Marco Farina

Dynamic multiobjective optimization problems: test cases, approximations, and applications

After demonstrating adequately the usefulness of evolutionary multiobjective optimization (EMO) algorithms in finding multiple Pareto-optimal solutions for static multiobjective optimization problems, there is now a growing need for solving dynamic multiobjective optimization problems in a similar manner. In this paper, we focus on addressing this issue by developing a number of test problems and by suggesting a baseline algorithm. Since in a dynamic multiobjective optimization problem, the resulting Pareto-optimal set is expected to change with time (or, iteration of the optimization process), a suite of five test problems offering different patterns of such changes and different difficulties in tracking the dynamic Pareto-optimal front by a multiobjective optimization algorithm is presented. Moreover, a simple example of a dynamic multiobjective optimization problem arising from a dynamic control loop is presented. An extension to a previously proposed direction-based search method is proposed for solving such problems and tested on the proposed test problems. The test problems introduced in this paper should encourage researchers interested in multiobjective optimization and dynamic optimization problems to develop more efficient algorithms in the near future.

A fuzzy definition of "optimality" for many-criteria optimization problems

When dealing with many-objectives optimization problems, the concepts of Pareto-optimality and Pareto-dominance are often inefficient in modeling and simulating human decision making. This leads to an unpractical size for the set of Pareto-optimal (PO) solutions, and an additional selection criteria among solutions is usually arbitrarily considered. In the paper, different fuzzy-based definitions of optimality and dominated solutions, being nonpreference based, are introduced and tested on analytical test cases, in order to show their validity and nearness to human decision making. Based on this definitions, different subsets of PO solution set can be computed using simple and clear information provided by the decision maker and using a parameter value ranging from zero to one. When the value of the above parameter is zero, the introduced definitions coincide with classical Pareto-optimality and dominance. When the parameter value is increased, different subset of PO solutions can be obtained corresponding to higher degrees of optimality.

Comparative study of evolution strategies combined with approximation techniques for practical electromagnetic optimization problems

The paper addresses the practical problem of reducing the number of necessary function calls involving time consuming finite-element solutions by combining various evolution techniques with approximation methods based on Response Surface Methodology. A new algorithm is proposed which offers significant improvement of performance while preserving the quality of the final result. Comparisons are made between the new algorithm and different standard strategies in terms of reliability, efficiency and cost.

A neural network based generalized response surface multiobjective evolutionary algorithm

The practical use of multiobjective optimization tools in industry is still an open issue. A strategy for the reduction of objective functions is often essential, at a fixed degree of Pareto optimal front (POF) approximation accuracy. An extension of single-objective NN-based generalized response surfaces (GRS) methods to POF approximation is proposed. Such an extension is not at all straightforward due to the complex relation between the POF and Pareto optimal set. As a consequence of such complexity, it is extremely difficult to identify a multiobjective analogue of the single-objective current optimum region. Consequently, the design domain search space zooming strategy, which is the core of the GRS method, is to be carefully reconsidered when POF approximation is concerned.

An improved technique for enhancing diversity in Pareto evolutionary optimization of electromagnetic devices

When a multiobjective optimization problem is tackled using Pareto optima theory, particular care has to be taken to obtain a full sampling of the Pareto Optimal Front. This leads to variability of individuals in both design space and objective space. We compare two different fitness assignment strategies based on two individual sharing procedures in the space domain and in the objective domain on some test cases.

An ALife-Inspired Evolutionary Algorithm for Dynamic Multiobjective Optimization Problems

Several important applications require a time-dependent (on-line) in which either the objective function or the problem parameters or both vary with time. Several studies are available in the literature about the use of genetic algorithms for time dependent fitness landscape in single-objective optimization problems. But when dynamic multi-objective optimization is concerned, very few studies can be found. Taking inspiration from Artificial Life (ALife), a strategy is proposed ensuring the approximation of Pareto-optimal set and front in case of unpredictable parameters changes. It is essentially an ALife-inspired evolutionary algorithm for variable fitness landscape search. We describe the algorithm and test it on some test cases.

Evolutionary multiobjective industrial design: the case of a racing car tire-suspension system

When dealing with multiobjective optimization (MO) of the tire-suspension system of a racing car, a large number of design variables and a large number of objectives have to be taken into account. Two different models have been used, both validated on data coming from an instrumented car, a differential equation-based physical model, and a neural network purely numerical model. Up to 23 objective functions have been defined, at least 14 of which are in strict conflict of each other. The equivalent scalar function based and the objective-as-constraint formulations are intentionally avoided due to their well-known limitations. A fuzzy definition of optima, being a generalization of Pareto optimality, is applied to the problem. The result of such an approach is that subsets of Pareto optimal solutions (on such a problem, a big portion of the entire search space) can be properly selected as a consequence of input from the designer. The obtained optimal solutions are compared with the reference vehicle and with the optima previously obtained with design of experiment techniques and different MO optimization strategies. The proposed strategy improves both the reference (actual) car and previously obtained optima (scalar preference function) in the majority of objectives with technically significant improvements. Moreover, the strategy offers an univoque criterion for the choice among tradeoff solutions in the 14-dimensional objective space. The problem is used as a test of a proposed optimal design strategy for industrial problems, integrating differential equation and neural networks modeling, design of experiments, MO, and fuzzy optimal-based decision making. Such a linked approach gives also a unified view of where to concentrate the computational effort.

Pricing derivatives by path integral and neural networks

Recent progress in the development of efficient computational algorithms to price financial derivatives is summarized. A first algorithm is based on a path integral approach to option pricing, while a second algorithm makes use of a neural network parameterization of option prices. The accuracy of the two methods is established from comparisons with the results of the standard procedures used in quantitative finance.

Dynamic multiobjective optimization problems: test cases, approximation, and applications

Parametric and dynamic multiobjective optimization problems for adaptive optimal control are carefully defined; some test problems are introduced for both continuous and discrete design spaces. A simple example of a dynamic multiobjective optimization problems arising from a dynamic control loop is given and an extension for dynamic situation of a previously proposed search direction based method is proposed and tested on the proposed test problems.

On the approximation of Maxwell''s eigenproblem in general 2D domains

In this paper we review some finite element methods to approximate the eigenvalues of Maxwell equations. The numerical schemes we are going to consider are based on two different variational formulations. Our aim is to compare the performances of the methods depending on the shape of the domain. We shall see that the nodal elements can give good results only using the penalized formulation and only if the domain is a convex or smooth polygon. In the case of domains with reentrant corners it turns out that the edge elements are efficient. Moreover we propose two new non standard finite elements in order to deal with the penalized formulation in presence of reentrant corners: the nonconforming nodal elements and the biquadratic elements with projection. EMAIL:: boffi@dimat.unipv.it KEYWORDS:: Fem, penalty method, nodal-edge elements, nonconforming elements, projection procedure