparing the best solution found by rdcoea and COEA, it is clear that the random decomposition scheme contributes to a less sensitive method to the underlying linkages between design variables as compared to traditional methods of decomposition. Apart from improved solution quality, it can be observed from MFE that rdcoea only requires half the number of evaluations required by COEA to find feasible solutions. The introduction of random decomposition also improves the consistency of rdcoea in finding feasible solutions. also provides enhanced performance in terms of best solution quality, FR, and MFE for all the test problems. C. Existing State-of-art Algorithms In this paper, the performance of SCCMA is validated against algorithms that are representative of the state-of-thearts in constraint-handling. The methods and their abbreviations are summarized in Table III-C. The basic mechanisms of stochastic ranking (SR), deterministic ranking (detr), dualstage generic framework, and adaptive tradeoff model are described as follows. Deterministic ranking: Ranking schemes of EAs can be modified to consider constraints. One of the more common approach is the deterministic ranking (detr) scheme [26]. This can be summarized by the following three rules: A feasible solution is preferred compared to the infeasible one, TABLE II EFFECTS OF COEVOLUTION, RANDOM DECOMPOSITION, AND SURROGATE-ASSISTED LOCAL SEARCH. THE BEST SOLUTION IS HIGHLIGHTED IN BOLD. Problem COEA rdcoea Best g01 MFE FR Best g03 MFE FR 1 1 Best g07 MFE FR between any two feasible solutions, the one having better objective value is preferred, and between any two infeasible solutions, the one having less constraint violation is preferred. From a survey of the literature, it has been demonstrated that preference of feasible solution approaches is generally more effective as compared to Penalty function based approaches. Stochastic ranking: Stochastic ranking (SR) [23] follows a similar set of rules. In order to overcome the limitation mentioned above, it introduces randomness in the comparison criteria. In [23], the decision on whether the objective value or constraint violation is used in comparison, which involves at least one infeasible solution, is determined by a certain probability threshold, P f. For instance, P f =0.5, means that there is an equal chance for both criteria to be used when performing such comparison. Dual Stage EA: Venkatraman and Yen [24] proposed a dualstage (DS) generic framework, which disregards the objective function in the first stage. The first stage is concerned with only the constraints and the entire search effort is directed toward finding a single feasible solution. Each individual of the population is ranked based on its constraint violation only. Upon finding a feasible solution, the second stage comes into play and a MO approach is applied to minimize both the objective function and constraint violation simultaneously. Adaptive Tradeoff Model Evolutionary Strategy: Wang et al. [25] suggested an adaptive tradeoff model which employs different strategies deping on the ratio of feasible to infeasible solutions. The MO approach is applied to minimize both the objective function and constraint violation simultaneously for better exploration when there are no infeasible solutions. Once feasible solutions exist in the population, a ranking scheme is adopted. In this ranking scheme, feasible solutions are compared based only on objective values while an aggregation of normalized objective function and constraint values are used when comparing infeasible solutions. D. Results The simulations for SCCMA are implemented using Matlab on an Intel Pentium GHz desktop computer. For test 747

5 TABLE III REPRESENTATIVE CONSTRAINT-HANDLING EAS. Stochastic Deterministic Dual-Stage Adaptive Tradeoff Model Ranking Ranking framework Evolutionary Strategy SR detr DS ATMES Parameter TABLE IV PARAMETER SETTING FOR SCCMA Settings Populations Subpopulation size 10 in SCCMA; Chromosome Real number coding; Selection Binary tournament selection; Crossover operator Simulated binary crossover; Crossover rate 1.0; NC 5.0; Mutation operator uniform mutation; Mutation rate 0.1; Maximum Evaluation 3000; functions with equality constraints, an ε value of is adopted in this paper. SCCMA is implemented using real number coding scheme, binary tournament selection, Simulated Binary Crossover (SBX), and uniform mutation. The parameter settings considered in the study are listed in Table IV. The obtained results are summarized in Table V. In terms of solution quality, most of the algorithms found the optimal solution for g01 and g03. With the exception of problem g09, SCCMA provides competitive or improved results over the other algorithms. This demonstrates that the proposed algorithm is useful in finding good feasible solutions. By comparing the results of SCCMA in Table V and rdcoea in Table III-B, it clearly shows that the incorporation of surrogate-assisted local search in SCCMA provides a means to improve algorithmic capability in finding feasible solutions. Apart from finding solutions close to the global optimum, it is able to find feasible solutions consistently. Moreover, SCCMA is observed to significantly reduce the number of evaluations required to find the first feasible solution as compared to other algorithms. The capability to find good feasible solutions with lower number of evaluations is particularly important for optimization problems with expensive evaluation functions. IV. CONCLUSION In this paper we have presented a new surrogate-assisted Memetic algorithm that integrates cooperative co-evolutionary mechanism into existing surrogate-assisted Memetic schemes for enhanced efficiency and constraint-handling. The proposed multi-objective ranking scheme exploits information from different constraint measures to guide the search while the surrogate-assisted sequential quadratic programming solver allows the algorithm to find feasible local optimum cheaply. Empirical results are presented for a series of commonly used benchmark problems to demonstrate that the proposed framework converges to good solution quality more efficiently than the standard evolutionary algorithms and co-evolutionary algorithms. REFERENCES [1] Y. S. Ong, P. B. Nair, and A. J. Keane, Evolutionary optimization of computationally expensive problems via surrogate modeling, AIAA Journal, vol. 41, no. 4, pp , [2] A. 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