Kaisa Miettinen

A preference-based evolutionary algorithm for multi-objective optimization

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.

Synchronous approach in interactive multiobjective optimization

We introduce a new approach in the methodology development for interactive multiobjective optimization. The presentation is given in the context of the interactive NIMBUS method, where the solution process is based on the classification of objective functions. The idea is to formulate several scalarizing functions, all using the same preference information of the decision maker. Thus, opposed to fixing one scalarizing function (as is done in most methods), we utilize several scalarizing functions in a synchronous way. This means that we as method developers do not make the choice between different scalarizing functions but calculate the results of different scalarizing functions and leave the final decision to the expert, the decision maker. Simultaneously, (s)he obtains a better view of the solutions corresponding to her/his preferences expressed once during each iteration. In this paper, we describe a synchronous variant of the NIMBUS method. In addition, we introduce a new version of its implementation WWW-NIMBUS operating on the Internet. WWW-NIMBUS is a software system capable of solving even computationally demanding nonlinear problems. The new version of WWW-NIMBUS can handle versatile types of multiobjective optimization problems and includes new desirable features increasing its user-friendliness.

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.

Introduction to Multiobjective Optimization: Noninteractive Approaches

We give an introduction to nonlinear multiobjective optimization by covering some basic concepts as well as outlines of some methods. Because Pareto optimal solutions cannot be ordered completely, we need extra preference information coming from a decision maker to be able to select the most preferred solution for a problem involving multiple conflicting objectives. Multiobjective optimization methods are often classified according to the role of a decision maker in the solution process. In this chapter, we concentrate on noninteractive methods where the decision maker either is not involved or specifies preference information before or after the actual solution process. In other words, the decision maker is not assumed to devote too much time in the solution process.

Ordinal criteria in stochastic multicriteria acceptability analysis (SMAA)

Abstract We suggest a method for providing descriptive information about the acceptability of decision alternatives in discrete co-operative group decision-making problems. The new SMAA-O method is a variant of the stochastic multicriteria acceptability analysis (SMAA). SMAA-O is designed for problems where criteria information for some or all criteria is ordinal; that is, experts (or decision-makers) have ranked the alternatives according to each (ordinal) criterion. Considerable savings can be obtained if rank information for some or all the criteria is sufficient for making decisions without significant loss of quality. The approach is particularly useful for group decision making when the group can agree on the use of an additive decision model but only partial preference information, or none at all, is available.

A Hybrid Framework for Evolutionary Multi-Objective Optimization

Evolutionary multi-objective optimization algorithms are widely used for solving optimization problems with multiple conflicting objectives. However, basic evolutionary multi-objective optimization algorithms have shortcomings, such as slow convergence to the Pareto optimal front, no efficient termination criterion, and a lack of a theoretical convergence proof. A hybrid evolutionary multi-objective optimization algorithm involving a local search module is often used to overcome these shortcomings. But there are many issues that affect the performance of hybrid evolutionary multi-objective optimization algorithms, such as the type of scalarization function used in a local search and frequency of a local search. In this paper, we address some of these issues and propose a hybrid evolutionary multi-objective optimization framework. The proposed hybrid evolutionary multi-objective optimization framework has a modular structure, which can be used for implementing a hybrid evolutionary multi-objective optimization algorithm. A sample implementation of this framework considering NSGA-II, MOEA/D, and MOEA/D-DRA as evolutionary multi-objective optimization algorithms is presented. A gradient-based sequential quadratic programming method as a single objective optimization method for solving a scalarizing function used in a local search is implemented. Hence, only continuously differentiable functions were considered for numerical experiments. The numerical experiments demonstrate the usefulness of our proposed framework.

Interactive multiobjective optimization system WWW-NIMBUS on the internet

Abstract NIMBUS is a multiobjective optimization method capable of solving nondifferentiable and nonconvex problems. We describe the NIMBUS algorithm and its implementation WWW-NIMBUS. To our knowledge WWW-NIMBUS is the first interactive multiobjective optimization system on the Internet. The main principles of its implementation are centralized computing and a distributed interface. Typically, the delivery and update of any software is problematic. Limited computer capacity may also be a problem. Via the Internet, there is only one version of the software to be updated and any client computer has the capabilities of a server computer. Further, the World-Wide Web (WWW) provides a graphical user interface. No special tools, compilers or software besides a WWW browser are needed. Scope and purpose Interaction between the decision maker and the solution algorithm is often necessary for finding solutions to optimization problems with several conflicting criteria. The Internet provides a versatile tool in realizing such an interaction. The Internet is easily available and sets minimal requirements to the computer facilities of the user. We describe an interactive optimization method and its implementation utilizing the Internet.

Numerical Comparison of Some Penalty-Based Constraint Handling Techniques in Genetic Algorithms

We study five penalty function-based constraint handling techniques to be used with genetic algorithms in global optimization. Three of them, the method of superiority of feasible points, the method of parameter free penalties and the method of adaptive penalties have already been considered in the literature. In addition, we introduce two new modifications of these methods. We compare all the five methods numerically in 33 test problems and report and analyze the results obtained in terms of accuracy, efficiency and reliability. The method of adaptive penalties turned out to be most efficient while the method of parameter free penalties was the most reliable.

On initial populations of a genetic algorithm for continuous optimization problems

Genetic algorithms are commonly used metaheuristics for global optimization, but there has been very little research done on the generation of their initial population. In this paper, we look for an answer to the question whether the initial population plays a role in the performance of genetic algorithms and if so, how it should be generated. We show with a simple example that initial populations may have an effect on the best objective function value found for several generations. Traditionally, initial populations are generated using pseudo random numbers, but there are many alternative ways. We study the properties of different point generators using four main criteria: the uniform coverage and the genetic diversity of the points as well as the speed and the usability of the generator. We use the point generators to generate initial populations for a genetic algorithm and study what effects the uniform coverage and the genetic diversity have on the convergence and on the final objective function values. For our tests, we have selected one pseudo and one quasi random sequence generator and two spatial point processes: simple sequential inhibition process and nonaligned systematic sampling. In numerical experiments, we solve a set of 52 continuous test functions from 16 different function families, and analyze and discuss the results.

Some Methods for Nonlinear Multi-objective Optimization

A general overview of nonlinear multiobjective optimization methods is given. The basic features of several methods are introduced so that an appropriate method could be found for different purposes. The methods are classified according to the role of a decision maker in the solution process. The main emphasis is devoted to interactive methods where the decision maker progressively provides preference information so that the most satisfactory solution can be found.