On the Use of Projected Gradients for Constrained Multiobjective Optimization Problems

  • Alfredo G. Hernandez-Diaz
  • Carlos A. Coello Coello
  • Luis V. Santana-Quintero
  • Fatima Perez
  • Julian Molina
  • Rafael Caballero
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5199)


Recent works have shown how hybrid variants of gradient-based methods and evolutionary algorithms perform better than a pure evolutionary method both for single-objective and multiobjective optimization. This same idea has been used with Evolutionary Multiobjective Optimization (EMO), obtaining also very promising results. In most cases, gradient information is used as part of the mutation operator (and only for unconstrained MOPs), in order to move every generated point to the exact Pareto front. In our approach, we use the Karush-Kuhn-Tucker optimality condition for constrained optimization problems to combine the information provided by the gradient vector of each objective function and the gradient vectors of constraint functions to obtain a feasible movement direction in those points near the border. In our approach, gradients of the objective functions will be approximated using quadratic regressions, trying to avoid local optima. The proposed algorithm is able to converge on several nonlinear constrained multiobjective optimization problems obtained from a benchmark, consuming few objective function evaluations (between 150 and 1000). Our results indicate that our proposed scheme may produce a significant reduction in the computational cost, while producing results of good quality, when it is incorporated into a hybrid MOEA or when it is used to seed an EMO algorithm.


Gradient-based method constrained optimization nonlinear multiobjective programming quadratic approximation 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Alfredo G. Hernandez-Diaz
    • 1
  • Carlos A. Coello Coello
    • 2
  • Luis V. Santana-Quintero
    • 2
  • Fatima Perez
    • 3
  • Julian Molina
    • 3
  • Rafael Caballero
    • 3
  1. 1.Department of Economics, Quantitative Methods and Economic HistoryPablo de Olavide UniversitySevilleSpain
  2. 2.Centro de Investigacion y de Estudios AvanzadosMexico D.F.Mexico
  3. 3.Department of Applied Economics(Mathematics)University of MalagaMalagaSpain

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