On Generalizing Lipschitz Global Methods for Multiobjective Optimization

Lipschitz global methods for single-objective optimization can represent the optimal solutions with desired accuracy. In this paper, we highlight some directions on how the Lipschitz global methods can be extended as faithfully as possible to multiobjective optimization problems. In particular, we present a multiobjective version of the Pijavskiǐ-Schubert algorithm.

[1]  Qingfu Zhang,et al.  MOEA/D: A Multiobjective Evolutionary Algorithm Based on Decomposition , 2007, IEEE Transactions on Evolutionary Computation.

[2]  Alberto Lovison,et al.  Global search perspectives for multiobjective optimization , 2013, J. Glob. Optim..

[3]  A. Neumaier.  Complete search in continuous global optimization and constraint satisfaction , 2004, Acta Numerica.

[4]  Kaisa Miettinen,et al.  PAINT: Pareto front interpolation for nonlinear multiobjective optimization , 2012, Comput. Optim. Appl..

[5]  B. Shubert.  A Sequential Method Seeking the Global Maximum of a Function , 1972 .

[6]  S. Smale,et al.  Global Analysis and Economics I: Pareto Optimum and a Generalization of Morse Theory† , 1975 .

[7]  Alberto Lovison,et al.  Singular Continuation: Generating Piecewise Linear Approximations to Pareto Sets via Global Analysis , 2010, SIAM J. Optim..

[8]  M. Dellnitz,et al.  Covering Pareto Sets by Multilevel Subdivision Techniques , 2005 .

[9]  Alberto Lovison,et al.  Adaptive sampling with a Lipschitz criterion for accurate metamodeling , 2010 .

[10]  Marco Laumanns,et al.  Scalable multi-objective optimization test problems , 2002, Proceedings of the 2002 Congress on Evolutionary Computation. CEC'02 (Cat. No.02TH8600).

[11]  S. Smale.  Global analysis and economics , 1975, Synthese.

[12]  C. D. Perttunen,et al.  Lipschitzian optimization without the Lipschitz constant , 1993 .

[13]  Y. Sergeyev,et al.  Lipschitz and Hölder global optimization using space-filling curves , 2010 .

[14]  Luís N. Vicente,et al.  Direct Multisearch for Multiobjective Optimization , 2011, SIAM J. Optim..

[15]  Filippo Pecci,et al.  Hierarchical stratification of Pareto sets , 2014, 1407.1755.

[16]  Alberto Lovison,et al.  PAINT–SiCon: constructing consistent parametric representations of Pareto sets in nonconvex multiobjective optimization , 2014, Journal of Global Optimization.

[17]  János D. Pintér,et al.  Nonlinear optimization with GAMS /LGO , 2007, J. Glob. Optim..

[18]  Yaroslav D. Sergeyev,et al.  A univariate global search working with a set of Lipschitz constants for the first derivative , 2009, Optim. Lett..

[19]  Ashutosh Tiwari,et al.  Multi-Objective Optimisation Problems: A Symbolic Algorithm for Performance Measurement of Evolutionary Computing Techniques , 2009, EMO.

[20]  S. A. Piyavskii.  An algorithm for finding the absolute extremum of a function , 1972 .

[21]  Kaisa Miettinen,et al.  Constructing a Pareto front approximation for decision making , 2011, Math. Methods Oper. Res..

[22]  Regina Hunter Mladineo.  An algorithm for finding the global maximum of a multimodal, multivariate function , 1986, Math. Program..