Nonlinear Analysis 71 (2009) e1818–e1825 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Interval Robust Multi-objective Algorithm G.L. Soares b,c,a , R.O. Parreiras b,∗ , L. Jaulin c , J.A. Vasconcelos a , C.A. Maia a a Universidade Federal de Minas Gerais (UFMG), Belo Horizonte, Brazil b Pontifícia Universidade Católica de Minas Gerais (PUCMinas), Belo Horizonte, Brazil c École Nationale Supérieure des Ingénieurs des Etudes des Techniques d’Armement (ENSIETA), Brest, France article info Keywords: Robust Multi-objective optimization Robust Pareto frontier Interval Analysis Genetic algorithm abstract This paper introduces a method for solving multi-objective optimization problems in uncertain environment. When the uncertainty factors of the optimization problem can be included into the mathematical model, through bounded intervals, [I]RMOA (Interval Robust Multi-objective Algorithm) can find an enclosure of the robust Pareto frontier. In this approach, the robust Pareto solutions are the ones that have the best performance when the worst case scenario, characterized by the uncertainty parameters, is considered. [I]RMOA has some positive aspects: it does not require the calculus of derivatives; it has only two input parameters; it is a reliable tool for solving different robust optimization problems, which can be nonlinear and discontinuous or have nonconvex Pareto frontier, for instance. The success of the method depends only on the quality of the objective inclusion functions and the precision parameters. Its disadvantage lies in the fact that it requires high computational effort, when high-dimensional problems are considered or when a very accurate enclosure is needed. Two analytical robust test functions are proposed to be treated by [I]RMOA and to validate the results provided by a stochastic multi-objective optimization method. The results are satisfactory. © 2009 Elsevier Ltd. All rights reserved. 1. Introduction The process of modeling the real system to be optimized involves several sources of uncertainties [1–3]. For instance, precision can be lost when constructing a computational model of the optimization parameters; the environment (factors such as humidity, pressure, and temperature) may produce effects on the real system, that are difficult to be quantified; there may be measurement imprecision in the estimative of the objective functions and of the optimization parameters. A possible approach for dealing with uncertainty factors consists in constructing an optimization model, which includes additional parameters that reflect the uncertainty effects [4–6]. The use of such an approach is interesting, as it allows one to adequately consider the uncertainty factors, since the beginning of the problem analysis. Stochastic and deterministic frameworks have been applied to treat optimization problems under uncertainties, such as [7,8] for stochastic methods, and [4,9] for deterministic ones. Intervals have been applied to compute the uncertainties as in [10]. In a different approach, this paper proposes [I]RMOA (Interval Robust Multi-objective Algorithm), which utilizes concepts from the Interval Analysis to compute the uncertainty parameters during the search for an enclosure for the robust Pareto solutions. Here, as in [11,9,7], the robust Pareto solutions are the ones that have the best performance (reflected by the objective function evaluations), when the worst case scenario, characterized by the uncertainty parameters, is considered. The proposed algorithm has some advantageous properties, such ∗ Corresponding author. E-mail addresses: gsoares@cpdee.ufmg.br (G.L. Soares), roberta.parreiras@terra.com.br (R.O. Parreiras), jaulinlu@ensieta.fr (L. Jaulin), joao@cpdee.ufmg.br (J.A. Vasconcelos), maia@cpdee.ufmg.br (C.A. Maia). 0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.02.077