Showing papers on "Multi-objective optimization published in 2002"

A fast and elitist multiobjective genetic algorithm: NSGA-II

Abstract: Multi-objective evolutionary algorithms (MOEAs) that use non-dominated sorting and sharing have been criticized mainly for: (1) their O(MN/sup 3/) computational complexity (where M is the number of objectives and N is the population size); (2) their non-elitism approach; and (3) the need to specify a sharing parameter. In this paper, we suggest a non-dominated sorting-based MOEA, called NSGA-II (Non-dominated Sorting Genetic Algorithm II), which alleviates all of the above three difficulties. Specifically, a fast non-dominated sorting approach with O(MN/sup 2/) computational complexity is presented. Also, a selection operator is presented that creates a mating pool by combining the parent and offspring populations and selecting the best N solutions (with respect to fitness and spread). Simulation results on difficult test problems show that NSGA-II is able, for most problems, to find a much better spread of solutions and better convergence near the true Pareto-optimal front compared to the Pareto-archived evolution strategy and the strength-Pareto evolutionary algorithm - two other elitist MOEAs that pay special attention to creating a diverse Pareto-optimal front. Moreover, we modify the definition of dominance in order to solve constrained multi-objective problems efficiently. Simulation results of the constrained NSGA-II on a number of test problems, including a five-objective, seven-constraint nonlinear problem, are compared with another constrained multi-objective optimizer, and the much better performance of NSGA-II is observed.

MOPSO: a proposal for multiple objective particle swarm optimization

Abstract: This paper introduces a proposal to extend the heuristic called "particle swarm optimization" (PSO) to deal with multiobjective optimization problems. Our approach uses the concept of Pareto dominance to determine the flight direction of a particle and it maintains previously found nondominated vectors in a global repository that is later used by other particles to guide their own flight. The approach is validated using several standard test functions from the specialized literature. Our results indicate that our approach is highly competitive with current evolutionary multiobjective optimization techniques.

Recent approaches to global optimization problems through Particle Swarm Optimization

Abstract: This paper presents an overview of our most recent results concerning the Particle Swarm Optimization (PSO) method. Techniques for the alleviation of local minima, and for detecting multiple minimizers are described. Moreover, results on the ability of the PSO in tackling Multiobjective, Minimax, Integer Programming and e1 errors-in-variables problems, as well as problems in noisy and continuously changing environments, are reported. Finally, a Composite PSO, in which the heuristic parameters of PSO are controlled by a Differential Evolution algorithm during the optimization, is described, and results for many well-known and widely used test functions are given.

Scalable multi-objective optimization test problems

Abstract: After adequately demonstrating the ability to solve different two-objective optimization problems, multi-objective evolutionary algorithms (MOEAs) must show their efficacy in handling problems having more than two objectives. In this paper, we suggest three different approaches for systematically designing test problems for this purpose. The simplicity of construction, scalability to any number of decision variables and objectives, knowledge of exact shape and location of the resulting Pareto-optimal front, and ability to control difficulties in both converging to the true Pareto-optimal front and maintaining a widely distributed set of solutions are the main features of the suggested test problems. Because of these features, they should be useful in various research activities on MOEAs, such as testing the performance of a new MOEA, comparing different MOEAs, and having a better understanding of the working principles of MOEAs.

Combining convergence and diversity in evolutionary multiobjective optimization

Abstract: Over the past few years, the research on evolutionary algorithms has demonstrated their niche in solving multiobjective optimization problems, where the goal is to find a number of Pareto-optimal solutions in a single simulation run. Many studies have depicted different ways evolutionary algorithms can progress towards the Pareto-optimal 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 e-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 modifications to the baseline algorithm are also suggested. The concept of e-dominance introduced in this paper is practical and should make the proposed algorithms useful to researchers and practitioners alike.

Particle swarm optimization method in multiobjective problems

Abstract: This paper constitutes a first study of the Particle Swarm Optimization (PSO) method in Multiobjective Optimization (MO) problems. The ability of PSO to detect Pareto Optimal points and capture the shape of the Pareto Front is studied through experiments on well-known non-trivial test functions. The Weighted Aggregation technique with fixed or adaptive weights is considered. Furthermore, critical aspects of the VEGA approach for Multiobjective Optimization using Genetic Algorithms are adapted to the PSO framework in order to develop a multi-swarm PSO that can cope effectively with MO problems. Conclusions are derived and ideas for further research are proposed.

Multiobjective optimization using dynamic neighborhood particle swarm optimization

Abstract: This paper presents a particle swarm optimization (PSO) 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. Several benchmark cases were tested and showed that PSO could efficiently find multiple Pareto optimal solutions.

Reliable robust flight tracking control: an LMI approach

Abstract: This paper studies the reliable robust tracking controller design problem against actuator faults and control surface impairment for aircraft. First, models of actuator faults and control surface impairment are presented. Then a reliable robust tracking controller design method is developed. This method is based on the mixed linear quadratic (LQ)//H/sub /spl infin// tracking performance index and multiobjective optimization in terms of linear matrix inequalities. Flight control examples are given, and both linear and nonlinear simulations are given.

The self-adaptive Pareto differential evolution algorithm

Abstract: The Pareto differential evolution (PDE) algorithm was introduced and showed competitive results. The behavior of PDE, as in many other evolutionary multiobjective optimization (EMO) methods, varies according to the crossover and mutation rates. In this paper, we present a new version of PDE with self-adaptive crossover and mutation. We call the new version self-adaptive Pareto differential evolution (SPDE). The emphasis of this paper is to analyze the dynamics and behavior of SPDE. The experiments also show that the algorithm is very competitive with other EMO algorithms.

A Swarm Metaphor for Multiobjective Design Optimization

Abstract: This paper presents a new optimization algorithm to solve multiobjective design optimization problems based on behavioral concepts similar to that of a real swarm. The individuals of a swarm update their flying direction through communication with their neighboring leaders with an aim to collectively attain a common goal. The success of the swarm is attributed to three fundamental processes: identification of a set of leaders, selection of a leader for information acquisition, and finally a meaningful information transfer scheme. The proposed algorithm mimics the above behavioral processes of a real swarm. The algorithm employs a multilevel sieve to generate a set of leaders, a probabilistic crowding radius-based strategy for leader selection and a simple generational operator for information transfer. Two test problems, one with a discontinuous Pareto front and the other with a multi-modal Pareto front is solved to illustrate the capabilities of the algorithm in handling mathematically complex problems. ...

Nondominated Sorting Genetic Algorithm for Optimal Phasor Maesurement Placement

Abstract: This 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, for the improvement of one of them leads to deterioration of another. Consequently, instead of a unique optimal solution, there exists a set of the best trade-offs between competing objectives, the so-called Pareto-optimal solutions. A specially tailored nondominated sorting genetic algorithm (NSGA) for the 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 the initial number of the PMU 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.

On the Global Convergence of a Filter--SQP Algorithm

Abstract: A mechanism for proving global convergence in SQP--filter methods for nonlinear programming (NLP) is described. Such methods are characterized by their use of the dominance concept of multiobjective optimization, instead of a penalty parameter whose adjustment can be problematic. The main point of interest is to demonstrate how convergence for NLP can be induced without forcing sufficient descent in a penalty-type merit function. The proof relates to a prototypical algorithm, within which is allowed a range of specific algorithm choices associated with the Hessian matrix representation, updating the trust region radius, and feasibility restoration.

Multipoint and Multi-Objective Aerodynamic Shape Optimization

Abstract: A Newton‐Krylov algorithm is presented for the aerodynamic optimization of singleand multi-element airfoil configurations. The flow is governed by the compressible Navier‐Stokes equations in conjunction with a one-equation turbulence model. The preconditioned generalized minimum residual method is applied to solve the discreteadjoint equation, leading to a fast computation of accurate objective function gradients. Optimization constraints are enforced through a penalty formulation, and the resulting unconstrained problem is solved via a quasi-Newton method. Design examples include lift-enhancement and multi-point lift-constrained drag minimization problems. Furthermore, the new algorithm is used to compute a Pareto front for a multi-objective problem, and the results are validated using a genetic algorithm. Overall, the new algorithm provides an ecient and robust approach for addressing the issues of complex aerodynamic

Multiobjective optimization using a Pareto differential evolution approach

Abstract: Differential evolution is a simple, fast, and robust evolutionary algorithm that has proven effective in determining the global optimum for several difficult single-objective optimization problems. In this paper, the differential evolution algorithm is extended to multiobjective optimization problems by using a Pareto-based approach. The algorithm performs well when applied to several test optimization problems from the literature.

On scalarizing functions in multiobjective optimization

Abstract: Scalarizing functions play an essential role in solving multiobjective optimization problems. Many different scalarizing functions have been suggested in the literature based on different approaches. Here we concentrate on classification and reference point-based functions. We present a collection of functions that have been used in interactive methods as well as some modifications. We compare their theoretical properties and numerical behaviour. In particular, we are interested in the relation between the information provided and the results obtained. Our aim is to select some of them to be used in our WWW-NIMBUS optimization system.

Preferences and their application in evolutionary multiobjective optimization

Abstract: The paper describes a new preference method and its use in multiobjective optimization. These preferences are developed with a goal to reduce the cognitive overload associated with the relative importance of a certain criterion within a multiobjective design environment involving large numbers of objectives. Their successful integration with several genetic-algorithm-based design search and optimization techniques (weighted sums, weighted Pareto, weighted co-evolutionary methods, and weighted scenarios) are described and theoretical results relating to complexity and sensitivity of the algorithm are presented and discussed. Its usefulness was demonstrated in a real-world project of conceptual airframe design.

Evolutionary Algorithms for Multi-Objective Optimization: Performance Assessments and Comparisons

Abstract: Evolutionary techniques for multi-objective (MO) optimization are currently gaining significant attention from researchers in various fields due to their effectiveness and robustness in searching for a set of trade-off solutions. Unlike conventional methods that aggregate multiple attributes to form a composite scalar objective function, evolutionary algorithms with modified reproduction schemes for MO optimization are capable of treating each objective component separately and lead the search in discovering the global Pareto-optimal front. The rapid advances of multi-objective evolutionary algorithms, however, poses the difficulty of keeping track of the developments in this field as well as selecting an existing approach that best suits the optimization problem in-hand. This paper thus provides a survey on various evolutionary methods for MO optimization. Many well-known multi-objective evolutionary algorithms have been experimented with and compared extensively on four benchmark problems with different MO optimization difficulties. Besides considering the usual performance measures in MO optimization, e.g., the spread across the Pareto-optimal front and the ability to attain the global trade-offs, the paper also presents a few metrics to examine the strength and weakness of each evolutionary approach both quantitatively and qualitatively. Simulation results for the comparisons are analyzed, summarized and commented.

Analytical and Computational Aspects of Collaborative Optimization for Multidisciplinary Design

Abstract: Analytical features of multidisciplinary optimization (MDO) problem formulations have significant practical consequences for the ability of nonlinear programming algorithms to solve the resulting computational optimization problems reliably and efficiently. We explore this important but frequently overlooked fact using the notion of disciplinary autonomy. Disciplinary autonomy is a desirable goal in formulating and solving MDO problems; however, the resulting system optimization problems are frequently difficult to solve. We illustrate the implications of MDO problem formulation for the tractability of the resulting design optimization problem by examining a representative class of MDO problem formulations known as collaborative optimization. We also discuss an alternative problem formulation, distributed analysis optimization, that yields a more tractable computational optimization problem.

On the optimal solution definition for many-criteria optimization problems

Abstract: When dealing with many-criteria decision making and many-objectives optimization problems the concepts of Pareto optimality and Pareto dominance are inefficient for modelling and simulating human decision making Different fuzzy-based definitions of optimality and dominated solution are introduced and tested on analytical test cases in order to show their validity and closeness to human decision making

Multiobjective robust design using physical programming

Abstract: This paper develops a flexible Physical Programming (PP) based Robust Design Optimization (RDO) method. RDO yields a system that performs with minimal variability in the face of input variations or uncertainties. RDO methods generally seek to minimize the variation of an Aggregate Objective Function (AOF), thereby also minimizing the effect of input variations on the physical system --- it is hoped. RDO seeks to maintain design feasibility under input variations. The optimization outcome depends on (i) the acceptable level of variation in performance, and (ii) the level of input variations. For RDO to be flexible, it is desirable to allow designers to express preference with respect to specific design metric values, and variation levels thereof. Similarly, RDO would ideally allow the designer either to prescribe parameter tolerance levels, or to maximize their allowable levels to minimize manufacturing cost. This paper's approach possesses the above desirable features, and provides numerical examples. This paper also discusses the benefits of formulating RDO in terms of physically meaningful design performance degradation levels, rather than in terms of the variation of an AOF.

Product variety optimization under modular architecture

Abstract: In this paper, product variety design is discussed under an optimization viewpoint. Product variety design means the challenge to simultaneously design multiple products for achieving higher optimality beyond ordinary design methods for a single product. When the possibilities of computational optimization for product variety design under fixed product architecture are explored, optimization is demanded to determine the contents of modules and their combinations under fixed modular architecture. This indicates that product variety optimization includes three classes of optimization problems: attribute assignment, module combination, and simultaneous design of both. Following problem classification, the domains and situations of such optimization problems are investigated. Then, two typical optimization examples are demonstrated through aircraft design for simultaneous optimal attribute assignment and through design of television receiver circuits for optimal module combination, respectively. The discussion concludes with the roles of problem classification and the direction of future works.

Optimal control of switched systems via non-linear optimization based on direct differentiations of value functions

Abstract: This paper presents an approach for solving optimal control problems of switched systems. In general, in such problems one needs to find both optimal continuous inputs and optimal switching sequences, since the system dynamics vary before and after every switching instant. After formulating a general optimal control problem, we propose a two stage optimization methodology. Since many practical problems only concern optimization where the number of switchings and the sequence of active subsystems are given, we concentrate on such problems and propose a method which uses nonlinear optimization and is based on direct differentiations of value functions. The method is then applied to general switched linear quadratic (GSLQ) problems. Examples illustrate the results.

Developments in structural-acoustic optimization for passive noise control

Abstract: Low noise constructions receive more and more attention in highly industrialized countries. Consequently, decrease of noise radiation challenges a growing community of engineers. One of the most efficient techniques for finding quiet structures consists in numerical optimization. Herein, we consider structural-acoustic optimization understood as an (iterative) minimum search of a specified objective (or cost) function by modifying certain design variables. Obviously, a coupled problem must be solved to evaluate the objective function. In this paper, we will start with a review of structural and acoustic analysis techniques using numerical methods like the finite- and/or the boundary-element method. This is followed by a survey of techniques for structural-acoustic coupling. We will then discuss objective functions. Often, the average sound pressure at one or a few points in a frequency interval accounts for the objective function for interior problems, wheareas the average sound power is mostly used for external problems. The analysis part will be completed by review of sensitivity analysis and special techniques. We will then discuss applications of structural-acoustic optimization. Starting with a review of related work in pure structural optimization and in pure acoustic optimization, we will categorize the problems of optimization in structural acoustics. A suitable distinction consists in academic and more applied examples. Academic examples iclude simple structures like beams, rectangular or circular plates and boxes; real industrial applications consider problems like that of a fuselage, bells, loudspeaker diaphragms and components of vehicle structures. Various different types of variables are used as design parameters. Quite often, locally defined plate or shell thickness or discrete point masses are chosen. Furthermore, all kinds of structural material parameters, beam cross sections, spring characteristics and shell geometry account for suitable design modifications. This is followed by a listing of constraints that have been applied. After that, we will discuss strategies of optimization. Starting with a formulation of the optimization problem we review aspects of multiobjective optimization, approximation concepts and optimization methods in general. In a final chapter, results are categorized and discussed. Very often, quite large decreases of noise radiation have been reported. However, even small gains should be highly appreciated in some cases of certain support conditions, complexity of simulation, model and large frequency ranges. Optimization outcomes are categorized with respect to objective functions, optimization methods, variables and groups of problems, the latter with particular focus on industrial applications. More specifically, a close-up look at vehicle panel shell geometry optimization is presented. Review of results is completed with a section on experimental validation of optimization gains. The conclusions bring together a number of open problems in the field.

Multiobjective optimization of SMB and varicol process for chiral separation

Abstract: The multiobjective optimization of continuous countercurrent chromatography separation units, such as simulated moving bed (SMB) and Varicol, is considered. The Varicol system is based on a nonsynchronous shift of the inlet and outlet ports instead of the synchronous one used in the SMB technology. The optimization problem is complicated by the relative large number of decision variables, including continuous variables, such as flow rates and lengths, as well as discontinuous ones, such as column number and configuration. It is also important to reformulate the optimization problem as multiobjective, since the factors affecting the cost of a given separation process are multiple and often in conflict with each other. A typical example is simultaneous maximization of the productivity of the process and the purity of the corresponding products. A new optimization procedure based on a genetic algorithm allows handling these complex optimization problems. An existing literature chiral separation model was used to illustrate the potential of this optimization procedure. This work also offered a unique opportunity to compare the optimal separation performance achievable with the SMB and Varicol technologies.