Multi-objective evolutionary algorithms which use non-dominated sorting and sharing have been mainly criticized for their (i) O(MN 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(MN ) 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 a number of 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. Moreover, we modify the definition of dominance in order to solve constrained multi-objective problems eciently. Simulation results of the constrained NSGA-II on a number of test problems, including a five-objective, seven-constraint non-linear problem, are compared with another constrained multi-objective optimizer and much better performance of NSGA-II is observed. Because of NSGA-II's low computational requirements, elitist approach, parameter-less niching approach, and simple constraint-handling strategy, NSGA-II should find increasing applications in the coming years.

Over the past few years, the research on evolutionary algorithms has demonstrated their niche in solving multiobjective optimization problems, where the goal is to �nd a number of Pareto-optimal solutions in a single simulation run. Many studies have depicted different ways evolutionary algorithms can progress towards the Paretooptimal set with a widely spread distribution of solutions. However, none of the multiobjective evolutionary algorithms (MOEAs) has a proof of convergence to the true Pareto-optimal solutions with a wide diversity among the solutions. In this paper, we discuss why a number of earlier MOEAs do not have such properties. Based on the concept of-dominance, new archiving strategies are proposed that overcome this fundamental problem and provably lead to MOEAs that have both the desired convergence and distribution properties. A number of modi�cations to the baseline algorithm are also suggested. The concept of-dominance introduced in this paper is practical and should make the proposed algorithms useful to researchers and practitioners alike.

For the first time, a running time analysis of a multi-objective evolutionary algorithm for a discrete optimization problem is given. To this end, a simple pseudo-Boolean problem (Lotz: leading ones - trailing zeroes) is defined and a population-based optimization algorithm (FEMO). We show, that the algorithm performs a black box optimization in #(n 2 log n) function evaluations where n is the number of binary decision variables. 1

Over the past few years, the research on evolutionary algorithms has demonstrated their niche in solving multi-objective optimization problems, where the goal is to find a number of Pareto-optimal solutions in a single simulation run. However, none of the multi-objective evolutionary algorithms (MOEAs) has a proof of convergence to the true Pareto-optimal solutions with a wide diversity among the solutions. In this paper we discuss why a number of earlier MOEAs do not have such properties. A new archiving strategy is proposed that maintains a subset of the generated solutions. It guarantees convergence and diversity according to welldefined criteria, i.e. #-dominance and #-Pareto optimality.

Over the past few years, the research on evolutionary algorithms

Abstract—Sesame is a software framework that aims at developing a modeling and simulation environment for the efficient design space exploration of heterogeneous embedded systems. Since Sesame recognizes separate application and architecture models within a single system simulation, it needs an explicit mapping step to relate these models for cosimulation. The design tradeoffs during the mapping stage, namely, the processing time, power consumption, and architecture cost, are captured by a multiobjective nonlinear mixed integer program. This paper aims at investigating the performance of multiobjective evolutionary algorithms (MOEAs) on solving large instances of the mapping problem. With two comparative case studies, it is shown that MOEAs provide the designer with a highly accurate set of solutions in a reasonable amount of time. Additionally, analyses for different crossover types, mutation usage, and repair strategies for the purpose of constraints handling are carried out. Finally, a number of multiobjective optimization results are simulated for verification. Index Terms—Design space exploration, evolutionary algorithms, mixed integer programming, multiobjective optimization, multiprocessor system-on-chip (SoC) design. I.

Introduction The Gaussian distribution is the predominant choice for modeling noise frequently observable in measurings of various kinds. Here, we hold the view that a noise distribution with unbounded support (like the Gaussian, Cauchy, Laplace, Logistic, and others) may be quite unrealistic. Actually it is at least equally plausible to assume that the noise cannot exceed certain limits due to technical characteristics of the involved measurement unit. Even if a distributional shape close to a Gaussian appears reasonable we can resort to a symmetrical Beta distribution which can converge weakly to a Gaussian distribution under continuously increasing but bounded support (see e.g. Evans et al. 1993, p. 36). This assumption will have significant theoretical and practical impacts on the evolutionary algorithms (EAs) considered here. Traditional measures for coping with noisy fitness functions in evolutionary algorithms include the resampling of the random fitness value with averagi

This paper presents a rigorous running time analysis of evolutionary algorithms on pseudo-Boolean multiobjective optimization problems. We propose and analyze dierent population-based algorithms, the simple evolutionary multiobjective optimizer SEMO and two improved versions, FEMO and GEMO. The analysis is carried out on two bi-objective model problems, LOTZ (Leading Ones Trailing Zeroes) and COCZ (Count Ones Count Zeroes) as well as on the scalable m-objective versions mLOTZ and mCOCZ. Results on the running time of the dierent population-based algorithms and for an alternative approach, a multistart (1+1)-EA based on the -constraint method, are derived The comparison reveals that for many problems, the simple algorithm SEMO is as ecient as the (1+1)-EA. For some problems, the improved variants FEMO and GEMO are provably better. For the analysis we propose and apply two general tools, an upper bound technique based on a decision space partition and a randomized graph search algorithm, which facilitate the analysis considerably.

Presently, the limit theory of evolutionary algorithms (EA) for mono-criterion optimization under certainty is well developed. The situation is different for the fields of evolutionary optimization under complete or partial uncertainty, multiple criteria and so forth. Since these problem classes may be seen as special cases of the task of finding the set of minimal (or maximal) elements in partially ordered sets, a limit theory for EAs that can cope with this kind of problem passes all properties and results on its special cases mentioned above. 1 Introduction The theory of evolutionary algorithms (EAs) in the framework of stochastic processes is best developed currently for the field of optimization of a single deterministic objective function (see e.g. [7] for a survey). There is also a steadily growing theory for EAs facing a (single) stochastically perturbed objective function as can be learned from the overview presented in [1]. In case of multiple objective functions, howe...

In contemporary credit portfolio management, the portfolio risk-return analysis of financial instruments using certain downside credit risk measures requires the computation of a set of Pareto-efficient portfolio structures in a non-linear, non-convex setting. For real-world problems, additional constraints, e. g. supervisory capital limits, have to be respected. Particularly for formerly non-traded instruments, e. g. corporate loans, a discrete set of decision alternatives has to be considered for each financial instrument. The main result of this paper is a new, fast and flexible framework for solving the above issues using a hybrid heuristic method that combines multi-objective evolutionary and problem-specific local search methods in a unique way. We explicitly incorporate computational complexity in some of our considerations and consider proper genetic modelling of portfolio credit risk related problems. Also, we analyse empirical results from a study based on our implementation of the proposed hybrid method in a specific portfolio credit risk model context. These results show that this method is superior in convergence speed to a non-hybrid evolutionary approach and that our implementation finds risk-return efficient sets within reasonable time. Key words: portfolio credit risk, downside risk, Credit-Value-at-Risk, constrained discrete portfolio optimisation, portfolio selection, hybrid multi-objective evolutionary algorithm JEL Classification: G11, G20, G31, G33