JOURNAL OF MULTI-CRITERIA DECISION ANALYSIS J. Multi-Crit. Decis. Anal. 13: 115–123 (2005) Published online in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/mcda.377 Interactive Reference Direction Approach Using Implicit Parametrization for Nonlinear Multiobjective Optimization KAISA MIETTINEN a, * and LEONEED KIRILOV b,y a Helsinki School of Economics, P.O. Box 1210, FI-00101 Helsinki, Finland b Institute of InformationTechnologies-BAS, Acad. G. Bonchev Str., bl. 29A,1113 So¢a, Bulgaria ABSTRACT A method for solving nonlinear multiobjective optimization problems is presented. In this interactive method, the dialog with a decision maker takes place in terms of desirable aspiration levels forming a reference point in the objective function space. In the computation phase, a lexicographic subproblem is used to generate a desired number of nondominated solutions. Thus, the decision maker can direct the interactive solution process both by specifying reference points and by setting the number of new solutions to be generated at each iteration. The proposed method gives the decision maker a possibility to investigate any part of the whole nondominated set according to his/her preferences and emphasizes learning possibilities. The functioning of the method is illustrated with a numerical example. Copyright # 2006 John Wiley & Sons, Ltd. KEY WORDS: multiple criteria programming; nonlinear optimization; interactive method; reference point; aspiration level 1. INTRODUCTION Decision making plays an important part of life. Most decision problems can be formulated or reformulated as multiple criteria decision-making models. Examples of such models include pro- blems of portfolio selection, project appraisal, planning production, optimal shape design, and many types of engineering situations, etc., see, for example, Stadler (1988), White (1990). When models involve continuous variables, we talk about multiobjective optimization. Many phenom- ena can be reliably described using nonlinear models. Therefore, developing methods and tools for solving nonlinear multiobjective optimization problems is important; see, for example, Miettinen (1999). Some basic properties have to be taken into account when solving multiobjective optimization problems. The most important characteristic is that these problems are mathematically ill-defined. This means that they have a set of mathematically ‘equally good’ optimal solutions in the objective function space, so-called, nondominated solutions. Thus, some additional information is needed in order to be able select one of them. A decision maker (DM) has an important role in specifying preference information in the solution process. We can typically distinguish between two different phases of the solution process: a computation phase and a dialog phase between the algorithm and the DM. One or several solutions are generated in the phase of computation. Often, this is done using a single objective optimization problem, a so-called scalarizing problem. The DM evaluates the current solution(s) and possibly additional information and sets his/her preferences in the phase of a dialog. According to the role of the DM, methods can be classified as: * no-preference methods with no dialog, * a priori methods, where the phase of dialog precedes the phase of computation, * a posteriori methods, where the phase of computation precedes the phase of the dialog and * interactive methods, where the two phases alternate. For further details see, for example, Miettinen (1999). Copyright # 2006 John Wiley & Sons, Ltd. *Correspondence to: Helsinki School of Economics, P.O. Box 1210, FI-00101 Helsinki, Finland. y E-mail: lkirilov@iinf.bas.bg