A new scalarization and numerical method for constructing the weak Pareto front of multi-objective optimization problems

A numerical technique is presented for constructing an approximation of the weak Pareto front of nonconvex multi-objective optimization problems, based on a new Tchebychev-type scalarization and its equivalent representations. First, existing results on the standard Tchebychev scalarization, the weak Pareto and Pareto minima, as well as the uniqueness of the optimal value in the Pareto front, are recalled and discussed for the case when the set of weak Pareto minima is the same as the set of Pareto minima, namely, when weak Pareto minima are also Pareto minima. Of the two algorithms we present, Algorithm 1 is based on this discussion. Algorithm 2, on the other hand, is based on the new scalarization incorporating rays associated with the weights of the scalarization in the value (or objective) space, as constraints. We prove two relevant results for the new scalarization. The new scalarization and the resulting Algorithm 2 are particularly effective in constructing an approximation of the weak Pareto sections of the front. We illustrate the working and capability of both algorithms by means of smooth and nonsmooth test problems with connected and disconnected Pareto fronts.

[1]  C. Yalçin Kaya,et al.  Optimization Over the Efficient Set of Multi-objective Convex Optimal Control Problems , 2010, J. Optim. Theory Appl..

[2]  Włodzimierz Ogryczak.  Comments on properties of the minmax solutions in goal programming , 2001, Eur. J. Oper. Res..

[3]  Regina S. Burachik,et al.  An update rule and a convergence result for a penalty function method , 2007 .

[4]  Matthias Ehrgott,et al.  Multicriteria Optimization , 2005 .

[5]  Gabriele Eichfelder,et al.  Scalarizations for adaptively solving multi-objective optimization problems , 2009, Comput. Optim. Appl..

[6]  Boglárka G.-Tóth,et al.  Obtaining the efficient set of nonlinear biobjective optimization problems via interval branch-and-bound methods , 2009, Comput. Optim. Appl..

[7]  Margaret M. Wiecek,et al.  Generating epsilon-efficient solutions in multiobjective programming , 2007, Eur. J. Oper. Res..

[8]  T. Q. Phong,et al.  Scalarizing Functions for Generating the Weakly Efficient Solution Set in Convex Multiobjective Problems , 2005, SIAM J. Optim..

[9]  P. Fantini,et al.  A method for generating a well-distributed Pareto set in nonlinear multiobjective optimization , 2005 .

[10]  Kalyanmoy Deb,et al.  A fast and elitist multiobjective genetic algorithm: NSGA-II , 2002, IEEE Trans. Evol. Comput..

[11]  Sheldon H. Jacobson,et al.  Finding preferred subsets of Pareto optimal solutions , 2008, Comput. Optim. Appl..

[12]  R. Burachik,et al.  Using a General Augmented Lagrangian Duality with Implications on Penalty Methods , 2010 .

[13]  C. Yalçin Kaya,et al.  On a Modified Subgradient Algorithm for Dual Problems via Sharp Augmented Lagrangian* , 2006, J. Glob. Optim..

[14]  DebK.,et al.  A fast and elitist multiobjective genetic algorithm , 2002 .

[15]  Jörg Fliege,et al.  Newton's Method for Multiobjective Optimization , 2009, SIAM J. Optim..

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

[17]  Jeffrey C. Lagarias,et al.  Convergence Properties of the Nelder-Mead Simplex Method in Low Dimensions , 1998, SIAM J. Optim..

[18]  Günter Rudolph,et al.  Pareto Set and EMOA Behavior for Simple Multimodal Multiobjective Functions , 2006, PPSN.

[19]  Enrico Miglierina,et al.  Box-constrained multi-objective optimization: A gradient-like method without "a priori" scalarization , 2008, Eur. J. Oper. Res..

[20]  Charles Audet,et al.  Multiobjective Optimization Through a Series of Single-Objective Formulations , 2008, SIAM J. Optim..

[21]  Musa A. Mammadov,et al.  An inexact modified subgradient algorithm for nonconvex optimization , 2010, Comput. Optim. Appl..

[22]  Frank Kursawe,et al.  A Variant of Evolution Strategies for Vector Optimization , 1990, PPSN.

[23]  Dinh The Luc,et al.  Generating the weakly efficient set of nonconvex multiobjective problems , 2008, J. Glob. Optim..

[24]  Gabriele Eichfelder,et al.  Adaptive Scalarization Methods in Multiobjective Optimization , 2008, Vector Optimization.

[25]  Gabriele Eichfelder,et al.  An Adaptive Scalarization Method in Multiobjective Optimization , 2008, SIAM J. Optim..

[26]  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..

[27]  S Ruzikal,et al.  SUCCESSIVE APPROACH TO COMPUTE THE BOUNDED PARETO FRONT OF PRACTICAL MULTIOBJECTIVE OPTIMIZATION PROBLEMS , 2009 .

[28]  Masahiro Tanaka,et al.  GA-based decision support system for multicriteria optimization , 1995, 1995 IEEE International Conference on Systems, Man and Cybernetics. Intelligent Systems for the 21st Century.