Showing papers on "Multi-objective optimization published in 2003"

Performance assessment of multiobjective optimizers: an analysis and review

Abstract: An important issue in multiobjective optimization is the quantitative comparison of the performance of different algorithms. In the case of multiobjective evolutionary algorithms, the outcome is usually an approximation of the Pareto-optimal set, which is denoted as an approximation set, and therefore the question arises of how to evaluate the quality of approximation sets. Most popular are methods that assign each approximation set a vector of real numbers that reflect different aspects of the quality. Sometimes, pairs of approximation sets are also considered. In this study, we provide a rigorous analysis of the limitations underlying this type of quality assessment. To this end, a mathematical framework is developed which allows one to classify and discuss existing techniques.

Theory of Vector Optimization

Abstract: We introduce several solution concepts for multicriteria optimization problems, give a characterization of approximately efficient elements and discuss a general scalarization procedure. Furthermore, we derive necessary and sufficient optimality conditions, a minimal point theorem, a vector-valued variational principle of Ekeland’s type, Lagrangean multiplier rules and duality statements. An overview on vector variational inequalities and vector equilibria is given. Moreover, we discuss the results for special classes of vector optimization problems (vector-valued location and approximation problems, multicriteria fractional programming and optimal control problems).

The normalized normal constraint method for generating the Pareto frontier

Abstract: The authors recently proposed the normal constraint (NC) method for generating a set of evenly spaced solutions on a Pareto frontier – for multiobjective optimization problems. Since few methods offer this desirable characteristic, the new method can be of significant practical use in the choice of an optimal solution in a multiobjective setting. This paper’s specific contribution is two-fold. First, it presents a new formulation of the NC method that incorporates a critical linear mapping of the design objectives. This mapping has the desirable property that the resulting performance of the method is entirely independent of the design objectives scales. We address here the fact that scaling issues can pose formidable difficulties. Secondly, the notion of a Pareto filter is presented and an algorithm thereof is developed. As its name suggests, a Pareto filter is an algorithm that retains only the global Pareto points, given a set of points in objective space. As is explained in the paper, the Pareto filter is useful in the application of the NC and other methods. Numerical examples are provided.

The balance between proximity and diversity in multiobjective evolutionary algorithms

Abstract: Over the last decade, a variety of evolutionary algorithms (EAs) have been proposed for solving multiobjective optimization problems. Especially more recent multiobjective evolutionary algorithms (MOEAs) have been shown to be efficient and superior to earlier approaches. An important question however is whether we can expect such improvements to converge onto a specific efficient MOEA that behaves best on a large variety of problems. In this paper, we argue that the development of new MOEAs cannot converge onto a single new most efficient MOEA because the performance of MOEAs shows characteristics of multiobjective problems. While we point out the most important aspects for designing competent MOEAs in this paper, we also indicate the inherent multiobjective tradeoff in multiobjective optimization between proximity and diversity preservation. We discuss the impact of this tradeoff on the concepts and design of exploration and exploitation operators. We also present a general framework for competent MOEAs and show how current state-of-the-art MOEAs can be obtained by making choices within this framework. Furthermore, we show an example of how we can separate nondomination selection pressure from diversity preservation selection pressure and discuss the impact of changing the ratio between these components.

Multiobjective Optimization: Principles and Case Studies

Abstract: I Principles of multiobjective optimization methods.- 1 Introduction: multiobjective optimization and domination.- 2 Scalar methods.- 3 Interactive methods.- 4 Fuzzy methods.- 5 Multiobjective methods using metaheuristics.- 6 Decision aid methods.- II Evaluation of methods, and criteria for choice of method.- 7 Performance measurement.- 8 Test functions for multiobjective optimization methods.- 9 An attempt to classify multiobjective optimization methods.- III Case studies.- 10 Case study 1: qualification of scientific software.- 11 Case study 2: study of the extension of a telecommunication network.- 12 Case study 3: multicriteria decision tools to deal with bids.- 13 Conclusion.- References.

Effective and efficient algorithm for multiobjective optimization of hydrologic models

Abstract: [1] Practical experience with the calibration of hydrologic models suggests that any single-objective function, no matter how carefully chosen, is often inadequate to properly measure all of the characteristics of the observed data deemed to be important. One strategy to circumvent this problem is to define several optimization criteria (objective functions) that measure different (complementary) aspects of the system behavior and to use multicriteria optimization to identify the set of nondominated, efficient, or Pareto optimal solutions. In this paper, we present an efficient and effective Markov Chain Monte Carlo sampler, entitled the Multiobjective Shuffled Complex Evolution Metropolis (MOSCEM) algorithm, which is capable of solving the multiobjective optimization problem for hydrologic models. MOSCEM is an improvement over the Shuffled Complex Evolution Metropolis (SCEM-UA) global optimization algorithm, using the concept of Pareto dominance (rather than direct single-objective function evaluation) to evolve the initial population of points toward a set of solutions stemming from a stable distribution (Pareto set). The efficacy of the MOSCEM-UA algorithm is compared with the original MOCOM-UA algorithm for three hydrologic modeling case studies of increasing complexity.

PISA: A platform and programming language independent interface for search algorithms

Abstract: This paper introduces an interface specification (PISA) that allows to separate the problem-specific part of an optimizer from the problem-independent part. We propose a view of the general optimization scenario, where the problem representation together with the variation operators is seen as an integral part of the optimization problem and can hence be easily separated from the selection operators. Both parts are implemented as independent programs, that can be provided as ready-to-use packages and arbitrarily combined. This makes it possible to specify and implement representation-independent selection modules, which form the essence of modern multiobjective optimization algorithms. The variation operators, on the other hand, have to be defined in one module together with the optimization problem, facilitating a customized problem description. Besides the specification, the paper contains a correctness proof for the protocol and measured efficiency results.

Reducing the run-time complexity of multiobjective EAs: The NSGA-II and other algorithms

Abstract: The last decade has seen a surge of research activity on multiobjective optimization using evolutionary computation and a number of well performing algorithms have been published. The majority of these algorithms use fitness assignment based on Pareto-domination: Nondominated sorting, dominance counting, or identification of the nondominated solutions. The success of these algorithms indicates that this type of fitness is suitable for multiobjective problems, but so far the use of Pareto-based fitness has lead to program run times in O(GMN/sup 2/), where G is the number of generations, M is the number of objectives, and N is the population size. The N/sup 2/ factor should be reduced if possible, since it leads to long processing times for large population sizes. This paper presents a new and efficient algorithm for nondominated sorting, which can speed up the processing time of some multiobjective evolutionary algorithms (MOEAs) substantially. The new algorithm is incorporated into the nondominated sorting genetic algorithm II (NSGA-II) and reduces the overall run-time complexity of this algorithm to O(GN log/sup M-1/N), much faster than the O(GMN/sup 2/) complexity published by Deb et al. (2002). Experiments demonstrate that the improved version of the algorithm is indeed much faster than the previous one. The paper also points out that multiobjective EAs using fitness based on dominance counting and identification of nondominated solutions can be improved significantly in terms of running time by using efficient algorithms known from computer science instead of inefficient O(MN/sup 2/) algorithms.

Performance scaling of multi-objective evolutionary algorithms

Abstract: MOEAs are getting immense popularity in the recent past, mainly because of their ability to find a wide spread of Pareto-optimal solutions in a single simulation run. Various evolutionary approaches to multi-objective optimization have been proposed since 1985. Some of fairly recent ones are NSGA-II, SPEA2, PESA (which are included in this study) and others. They all have been mainly applied to two to three objectives. In order to establish their superiority over classical methods and demonstrate their abilities for convergence and maintenance of diversity, they need to be tested on higher number of objectives. In this study, these state-of-the-art MOEAs have been investigated for their scalability with respect to the number of objectives (2 to 8). They have also been compared on the basis of -(1) Their ability to converge to Pareto front, (2) Diversity of obtained non-dominated solutions and (3) Their running time. Four scalable test problems (DTLZ1, 2, 3 and 6) are used for the comparative study.

The measure of pareto optima: Applications to multi-objective metaheuristics

Abstract: This article describes a set function that maps a set of Pareto optimal points to a scalar. A theorem 1 is presented that shows that the maximization of this scalar value constitutes the necessary and sufficient condition for the function's arguments to be maximally diverse Pareto optimal solutions of a discrete, multi-objective, optimization problem. This scalar quantity, a hypervolume based on a Lebesgue measure, is therefore the best metric to assess the quality of multiobjective optimization algorithms. Moreover, it can be used as the objective function in simulated annealing (SA) to induce convergence in probability to the Pareto optima. An efficient, polynomial-time algorithm for calculating this scalar and an analysis of its complexity is also presented.

Engineering optimization with particle swarm

Abstract: The paper presents a modified particle swarm optimization (PSO) algorithm for engineering optimization problems with constraints. PSO is started with a group of feasible solutions and a feasibility function is used to check if the newly explored solutions satisfy all the constraints. All the particles keep only those feasible solutions in their memory. Several engineering design optimization problems were tested and the results show that PSO is an efficient and general approach to solve most nonlinear optimization problems with inequity constraints.

PISA: a platform and programming language independent interface for search algorithms

Abstract: This paper introduces an interface specification (PISA) that allows to separate the problem-specific part of an optimizer from the problem-independent part. We propose a view of the general optimization scenario, where the problem representation together with the variation operators is seen as an integral part of the optimization problem and can hence be easily separated from the selection operators. Both parts are implemented as independent programs, that can be provided as ready-to-use packages and arbitrarily combined. This makes it possible to specify and implement representation-independent selection modules, which form the essence of modern multiobjective optimization algorithms. The variation operators, on the other hand, have to be defined in one module together with the optimization problem, facilitating a customized problem description. Besides the specification, the paper contains a correctness proof for the protocol and measured efficiency results.

Nondominated sorting genetic algorithm for optimal phasor measurement placement

Abstract: The paper considers a phasor measurement unit (PMU) placement problem requiring simultaneous optimization of two conflicting objectives, such as minimization of the number of PMUs and maximization of the measurement redundancy. The objectives are in conflict, since the improvement of one of them leads to the deterioration of another. Instead of unique optimal solution, it exists a set of best trade-offs between competing objectives, the so-called Pareto-optimal solutions. A specially tailored nondominated sorting genetic algorithm (NSGA) for PMU placement problem is proposed as a methodology to find these Pareto-optimal solutions. The algorithm is combined with the graph-theoretical procedure and a simple GA to reduce initial number of the PMU's candidate locations. The NSGA parameters are carefully set by performing a number of trial runs and evaluating the NSGA performances based on the number of distinct Pareto-optimal solutions found in the particular run and the distance of the obtained Pareto front from the optimal one. Illustrative results on the 39-bus and 118-bus IEEE systems are presented.

Properties of an adaptive archiving algorithm for storing nondominated vectors

Abstract: Search algorithms for Pareto optimization are designed to obtain multiple solutions, each offering a different trade-off of the problem objectives. To make the different solutions available at the end of an algorithm run, procedures are needed for storing them, one by one, as they are found. In a simple case, this may be achieved by placing each point that is found into an "archive" which maintains only nondominated points and discards all others. However, even a set of mutually nondominated points is potentially very large, necessitating a bound on the archive's capacity. But with such a bound in place, it is no longer obvious which points should be maintained and which discarded; we would like the archive to maintain a representative and well-distributed subset of the points generated by the search algorithm, and also that this set converges. To achieve these objectives, we propose an adaptive archiving algorithm, suitable for use with any Pareto optimization algorithm, which has various useful properties as follows. It maintains an archive of bounded size, encourages an even distribution of points across the Pareto front, is computationally efficient, and we are able to prove a form of convergence. The method proposed here maintains evenness, efficiency, and cardinality, and provably converges under certain conditions but not all. Finally, the notions underlying our convergence proofs support a new way to rigorously define what is meant by "good spread of points" across a Pareto front, in the context of grid-based archiving schemes. This leads to proofs and conjectures applicable to archive sizing and grid sizing in any Pareto optimization algorithm maintaining a grid-based archive.

Introduction to applied optimization

Abstract: Provides well-written self-contained chapters, including problem sets and exercises, making it ideal for the classroom setting; Introducesapplied optimization to the hazardous waste blending problem; Explores linear programming, nonlinear programming, discrete optimization, global optimization, optimization under uncertainty, multi-objective optimization, optimal control and stochastic optimal control; Includes an extensive bibliography at the end of each chapter and an index; GAMS files of case studies for Chapters 2, 3, 4, 5, and 7are linked to http://www.springer.com/math/book/978-0-387-76634-8; Solutions manual available upon adoptions.

Pareto-based multi-objective differential evolution

Abstract: Evolutionary multiobjective optimization (EMOO) finds a set of Pareto solutions rather than any single aggregated optimal solution for a multiobjective problem. The purpose is to describe a newly developed evolutionary approach-Pareto-based multiobjective differential evolution (MODE). The concept of differential evolution, which is well-known in the continuous single-objective domain for its fast convergence and adaptive parameter setting, is extended to the multiobjective problem domain. A Pareto-based approach is proposed to implement the differential vectors. A set of benchmark test functions is used to validate this new approach. We compare the computational results with those obtained in the literature, specifically by strength Pareto evolutionary algorithm (SPEA). It is shown that this new approach tends to be more effective in finding the Pareto front in the sense of accuracy and approximate representation of the real Pareto front with comparable efficiency.

A multiobjective evolutionary algorithm for solving vehicle routing problem with time windows

Abstract: Vehicle routing problem with time windows (VRPTW) involves the routing of a set of vehicles with limited capacity from a central depot to a set of geographically dispersed customers with known demands and predefined time windows. This paper proposes a hybrid multiobjective evolutionary algorithm (HMOEA) that incorporates various heuristics for local exploitation in the evolutionary search and the concept of Pareto's optimality for solving multiobjective optimization in VRPTW problems. The proposed HMOEA optimizes all routing constraints and objectives simultaneously, which improves the routing solutions in many aspects, such as lower routing cost, wider scattering area and better convergence trace.

Particle swarm with extended memory for multiobjective optimization

Abstract: This paper presents a modified dynamic neighborhood particle swarm optimization (DNPSO) algorithm for multiobjective optimization problems. PSO is modified by using a dynamic neighborhood strategy, new particle memory updating, and one-dimension optimization to deal with multiple objectives. An extended memory is introduced to store global Pareto optimal solutions to reduce computation time. Several benchmark cases were tested and the results show that the modified DNPSO is much more efficient than the original DNPSO and other multiobjective optimization techniques.

Differential evolution for multi-objective optimization

Abstract: Two test problems on multiobjective optimization (one simple general problem and the second one on an engineering application of cantilever design problem) are solved using differential evolution (DE). DE is a population based search algorithm, which is an improved version of genetic algorithm (GA), Simulations carried out involved solving (1) both the problems using Penalty function method, and (2) first problem using Weighing factor method and finding Pareto optimum set for the chosen problem, DE found to be robust and faster in optimization. To consolidate the power of DE, the classical Himmelblau function, with bounds on variables, is also solved using both DE and GA. DE found to give the exact optimum value within less generations compared to simple GA.

Using unconstrained elite archives for multiobjective optimization

Abstract: Multiobjective evolutionary algorithms (MOEAs) have been the subject of numerous studies over the past 20 years. Recent work has highlighted the use of an active archive of elite, nondominated solutions to improve the optimization speed of these algorithms. However, preserving all elite individuals is costly in time (due to the linear comparison with all archived solutions needed before a new solution can be inserted into the archive). Maintaining an elite population of a fixed maximum size (by clustering or other means) alleviates this problem, but can cause retreating (or oscillitory) and shrinking estimated Pareto fronts - which can affect the efficiency of the search process. New data structures are introduced to facilitate the use of an unconstrained elite archive, without the need for a linear comparison to the elite set for every new individual inserted. The general applicability of these data structures is shown by their use in an evolution-strategy-based MOEA and a genetic-algorithm-based MOEA. It is demonstrated that MOEAs using the new data structures run significantly faster than standard, unconstrained archive MOEAs, and result in estimated Pareto fronts significantly ahead of MOEAs using a constrained archive. It is also shown that the use of an unconstrained elite archive permits robust criteria for algorithm termination to be used, and that the use of the data structure can also be used to increase the speed of algorithms using /spl epsi/-dominance methods.

Genetic algorithm for the flexible job-shop scheduling problem

Abstract: In this paper, we are interested in the multiobjective optimization of the schedule performance in the flexible job shops. The flexible job shop scheduling problem (FJSP) is known in the literature as one of the hardest combinatorial optimization problems and presents many objectives to be optimized. In this way, we aim to solve such a problem according to a set of some criteria, which characterize the feasible solutions of such a problem. The studied criteria are the following: the makespan, the workload of the critical machine, and the total workload of all the machines. Our study relates to the determination of a practical method using genetic algorithm in order to obtain the best performance of the production system. The solution performance is evaluated by comparing the values of the different values of the criteria with the corresponding lower bounds.

Designing evolutionary algorithms for dynamic optimization problems

Abstract: Most research in evolutionary computation focuses on optimization of static, non-changing problems. Many real-world optimization problems, however, are dynamic, and optimization methods are needed that are capable of continuously adapting the solution to a changing environment. If the optimization problem is dynamic, the goal is no longer to find the extrema, but to track their progression through the space as closely as possible. In this chapter, we suggest a classification of dynamic optimization problems, and survey and classify a number of the most widespread techniques that have been published in the literature so far to make evolutionary algorithms suitable for changing optimization problems. After this introduction to the basics, we will discuss in more detail two specific approaches, pointing out their deficiencies and potential. The first approach is based on memorization, the other one uses a novel multi-population structure.

Trade-off between performance and robustness: an evolutionary multiobjective approach

Abstract: In real-world applications, it is often desired that a solution is not only of high performance, but also of high robustness. In this context, a solution is usually called robust, if its performance only gradually decreases when design variables or environmental parameters are varied within a certain range. In evolutionary optimization, robust optimal solutions are usually obtained by averaging the fitness over such variations. Frequently, maximization of the performance and increase of the robustness are two conflicting objectives, which means that a trade-off exists between robustness and performance. Using the existing methods to search for robust solutions, this trade-off is hidden and predefined in the averaging rules. Thus, only one solution can be obtained. In this paper, we treat the problem explicitly as a multi objective optimization task, thereby clearly identifying the trade-off between performance and robustness in the form of the obtained Pareto front. We suggest two methods for estimating the robustness of a solution by exploiting the information available in the current population of the evolutionary algorithm, without any additional fitness evaluations. The estimated robustness is then used as an additional objective in optimization. Finally, the possibility of using this method for detecting multiple optima of multimodal functions is briefly discussed.

Dynamic multiobjective evolutionary algorithm: adaptive cell-based rank and density estimation

Abstract: This paper proposes a new evolutionary approach to multiobjective optimization problems - the dynamic multiobjective evolutionary algorithm (DMOEA). In DMOEA, a novel cell-based rank and density estimation strategy is proposed to efficiently compute dominance and diversity information when the population size varies dynamically. In addition, a population growing and declining strategies are designed to determine if an individual will survive or be eliminated based on some qualitative indicators. Meanwhile, an objective space compression strategy is devised to continuously refine the quality of the resulting Pareto front. By examining the selected performance metrics on three recently designed benchmark functions, DMOEA is found to be competitive with or even superior to five state-of-the-art MOEAs in terms of maintaining the diversity of the individuals along the tradeoff surface, tending to extend the Pareto front to new areas, and finding a well-approximated Pareto optimal front. Moreover, DMOEA is evaluated by using different parameter settings on the chosen test functions to verify its robustness of converging to an optimal population size, if it exists. Simulations show that DMOEA has the potential of autonomously determining the optimal population size, which is found insensitive to the initial population size chosen.

Multiobjective optimization using parallel vector evaluated particle swarm optimization

Abstract: This paper studies a parallel version of the Vector Evaluated Particle Swarm Optimization (VEPSO) method for multiobjective problems. Experiments on well known and widely used test problems are performed, aiming at investigating both the efficiency of VEPSO as well as the advantages of the parallel implementation. The obtained results are compared with the corresponding results of the Vector Evaluated Genetic Algorithm approach, yielding the superiority of VEPSO.