Showing papers on "Multi-objective optimization published in 2009"

Multiobjective Optimization Problems With Complicated Pareto Sets, MOEA/D and NSGA-II

Abstract: Partly due to lack of test problems, the impact of the Pareto set (PS) shapes on the performance of evolutionary algorithms has not yet attracted much attention. This paper introduces a general class of continuous multiobjective optimization test instances with arbitrary prescribed PS shapes, which could be used for studying the ability of multiobjective evolutionary algorithms for dealing with complicated PS shapes. It also proposes a new version of MOEA/D based on differential evolution (DE), i.e., MOEA/D-DE, and compares the proposed algorithm with NSGA-II with the same reproduction operators on the test instances introduced in this paper. The experimental results indicate that MOEA/D could significantly outperform NSGA-II on these test instances. It suggests that decomposition based multiobjective evolutionary algorithms are very promising in dealing with complicated PS shapes.

SMPSO: A new PSO-based metaheuristic for multi-objective optimization

Abstract: In this work, we present a new multi-objective particle swarm optimization algorithm (PSO) characterized by the use of a strategy to limit the velocity of the particles. The proposed approach, called Speed-constrained Multi-objective PSO (SMPSO) allows to produce new effective particle positions in those cases in which the velocity becomes too high. Other features of SMPSO include the use of polynomial mutation as a turbulence factor and an external archive to store the non-dominated solutions found during the search. Our proposed approach is compared with respect to five multi-objective metaheuristics representative of the state-of-the-art in the area. For the comparison, two different criteria are adopted: the quality of the resulting approximation sets and the convergence speed to the Pareto front. The experiments carried out indicate that SMPSO obtains remarkable results in terms of both, accuracy and speed.

Multiobjective Optimization for DG Planning With Load Models

Abstract: This paper presents a multiobjective performance index-based size and location determination of distributed generation in distribution systems with different load models. Normally, a constant power (real and reactive) load model is assumed in most of the studies made in the literature. It is shown that load models can significantly affect the optimal location and sizing of distributed generation (DG) resources in distribution systems. The simulation technique based on genetic algorithms is studied. The studies have been carried out on 16-bus and 37-bus distribution systems.

A Competitive-Cooperative Coevolutionary Paradigm for Dynamic Multiobjective Optimization

Abstract: In addition to the need for satisfying several competing objectives, many real-world applications are also dynamic and require the optimization algorithm to track the changing optimum over time. This paper proposes a new coevolutionary paradigm that hybridizes competitive and cooperative mechanisms observed in nature to solve multiobjective optimization problems and to track the Pareto front in a dynamic environment. The main idea of competitive-cooperative coevolution is to allow the decomposition process of the optimization problem to adapt and emerge rather than being hand designed and fixed at the start of the evolutionary optimization process. In particular, each species subpopulation will compete to represent a particular subcomponent of the multiobjective problem, while the eventual winners will cooperate to evolve for better solutions. Through such an iterative process of competition and cooperation, the various subcomponents are optimized by different species subpopulations based on the optimization requirements of that particular time instant, enabling the coevolutionary algorithm to handle both the static and dynamic multiobjective problems. The effectiveness of the competitive-cooperation coevolutionary algorithm (COEA) in static environments is validated against various multiobjective evolutionary algorithms upon different benchmark problems characterized by various difficulties in local optimality, discontinuity, nonconvexity, and high-dimensionality. In addition, extensive studies are also conducted to examine the capability of dynamic COEA (dCOEA) in tracking the Pareto front as it changes with time in dynamic environments.

A preference-based evolutionary algorithm for multi-objective optimization

Abstract: In this paper, we discuss the idea of incorporating preference information into evolutionary multi-objective optimization and propose a preference-based evolutionary approach that can be used as an integral part of an interactive algorithm One algorithm is proposed in the paper At each iteration, the decision maker is asked to give preference information in terms of his or her reference point consisting of desirable aspiration levels for objective functions The information is used in an evolutionary algorithm to generate a new population by combining the fitness function and an achievement scalarizing function In multi-objective optimization, achievement scalarizing functions are widely used to project a given reference point into the Pareto optimal set In our approach, the next population is thus more concentrated in the area where more preferred alternatives are assumed to lie and the whole Pareto optimal set does not have to be generated with equal accuracy The approach is demonstrated by numerical examples

Self-Adaptive Multimethod Search for Global Optimization in Real-Parameter Spaces

Abstract: Many different algorithms have been developed in the last few decades for solving complex real-world search and optimization problems. The main focus in this research has been on the development of a single universal genetic operator for population evolution that is always efficient for a diverse set of optimization problems. In this paper, we argue that significant advances to the field of evolutionary computation can be made if we embrace a concept of self-adaptive multimethod optimization in which multiple different search algorithms are run concurrently, and learn from each other through information exchange using a common population of points. We present an evolutionary algorithm, entitled A Multialgorithm Genetically Adaptive Method for Single Objective Optimization (AMALGAM-SO), that implements this concept of self adaptive multimethod search. This method simultaneously merges the strengths of the covariance matrix adaptation (CMA) evolution strategy, genetic algorithm (GA), and particle swarm optimizer (PSO) for population evolution and implements a self-adaptive learning strategy to automatically tune the number of offspring these three individual algorithms are allowed to contribute during each generation. Benchmark results in 10, 30, and 50 dimensions using synthetic functions from the special session on real-parameter optimization of CEC 2005 show that AMALGAM-SO obtains similar efficiencies as existing algorithms on relatively simple unimodal problems, but is superior for more complex higher dimensional multimodal optimization problems. The new search method scales well with increasing number of dimensions, converges in the close proximity of the global minimum for functions with noise induced multimodality, and is designed to take full advantage of the power of distributed computer networks.

Hybrid Nelder-Mead simplex search and particle swarm optimization for constrained engineering design problems

Abstract: Constrained optimization problems are very important in that they frequently appear in the real world. A constrained optimization problem consists of the optimization of a function subject to constraints, in which both the function and constraints may be nonlinear. Constraint handling is one of the major concerns when solving constrained optimization problems by hybrid Nelder-Mead simplex search method and particle swarm optimization, denoted as NM-PSO. This paper proposes embedding constraint handling methods, which include the gradient repair method and constraint fitness priority-based ranking method, in NM-PSO as a special operator to deal with satisfying constraints. Experiments using three benchmark function and three engineering design problems are presented and compared with the best known solutions reported in the literature. The comparison results with other evolutionary optimization methods demonstrate that NM-PSO with the embedded constraint operator proves to be extremely effective and efficient at locating optimal solutions.

Newton's Method for Multiobjective Optimization

Abstract: We propose an extension of Newton's method for unconstrained multiobjective optimization (multicriteria optimization). This method does not use a priori chosen weighting factors or any other form of a priori ranking or ordering information for the different objective functions. Newton's direction at each iterate is obtained by minimizing the max-ordering scalarization of the variations on the quadratic approximations of the objective functions. The objective functions are assumed to be twice continuously differentiable and locally strongly convex. Under these hypotheses, the method, as in the classical case, is locally superlinear convergent to optimal points. Again as in the scalar case, if the second derivatives are Lipschitz continuous, the rate of convergence is quadratic. Our convergence analysis uses a Kantorovich-like technique. As a byproduct, existence of optima is obtained under semilocal assumptions.

Scheduling of Demand Side Resources Using Binary Particle Swarm Optimization

Abstract: Interruptible loads represent highly valuable demand side resources within the electricity industry. However, maximizing their potential value in terms of system security and scheduling is a considerable challenge because of their widely varying and potentially complex operational characteristics. This paper investigates the use of binary particle swarm optimization (BPSO) to schedule a significant number of varied interruptible loads over 16 h. The scheduling objective is to achieve a system requirement of total hourly curtailments while satisfying the operational constraints of the available interruptible loads, minimizing the total payment to them and minimizing the frequency of interruptions imposed upon them. This multiobjective optimization problem was simplified by using a single aggregate objective function. The BPSO algorithm proved capable of achieving near-optimal solutions in manageable computational time-frames for this relatively complex, nonlinear and noncontinuous problem. The effectiveness of the approach was further improved by dividing the swarm into several subswarms. The proposed scheduling technique demonstrated useful performance for a relatively challenging scheduling task, and would seem to offer some potential advantages in scheduling significant numbers of widely varied and technically complex interruptible loads.

Particle swarm optimization approach to portfolio optimization

Abstract: The survey of the relevant literature showed that there have been many studies for portfolio optimization problem and that the number of studies which have investigated the optimum portfolio using heuristic techniques is quite high. But almost none of these studies deals with particle swarm optimization (PSO) approach. This study presents a heuristic approach to portfolio optimization problem using PSO technique. The test data set is the weekly prices from March 1992 to September 1997 from the following indices: Hang Seng in Hong Kong, DAX 100 in Germany, FTSE 100 in UK, S&P 100 in USA and Nikkei in Japan. This study uses the cardinality constrained mean-variance model. Thus, the portfolio optimization model is a mixed quadratic and integer programming problem for which efficient algorithms do not exist. The results of this study are compared with those of the genetic algorithms, simulated annealing and tabu search approaches. The purpose of this paper is to apply PSO technique to the portfolio optimization problem. The results show that particle swarm optimization approach is successful in portfolio optimization.

Approximating the Set of Pareto-Optimal Solutions in Both the Decision and Objective Spaces by an Estimation of Distribution Algorithm

Abstract: Most existing multiobjective evolutionary algorithms aim at approximating the Pareto front (PF), which is the distribution of the Pareto-optimal solutions in the objective space. In many real-life applications, however, a good approximation to the Pareto set (PS), which is the distribution of the Pareto-optimal solutions in the decision space, is also required by a decision maker. This paper considers a class of multiobjective optimization problems (MOPs), in which the dimensionalities of the PS and the PF manifolds are different so that a good approximation to the PF might not approximate the PS very well. It proposes a probabilistic model-based multiobjective evolutionary algorithm, called MMEA, for approximating the PS and the PF simultaneously for an MOP in this class. In the modeling phase of MMEA, the population is clustered into a number of subpopulations based on their distribution in the objective space, the principal component analysis technique is used to estimate the dimensionality of the PS manifold in each subpopulation, and then a probabilistic model is built for modeling the distribution of the Pareto-optimal solutions in the decision space. Such a modeling procedure could promote the population diversity in both the decision and objective spaces. MMEA is compared with three other methods, KP1, Omni-Optimizer and RM-MEDA, on a set of test instances, five of which are proposed in this paper. The experimental results clearly suggest that, overall, MMEA performs significantly better than the three compared algorithms in approximating both the PS and the PF.

MOCell: A cellular genetic algorithm for multiobjective optimization

Abstract: This paper introduces a new cellular genetic algorithm for solving multiobjective continuous optimization problems. Our approach is characterized by using an external archive to store nondominated solutions and a feedback mechanism in which solutions from this archive randomly replace existing individuals in the population after each iteration. The result is a simple and elitist algorithm called MOCell. Our proposal has been evaluated with both constrained and unconstrained problems and compared against NSGA-II and SPEA2, two state-of-the-art evolutionary multiobjective optimizers. For the studied benchmark, our experiments indicate that MOCell obtains competitive results in terms of convergence and hypervolume, and it clearly outperforms the other two compared algorithms concerning the diversity of the solutions along the Pareto front. © 2009 Wiley Periodicals, Inc.

A Multi-Objective Framework for Transmission Expansion Planning in Deregulated Environments

Abstract: Deregulation of power system has introduced new objectives and requirements for transmission expansion problem. In this paper, a static transmission expansion methodology is presented using a multi-objective optimization framework. Investment cost, reliability (both adequacy and security), and congestion cost are considered in the optimization as three objectives. To overcome the difficulties in solving the nonconvex and mixed integer nature of the optimization problems, the genetic based NSGA II algorithm is used followed by a fuzzy decision making analysis to obtain the final optimal solution. The planning methodology has been demonstrated on the IEEE 24-bus test system to show the feasibility and capabilities of the proposed algorithm. Also, in order to compare the historical expansion plan and the expansion plan developed by the proposed methodology, it was applied to the real life system of northeastern part of Iranian national 400-kV transmission grid.

Multiobjective Evolutionary Algorithm With Controllable Focus on the Knees of the Pareto Front

Abstract: The optimal solutions of a multiobjective optimization problem correspond to a nondominated front that is characterized by a tradeoff between objectives. A knee region in this Pareto-optimal front, which is visually a convex bulge in the front, is important to decision makers in practical contexts, as it often constitutes the optimum in tradeoff, i.e. substitution of a given Pareto-optimal solution with another solution on the knee region yields the largest improvement per unit degradation. This paper presents a selection scheme that enables a multiobjective evolutionary algorithm (MOEA) to obtain a nondominated set with controllable concentration around existing knee regions of the Pareto front. The preference- based focus is achieved by optimizing a set of linear weighted sums of the original objectives, and control of the extent of the focus is attained by careful selection of the weight set based on a user-specified parameter. The fitness scheme could be easily adopted in any Pareto-based MOEA with little additional computational cost. Simulations on various two- and three- objective test problems demonstrate the ability of the proposed method to guide the population toward existing knee regions on the Pareto front. Comparison with general-purpose Pareto based MOEA demonstrates that convergence on the Pareto front is not compromised by imposing the preference-based bias. The performance of the method in terms of an additional performance metric introduced to measure the accuracy of resulting convergence on the desired regions validates the efficacy of the method.

Reliability-Based Optimization Using Evolutionary Algorithms

Abstract: Uncertainties in design variables and problem parameters are often inevitable and must be considered in an optimization task if reliable optimal solutions are sought. Besides a number of sampling techniques, there exist several mathematical approximations of a solution's reliability. These techniques are coupled in various ways with optimization in the classical reliability-based optimization field. This paper demonstrates how classical reliability-based concepts can be borrowed and modified and, with integrated single and multiobjective evolutionary algorithms, used to enhance their scope in handling uncertainties involved among decision variables and problem parameters. Three different optimization tasks are discussed in which classical reliability-based optimization procedures usually have difficulties, namely (1) reliability-based optimization problems having multiple local optima, (2) finding and revealing reliable solutions for different reliability indices simultaneously by means of a bi-criterion optimization approach, and (3) multiobjective optimization with uncertainty and specified system or component reliability values. Each of these optimization tasks is illustrated by solving a number of test problems and a well-studied automobile design problem. Results are also compared with a classical reliability-based methodology.

Objective reduction in evolutionary multiobjective optimization: Theory and applications

Abstract: Many-objective problems represent a major challenge in the field of evolutionary multiobjective optimization---in terms of search efficiency, computational cost, decision making, visualization, and so on. This leads to various research questions, in particular whether certain objectives can be omitted in order to overcome or at least diminish the difficulties that arise when many, that is, more than three, objective functions are involved. This study addresses this question from different perspectives. First, we investigate how adding or omitting objectives affects the problem characteristics and propose a general notion of conflict between objective sets as a theoretical foundation for objective reduction. Second, we present both exact and heuristic algorithms to systematically reduce the number of objectives, while preserving as much as possible of the dominance structure of the underlying optimization problem. Third, we demonstrate the usefulness of the proposed objective reduction method in the context of both decision making and search for a radar waveform application as well as for well-known test functions.

ArcheOpterix: An extendable tool for architecture optimization of AADL models

Abstract: For embedded systems quality requirements are equally if not even more important than functional requirements. The foundation for the fulfillment of these quality requirements has to be set in the architecture design phase. However, finding a suitable architecture design is a difficult task for software and system architects. Some of the reasons for this are an ever-increasing complexity of today's systems, strict design constraints and conflicting quality requirements. To simplify the task, this paper presents an extendable Eclipse-based tool, called ArcheOpterix, which provides a framework to implement evaluation techniques and optimization heuristics for AADL specifications. Currently, evolutionary strategies have been implemented to identify optimized deployment architectures with respect to multiple quality objectives and design constraints. Experiments with a set of initial deployment architectures provide evidence that the tool can successfully find architecture specifications with better quality.

Quantum behaved Particle Swarm Optimization (QPSO) for multi-objective design optimization of composite structures

Abstract: We present a new, generic method/model for multi-objective design optimization of laminated composite components using a novel multi-objective optimization algorithm developed on the basis of the Quantum behaved Particle Swarm Optimization (QPSO) paradigm. QPSO is a co-variant of the popular Particle Swarm Optimization (PSO) and has been developed and implemented successfully for the multi-objective design optimization of composites. The problem is formulated with multiple objectives of minimizing weight and the total cost of the composite component to achieve a specified strength. The primary optimization variables are - the number of layers, its stacking sequence (the orientation of the layers) and thickness of each layer. The classical lamination theory is utilized to determine the stresses in the component and the design is evaluated based on three failure criteria; Failure Mechanism based Failure criteria, Maximum stress failure criteria and the Tsai-Wu Failure criteria. The optimization method is validated for a number of different loading configurations - uniaxial, biaxial and bending loads. The design optimization has been carried for both variable stacking sequences as well as fixed standard stacking schemes and a comparative study of the different design configurations evolved has been presented. Also, the performance of QPSO is compared with the conventional PSO.

Application of NSGA-II Algorithm to Generation Expansion Planning

Abstract: This paper describes use of a multiobjective optimization method, elitist nondominated sorting genetic algorithm version II (NSGA-II), to the generation expansion planning (GEP) problem. The proposed model provides for decision maker choice from among the different trade-off solutions. Two different problem formulations are considered. In one formulation, the first objective is to minimize cost; the second objective is to minimize sum of normalized constraint violations. In the other formulation, the first objective is to minimize investment cost; the second objective is to minimize outage cost (or maximize reliability). Virtual mapping procedure is introduced to improve the performance of NSGA-II. The GEP problem considered is a test system for a six-year planning horizon having five types of candidate units. The results are compared and validated.

Multi-Objective Particle Swarm Optimizers: An Experimental Comparison

Abstract: Particle Swarm Optimization (PSO) has received increasing attention in the optimization research community since its first appearance in the mid-1990s. Regarding multi-objective optimization, a considerable number of algorithms based on Multi-Objective Particle Swarm Optimizers (MOPSOs) can be found in the specialized literature. Unfortunately, no experimental comparisons have been made in order to clarify which MOPSO version shows the best performance. In this paper, we use a benchmark composed of three well-known problem families (ZDT, DTLZ, and WFG) with the aim of analyzing the search capabilities of six representative state-of-the-art MOPSOs, namely, NSPSO, SigmaMOPSO, OMOPSO, AMOPSO, MOPSOpd, and CLMOPSO. We additionally propose a new MOPSO algorithm, called SMPSO, characterized by including a velocity constraint mechanism, obtaining promising results where the rest perform inadequately.