PAINT–SiCon: constructing consistent parametric representations of Pareto sets in nonconvex multiobjective optimization

We introduce a novel approximation method for multiobjective optimization problems called PAINT–SiCon. The method can construct consistent parametric representations of Pareto sets, especially for nonconvex problems, by interpolating between nondominated solutions of a given sampling both in the decision and objective space. The proposed method is especially advantageous in computationally expensive cases, since the parametric representation of the Pareto set can be used as an inexpensive surrogate for the original problem during the decision making process.

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

[2]  W. Marsden.  I and J , 2012 .

[3]  Arnaldo V. Moura,et al.  LATIN'98: Theoretical Informatics , 1998, Lecture Notes in Computer Science.

[4]  O Shoval,et al.  Evolutionary Trade-Offs, Pareto Optimality, and the Geometry of Phenotype Space , 2012, Science.

[5]  Steven Fortune,et al.  Voronoi Diagrams and Delaunay Triangulations , 2004, Handbook of Discrete and Computational Geometry, 2nd Ed..

[6]  Murat Köksalan,et al.  Generating a Representative Subset of the Nondominated Frontier in Multiple Criteria Decision Making , 2009, Oper. Res..

[7]  W. De Melo,et al.  Stability and optimization of several functions , 1976 .

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

[9]  C. Hillermeier.  Generalized Homotopy Approach to Multiobjective Optimization , 2001 .

[10]  Ching-Lai Hwang,et al.  Fuzzy Multiple Attribute Decision Making - Methods and Applications , 1992, Lecture Notes in Economics and Mathematical Systems.

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

[12]  Oliver Schütze,et al.  On Continuation Methods for the Numerical Treatment of Multi-Objective Optimization Problems , 2005, Practical Approaches to Multi-Objective Optimization.

[13]  John W. Eaton,et al.  Gnu Octave Manual , 2002 .

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

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

[16]  Layne T. Watson,et al.  Multi-Objective Control-Structure Optimization via Homotopy Methods , 1993, SIAM J. Optim..

[17]  C. Hillermeier.  Nonlinear Multiobjective Optimization: A Generalized Homotopy Approach , 2001 .

[18]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

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

[20]  Kaisa Miettinen,et al.  Nonlinear multiobjective optimization , 1998, International series in operations research and management science.

[21]  Claus Hillermeier,et al.  Nonlinear Multiobjective Optimization , 2001 .

[22]  Ian Millington.  Stability and Optimization , 2007 .

[23]  G. C. Shephard,et al.  Convex Polytopes , 1969, The Mathematical Gazette.

[24]  Kurt M. Anstreicher,et al.  Linear Programming in O([n3/ln n]L) Operations , 1999, SIAM J. Optim..

[25]  John E. Dennis,et al.  Normal-Boundary Intersection: A New Method for Generating the Pareto Surface in Nonlinear Multicriteria Optimization Problems , 1998, SIAM J. Optim..

[26]  W. de Melo,et al.  On the structure of the pareto set of generic mappings , 1976 .

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

[28]  E. Allgower,et al.  Piecewise linear methods for nonlinear equations and optimization , 2000 .

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

[30]  C. Hwang.  Multiple Objective Decision Making - Methods and Applications: A State-of-the-Art Survey , 1979 .

[31]  Kathrin Klamroth,et al.  Integrating Approximation and Interactive Decision Making in Multicriteria Optimization , 2008, Oper. Res..

[32]  S. Ruzika,et al.  Approximation Methods in Multiobjective Programming , 2005 .

[33]  A. Wierzbicki.  On the completeness and constructiveness of parametric characterizations to vector optimization problems , 1986 .

[34]  M. McLure.  One Hundred Years from Today: Vilfredo Pareto, Manuale di Economia Politica con una Introduzione alla Scienza Sociale, Milan: Societa Editrice Libraria. 1906 , 2006 .

[35]  Herbert Edelsbrunner,et al.  Shape Reconstruction with Delaunay Complex , 1998, LATIN.

[36]  J. Mather.  Notes on Topological Stability , 2012 .

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

[38]  J. van Leeuwen,et al.  Discrete and Computational Geometry , 2002, Lecture Notes in Computer Science.

[39]  Herbert Edelsbrunner,et al.  Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms , 1988, SCG '88.

[40]  R. Milo,et al.  Efficiency in Evolutionary Trade-Offs , 2012, Science.

[41]  Y. Wan,et al.  On the algebraic criteria for local Pareto optima—I , 1977 .

[42]  R. Thom.  Les singularites des applications differentiables , 1956 .

[43]  M. Golubitsky,et al.  Stable mappings and their singularities , 1973 .

[44]  Yieh Hei Wan.  On the algebraic criteria for local Pareto optima. II , 1978 .

[45]  H. P. Benson,et al.  Towards finding global representations of the efficient set in multiple objective mathematical programming , 1997 .

[46]  Kaisa Miettinen,et al.  Decision Making on Pareto Front Approximations with Inherent Nondominance , 2011 .

[47]  Vilfredo Pareto,et al.  Manuale di economia politica : con una introduzione alla scienza sociale , 1906 .