A novel elitist multiobjective optimization algorithm: multiobjective extremal optimization Min-Rong Chen*, Yong-Zai Lu Department of Automation, Shanghai Jiao Tong University, Dongchuan Road 800, Class A0403291, 200240 Shanghai, China Abstract: Recently, a general-purpose local-search heuristic method called Extremal Optimization (EO) has been successfully applied to some NP-hard combinatorial optimization problems. This paper presents an investigation on EO with its application in multiobjective optimization and proposes a new novel elitist (1+ λ ) multiobjective algorithm, called Multiobjective Extremal Optimization (MOEO). In order to extend EO to solve the multiobjective optimization problems, the Pareto dominance strategy is introduced to the fitness assignment of the proposed approach. We also present a new hybrid mutation operator that enhances the exploratory capabilities of our algorithm. The proposed approach is validated using five popular benchmark functions. The simulation results indicate that the proposed approach is highly competitive with the state-of-the-art multiobjective evolutionary algorithms. Thus MOEO can be considered a good alternative to solve multiobjective optimization problems. Keywords: Multiple objective programming; Metaheuristics; Extremal optimization; Self-organized criticality 1. Introduction Most real-world engineering optimization problems are multiobjective in nature, since they normally have several (possibly conflicting) objectives that must be satisfied at the same time. Instead of aiming at finding a single solution, the multiobjective optimization methods try to produce a set of good trade-off solutions from which the decision maker may select one. The Operations Research (OR) community has developed a lot of mathematical programming techniques to solve multiobejctive optimization problems (MOPs) since the 1950s [1, 2]. However, mathematical programming techniques have some limitations when dealing with MOPs [3]. For example, many of them may not work when the Pareto front is concave or disconnected. Some require differentiability of the objective functions and the constraints. In addition, most of them only generate a single solution from each __________________ *Corresponding author. E-mail address: [email protected] run. During the past two decades, a considerable amount of multiobjective evolutionary algorithms (MOEAs) have been presented to solve MOPs [3-7]. Evolutionary algorithms seem particularly suitable to solve MOPs, because they can deal with a set of possible solutions (or so-called population) simultaneously. This allows us to find several members of the Pareto-optimal set in a single run of the algorithm [3]. There is also no requirement for differentiability of the objective functions and the constraints. Moreover, evolutionary algorithms are susceptible to the shape of the Pareto front and can easily deal with discontinuous or concave Pareto fronts. Recently, a general-purpose local-search heuristic algorithm named Extremal Optimization (EO) was proposed by Boettcher and Percus [8]. EO is based on the Bak- Sneppen (BS) model [9], which shows the emergence of self-organized criticality (SOC) [10] in ecosystems. The evolution in this model is driven by a process where the weakest species in the population, together with its nearest neighbors, is always forced to mutate. The dynamics of this extremal process exhibits the characteristics of SOC, such as punctuated equilibrium [9]. EO opens the door to applying non-equilibrium process, while the simulated annealing (SA) applies equilibrium statistical mechanics. In contrast to genetic algorithm (GA) which operates on an entire “gene-pool” of huge number of possible solutions, EO successively eliminates those worst components in the sub- optimal solutions. Its large fluctuations provide significant hill-climbing ability, which enables EO to perform well particularly at the phase transitions. EO has been successfully applied to some NP-hard combinatorial optimization problems such as graph bi-partitioning [8], graph coloring [11], spin glasses [12], MAXSAT [13], production scheduling [14,15], function optimization [16] and dynamic combinatorial problems [17]. So far there have been some papers studying on the multiobjective optimization using extremal dynamics. Ahmed and Elettreby [18,19] introduced random version of BS model. They also generalized the single objective BS model to a multiobjective one by weighted sum aggregation method. The method is easy to implement but its most serious drawback is that it cannot generate proper members of the Pareto-optimal set when the Pareto front is concave regardless of the weights used [20]. Galski et al. [21] applied the Generalized Extremal Optimization (GEO) algorithm [22] to design a spacecraft thermal control system. The design procedure was tackled as a multiobjective optimization problem and they also resorted to the weighted sum aggregation method to solve the problem. In order to extend GEO to solve the MOPs effectively, Galski et al. [23] further present a revised multiobjective version of the GEO algorithm, called M-GEO. The M-GEO algorithm does not use the weighted sum aggregation method. Instead, the Pareto dominance concept was introduced to M-GEO in order to find out the approximate Pareto front, and at the same time the approximate Pareto front was stored and updated each run. Since the fitness assignment in the M-GEO is not based on the Pareto dominance strategy, M-GEO belongs to the non-Pareto approach [23]. M-GEO was successfully applied to the inverse design of a remote sensing satellite constellation [23]. In this work, we develop a novel elitist multiobjective optimization method, called Multiobjective Extremal Optimization (MOEO). Our approach does not use the weighted sum aggregation method to solve MOPs. Instead, we adopt the fitness assignment method based on the Pareto dominance strategy. Thus, MOEO is a Pareto-based approach. It is interesting to note that, similar to the (1+λ ) Pareto Archived Evolution Strategy (PAES) [24], MOEO is also a single-parent λ -offspring multiobjective optimization algorithm. Furthermore, we propose a new hybrid mutation operator that enhances the exploratory capabilities of our algorithm. Our approach has been validated using five benchmark functions reported in the specialized literature and compared with four competitive MOEAs: the Nondominated Sorting Genetic Algorithm-II (NSGA-II) [25], the Pareto Archived Evolution Strategy (PAES) [24], the Strength Pareto Evolutionary Algorithm (SPEA) [26] and the Strength Pareto Evolutionary Algorithm2 (SPEA2) [27]. The simulation results demonstrate that the proposed approach is highly competitive with the state- of-the-art MOEAs. Hence, MOEO may be a good alternative to solve the multiobjective optimization problems. This paper is organized as follows. In Section 2, we give the problem formulation of multiobjective optimization problem. Section 3 describes four state-of-the-art elitist multiobjective evolutionary algorithms. The extremal optimization algorithm is introduced in Section 4. In Section 5, we propose the MOEO algorithm and describe its main components in detail. In Section 6, the proposed approach is validated using five popular benchmark functions. In addition, the simulation results are compared with those of four state-of-the-art multiobjective evolutionary algorithms. Finally, Section 7 concludes the paper with an outlook on future work. 2. Problem formulation Without loss of generality, the MOPs are mathematically defined as follows: Find x which minimizes Fx()= ( f12 (),(),, x f x fk ()) x subject to : gx()≥= 0， i 1,2,, m i (1) hxj ( )== 0, j 1,2, , p T where xxxx=∈Ω(,12 , ,n ) is a vector of decision variables, each decision variable is bounded by lower and upper limits lxullll≤≤, = 1, , n, k is the number of objectives, m is the number of inequality constraints and p is the number of equality constraints. The following four concepts are of importance [28]: Definition 1. Pareto dominance: A vector uu= (, , u ) is said to dominate another 1 k vector vv= (,1 , vk )(denoted by uv≺ ) if and only if u is partially less than v , i.e., ∀∈ikuvjkuv{1, , } :ii ≤ ∧∃∈ ( {1, , } : j < j ) . Definition 2. Pareto optimality: A solution x ∈Ω is said to be Pareto optimal with respect to Ω if and only if there is no x' ∈Ω for which vFx==('' ) ( fx ( ), , fx ( ' )) 1 k dominates uFx==( ) ( fx1 ( ), , fxk ( )) . The phrase “Pareto optimal” is taken to mean with respect to the entire decision variable space unless otherwise specified. Definition 3. Pareto-optimal set: The Pareto optimal set PS is defined as the set of all '' Pareto optimal solutions, i.e., PxS =∈Ω¬∃∈Ω{| x :()()} FxFx≺ . Definition 4. Pareto-optimal front: The Pareto-optimal front PF is defined as the set of all objective functions values corresponding to the solutions in PS , i.e., PFxfxfxxPFkS=={ () (1 (),, ())| ∈ }. 3. Elitist multiobjective evolutionary algorithm As opposed to single-objective optimization, where the best solution is always copied into the next population, the incorporation of elitism in MOEAs is substantially more complex. Often used is the concept of maintaining an external archive of solutions that are nondominated among all individuals generated so far. Another promising elitism approach provides the so-called ()µ + λ selection, where parents and offspring compete against each other. In the study of Zitzler et al. [29], it was clearly shown that elitism helps in