On Continuation Methods for the Numerical Treatment of Multi-Objective Optimization Problems

In this report we describe how continuation methods can be used for the numerical treatment of multi-objective optimization problems (MOPs): starting with a given Karush-Kuhn-Tucker point (KKT-point) x of an MOP, these techniques can be applied to detect further KKT-points in the neighborhood of x. In the next step, again further points are computed starting with these new-found KKT-points, and so on. In order to maintain a good spread of these solutions we use boxes for the representation of the computed parts of the solution set. Based on this background, we propose a new predictor-corrector variant, and show some numerical results indicating the strength of the method, in particular in higher dimensions. Further, the data structure allows for an efficient computation of MOPs with more than two objectives, which has not been considered so far in most existing continuation methods.

[1]  J. Marchal.  Cours d'economie politique , 1950 .

[2]  John A. Nelder,et al.  A Simplex Method for Function Minimization , 1965, Comput. J..

[3]  D. Anderson,et al.  Algorithms for minimization without derivatives , 1974 .

[4]  E. Allgower,et al.  An Algorithm for Piecewise-Linear Approximation of an Implicitly Defined Manifold , 1985 .

[5]  J. Jahn.  Mathematical vector optimization in partially ordered linear spaces , 1986 .

[6]  W. Rheinboldt.  Numerical analysis of parametrized nonlinear equations , 1986 .

[7]  Ju Airgen Guddat,et al.  Multiobjective and stochastic optimization based on parametric optimization , 1987 .

[8]  W. Rheinboldt.  On the computation of multi-dimensional solution manifolds of parametrized equations , 1988 .

[9]  R. S. Laundy,et al.  Multiple Criteria Optimisation: Theory, Computation and Application , 1989 .

[10]  Ralph E. Steuer.  The Tchebycheff Procedure of Interactive Multiple Objective Programming , 1989 .

[11]  E. Allgower,et al.  Numerical Continuation Methods , 1990 .

[12]  R. T. Haftka,et al.  Tracing the Efficient Curve for Multi-objective Control-Structure Optimization , 1991 .

[13]  Dimitri P. Bertsekas,et al.  Nonlinear Programming , 1997 .

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

[15]  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..

[16]  Kalyanmoy Deb,et al.  A Fast Elitist Non-dominated Sorting Genetic Algorithm for Multi-objective Optimisation: NSGA-II , 2000, PPSN.

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

[18]  Kalyanmoy Deb,et al.  Evolutionary Multi-Criterion Optimization: First International Conference, EMO 2001, Zurich, Switzerland, March 7-9, 2001 Proceedings , 2001 .

[19]  Marco Laumanns,et al.  SPEA2: Improving the strength pareto evolutionary algorithm , 2001 .

[20]  S. Schäffler,et al.  Stochastic Method for the Solution of Unconstrained Vector Optimization Problems , 2002 .

[21]  M. Dellnitz,et al.  Finding zeros by multilevel subdivision techniques , 2002 .

[22]  C.A. Coello Coello,et al.  MOPSO: a proposal for multiple objective particle swarm optimization , 2002, Proceedings of the 2002 Congress on Evolutionary Computation. CEC'02 (Cat. No.02TH8600).

[23]  Jonathan E. Fieldsend,et al.  A Multi-Objective Algorithm based upon Particle Swarm Optimisation, an Efficient Data Structure and , 2002 .

[24]  Michael E. Henderson,et al.  Multiple Parameter Continuation: Computing Implicitly Defined k-Manifolds , 2002, Int. J. Bifurc. Chaos.

[25]  C. Fonseca.  Evolutionary multi-criterion optimization : Second International Conference, EMO 2003, Faro, Portugal, April 8-11, 2003 : proceedings , 2003 .

[26]  J. van Leeuwen,et al.  Evolutionary Multi-Criterion Optimization , 2003, Lecture Notes in Computer Science.

[27]  Jürgen Teich,et al.  Covering Pareto Sets by Multilevel Evolutionary Subdivision Techniques , 2003, EMO.

[28]  Oliver Schütze,et al.  A New Data Structure for the Nondominance Problem in Multi-objective Optimization , 2003, EMO.

[29]  Sanaz Mostaghim.  Multi-objective evolutionary algorithms: data structures, convergence, and diversity , 2004 .

[30]  Peter Deuflhard,et al.  Newton Methods for Nonlinear Problems , 2004 .

[31]  J. Fliege,et al.  Constructing approximations to the efficient set of convex quadratic multiobjective problems , 2004 .

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

[33]  K. Schittkowski,et al.  NONLINEAR PROGRAMMING , 2022 .