Interactive MCDM Support System in the Internet

NIMBUS is an interactive multiobjective optimization system. Among other things, it is capable of solving complicated real-world applications involving nondifferentiable and nonconvex functions. We describe an implementation of NIMBUS operating in the Internet, where the World- Wide Web (WWW) provides a graphical user interface. Different kind of visualizations of alternatives produced are available for aiding in the solution process.

NIMBUS — Interactive Method for Nondifferentiable Multiobjective Optimization Problems

An interactive method, NIMBUS, for nondifferentiable multiobjective optimization problems is introduced. We assume that every objective function is to be minimized The idea of NIMBUS is that the decision maker can easily indicate what kind of improvements are desired and what kind of impairments are tolerable at the point considered.

On the methods of nonsmooth optimization

In this paper we shall give a short derivation of the most promising methods in nonsmooth optimization, namely bundle methods. We introduce the basic bundle idea due to Lemarechal and several modifications by Kiwiel, Schramm and Zowe. To the end we shall give some numerical results comparing the efficience of these methods. As test problems we have used well-known test problems from litterature and in addition we shall give some contributions to nonsmooth optimal control problems.

Interactive Method NIMBUS for Nondifferentiable Multiobjective Optimization Problems

An interactive method, NIMBUS, for nondifferentiable multiobjective optimization problems is introduced. The method is capable of handling several nonconvex locally Lipschitzian objective functions subject to nonlinear (possibly nondifferentiable) constraints. The idea of NIMBUS is that the decision maker can easily indicate what kind of improvements are desired and what kind of impairments are tolerable at the point considered. The decision maker is asked to classify the objective functions into five different classes: those to be improved, those to be improved down to some aspiration level, those to be accepted as they are, those to be impaired till some upper bound, and those allowed to …

Experiments with classification-based scalarizing functions in interactive multiobjective optimization

In multiobjective optimization methods, the multiple conflicting objectives are typically converted into a single objective optimization problem with the help of scalarizing functions and such functions may be constructed in many ways. We compare both theoretically and numerically the performance of three classification-based scalarizing functions and pay attention to how well they obey the classification information. In particular, we devote special interest to the differences the scalarizing functions have in the computational cost of guaranteeing Pareto optimality. It turns out that scalarizing functions with or without so-called augmentation terms have significant differences in this re…

Tangent and Normal Cones in Nonconvex Multiobjective Optimization

Trade-off information is important in multiobjective optimization. It describes the relationships of changes in objective function values. For example, in interactive methods we need information about the local behavior of solutions when looking for improved search directions.

Interactive multiobjective optimization system NIMBUS applied to nonsmooth structural design problems

We shortly describe an interactive method, called NIMBUS, for multiobjective optimization involving nondifferentiable and nonconvex functions. We illustrate the functioning of NIMBUS by numerical examples in the area of structural design. We consider a beam with varying thickness and our aim is to find a thickness distribution in such a way that the resulting structure is as good as possible.

Comparative evaluation of some interactive reference point-based methods for multi-objective optimisation

Many real-world optimisation applications include several conflicting objectives of possibly nondifferentiable character. However, the lack of computationally efficient, interactive methods for nondifferentiable multi-objective optimisation problems is apparent. To satisfy this demand, a method called NIMBUS has been developed. Two versions of the basic method are presented and compared both theoretically and computationally. In order to give variety to the comparison, a related approach, called reference direction method is included. Theoretically, the methods differ in handling the information requested from the user. Numerical experiments indicate differences in computational efficiency …

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 t…

Nonsmooth Penalty Techniques in Control of the Continuous Casting Process

We introduce a mathematical model which is used to simulate the continuous casting process and to control the secondary cooling water sprays. The main object is to minimize the defects in the final products. The problem is formulated as an optimal control problem where the cost function is constructed according to certain metallurgical criteria and constraints. The temperature distribution of the strand is calculated by solving a nonlinear heat equation with free boundaries between solid and liquid phases.

On interactive multiobjective optimization with NIMBUS® in chemical process design

We study multiobjective optimization problems arising from chemical process simulation. The interactive multiobjective optimization method NIMBUS®, developed at the University of Jyvaskyla, is combined with the BALAS® process simulator, developed at the VTT Technical Research Center of Finland, in order to provide a new interactive tool for designing chemical processes. Continuous interaction between the method and the designer provides a new efficient approach to explore Pareto optimal solutions and helps the designer to learn about the behaviour of the process. As an example of how the new tool can be used, we report the results of applying it in a heat recovery system design problem rela…

Large-scale nonsmooth optimization: new variable metric bundle algorithm with limited memory

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 …

Limited memory bundle algorithm for inequality constrained nondifferentiable optimization

Many practical optimization problems involve nonsmooth (that is, not necessarily differentiable) functions of hundreds or thousands of variables with various constraints. In this paper, we describe a new efficient adaptive limited memory interior point bundle method for large, possible nonconvex, nonsmooth inequality constrained optimization. The method is a hybrid of the nonsmooth variable metric bundle method and the smooth limited memory variable metric method, and the constraint handling is based on the primal-dual feasible direction interior point approach. The preliminary numerical experiments to be presented confirm the effectiveness of the method.

Limited memory bundle algorithm for large bound constrained nonsmooth minization problems

Typically, practical optimization problems involve nonsmooth functions of hundreds or thousands of variables. As a rule, the variables in such problems are restricted to certain meaningful intervals. In this paper, we propose an efficient adaptive limited memory bundle method for large-scale nonsmooth, possibly nonconvex, bound constrained optimization. The method combines the nonsmooth variable metric bundle method and the smooth limited memory variable metric method, while the constraint handling is based on the projected gradient method and the dual subspace minimization. The preliminary numerical experiments to be presented confirm the usability of the method.

Mental contents in interacting with a multiobjective optimization program

User psychology aims at understanding human-machine interaction from a psychological point of view. Its ultimate goal is to provide knowledge about human psychological properties for interaction designers. In this article, we are particularly interested in applying the theoretical concepts of mental contents (i.e., the information contents of users’ mental representations), in studying interaction with professional software. The immediate motivation for adopting such an approach arises from problems met in designing interaction processes in multiobjective optimization software. These types of software are meant to support complex thought and decision-making processes and this is why interac…