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Evolving Continuous Pareto Regions
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In this chapter we propose a new evolutionary elitist approach combining a non-standard solution representation and an evolutionary optimization technique. The proposed method permits detection of continuous decision regions. In our approach an individual (a solution) is either a closed interval or a point. The individuals in the final population give a realistic representation of the Pareto-optimal set. Each solution in this population corresponds to a decision region of the Pareto-optimal set. The proposed technique is an elitist one. It uses a unique population. The current population contains non-dominated solutions already computed.
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... Multi-response optimization is related with multi-objective global optimization that can be defined as follows (Dumitrescu, Grosan, & Oltean, 2005): Defining X as the search space where there are n objective functions, optimize F(X) = (f 1 (X), f 2 (X), f 3 (X), . . . , f n (X)) where X 2 X. ...
The joining of Advanced High Strength Steel (AHSS) Martensitic type is being introduced in automotive industry; however, the optimization of the welding process is required to meet customer quality requirements. Two neural networks are built for modeling the relationship between the welding parameters and the output response of the process. An evolutionary algorithm is used for multi-objective optimization considering the neural networks as objective functions. The results consist of a set of solutions that approximate the Pareto optimal set. The related response of this set is known as the Pareto front. The set of solutions are validated in the real process satisfying the security and quality requirements.
In multi-objective evolutionary optimization it is desirable to obtain a set of solutions usually called the Pareto optimal set. In this paper some preliminary results are presented to approximate a Pareto optimal set considering an evolutionary adjustment of the parameters of a normal distribution function. The univariable and multivariable normal distribution functions are considered. Four test Junctions are used considering two forms of evaluation. Results indicate a novel way to represent the Pareto front and suggest different ideas for this area of optimization.
In multi-objective evolutionary optimization it is desirable to obtain a set of solutions usually called the Pareto optimal set. In this paper, some preliminary results are presented to approximate a Pareto optimal set considering an evolutionary adjustment of the parameters of a normal distribution function. The univariable and multivariable normal distribution functions are considered. Four test functions are used considering two forms of evaluation. Results indicate a novel way to represent the Pareto front and suggest different ideas for this area of optimization.
We introduce a new multiobjective evolutionary algorithm called PESA (the Pareto Envelope-based Selection Algorithm), in which selection and diversity maintenance are controlled via a simple hyper-grid based scheme. PESA's selection method is relatively unusual in comparison with current well known multiobjective evolutionary algorithms, which tend to use counts based on the degree to which solutions dominate others in the population. The diversity maintenance method is similar to that used by certain other methods. The main attraction of PESA is the integration of selection and diversity maintenance, whereby essentially the same technique is used for both tasks. The resulting algorithm is simple to describe, with full pseudocode provided here and real code available from the authors. We compare PESA with two recent strong-performing MOEAs on some multiobjective test problems recently proposed by Deb. We find that PESA emerges as the best method overall on these problems.
Multi-objective evolutionary algorithms which use non-dominated sorting and sharing have been mainly criticized for their (i) O(MN³) computational complexity (where M is the number of objectives and N is the population size), (ii) non-elitism approach, and (iii) the need for specifying a sharing parameter. In this paper, we suggest a non-dominated sorting based multi-objective evolutionary algorithm (we called it the Non-dominated Sorting GA-II or NSGA-II) which alleviates all the above three difficulties. Specifically, a fast non-dominated sorting approach with O(MN²) computational complexity is presented. Second, a selection operator is presented which creates a mating pool by combining the parent and child populations and selecting the best (with respect to fitness and spread) N solutions. Simulation results on five difficult test problems show that the proposed NSGA-II, in most problems, is able to find much better spread of solutions and better convergence near the true Pareto-optimal front compared to PAES and SPEA—two other elitist multi-objective EAs which pay special attention towards creating a diverse Pareto-optimal front. Because of NSGA-II's low computational requirements, elitist approach, and parameter-less sharing approach, NSGA-II should find increasing applications in the years to come.
In trying to solve multiobjective optimization problems, many traditional methods scalarize the objective vector into a single objective. In those cases, the obtained solution is highly sensitive to the weight vector used in the scalarization process and demands that the user have knowledge about the underlying problem. Moreover, in solving multiobjective problems, designers may be interested in a set of Pareto-optimal points, instead of a single point. Since genetic algorithms (GAs) work with a population of points, it seems natural to use GAs in multiobjective optimization problems to capture a number of solutions simultaneously. Although a vector evaluated GA (VEGA) has been implemented by Schaffer and has been tried to solve a number of multiobjective problems, the algorithm seems to have bias toward some regions. In this paper, we investigate Goldberg's notion of nondominated sorting in GAs along with a niche and speciation method to find multiple Pareto-optimal points simultaneously. The proof-of-principle results obtained on three problems used by Schaffer and others suggest that the proposed method can be extended to higher dimensional and more difficult multiobjective problems. A number of suggestions for extension and application of the algorithm are also discussed.
