On a practical notion of Geoffrion proper optimality in multicriteria optimization

ABSTRACT Geoffrion proper optimality is a widely used optimality notion in multicriteria optimization that prevents exact solutions having unbounded trade-offs. As algorithms for multicriteria optimization usually give only approximate solutions, we analyze the notion of approximate Geoffrion proper optimality. We show that in the limit, approximate Geoffrion proper optimality may converge to solutions having unbounded trade-offs. Therefore, we introduce a restricted notion of approximate Geoffrion proper optimality and prove that this restricted notion alleviates the problem of solutions having unbounded trade-offs. Furthermore, using a characterization based on the infeasibility of a system of inequalities, we investigate the convergence properties of different approximate optimality notions in multicriteria optimization. These convergence properties are important for algorithmic reasons. The restricted notion of approximate Geoffrion proper optimality seems to be the only approximate optimality notion that shows favourable convergence properties. This notion bounds the trade-offs globally and can be used in multicriteria decision-making algorithms as well. Due to these, it seems to be a practical optimality notion.

[1]  Existence of ∈-Minima for Vector Optimization Problems , 2001 .

[2]  Majid Soleimani-Damaneh,et al.  On Benson’s scalarization in multiobjective optimization , 2016, Optim. Lett..

[3]  Jen-Chwan Liu.  ϵ-properly efficient solutions to nondifferentiable multiobjective programming problems , 1999 .

[4]  E. Makarov,et al.  Unified Representation of Proper Efficiency by Means of Dilating Cones , 1999 .

[5]  Hartmut Schmeck,et al.  A Framework for Incorporating Trade-Off Information Using Multi-Objective Evolutionary Algorithms , 2010, PPSN.

[6]  Aimo A. Törn,et al.  A sampling-search-clustering approach for exploring the feasible/efficient solutions of MCDM problems , 1980, Comput. Oper. Res..

[7]  椹木 義一,et al.  Theory of multiobjective optimization , 1985 .

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

[9]  A. M. Geoffrion.  Proper efficiency and the theory of vector maximization , 1968 .

[10]  J. Borwein,et al.  Super efficiency in vector optimization , 1993 .

[11]  J. Borwein.  Proper Efficient Points for Maximizations with Respect to Cones , 1977 .

[12]  H. P. Benson,et al.  Existence of efficient solutions for vector maximization problems , 1978 .

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

[14]  D. Dentcheva,et al.  On several concepts for ɛ-efficiency , 1994 .

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

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

[17]  Tamaki Tanaka.  Approximately Efficient Solutions for Vector Optimization Problems(Optimization Theory and its Applications in Mathematical Systems) , 1995 .

[18]  Alexander Engau,et al.  Definition and Characterization of Geoffrion Proper Efficiency for Real Vector Optimization with Infinitely Many Criteria , 2015, J. Optim. Theory Appl..

[19]  H. P. Benson,et al.  An improved definition of proper efficiency for vector maximization with respect to cones , 1979 .

[20]  A. Guerraggio,et al.  Geoffrion type characterization of higher-order properly efficient points in vector optimization , 2007 .

[21]  Yu. K. Mashunin,et al.  Vector Optimization , 2017, Encyclopedia of Machine Learning and Data Mining.

[22]  Margaret M. Wiecek,et al.  Relative importance of criteria in multiobjective programming: A cone-based approach , 2010, Eur. J. Oper. Res..

[23]  M. I. Henig.  Proper efficiency with respect to cones , 1982 .

[24]  Gabriele Eichfelder,et al.  Variable Ordering Structures in Vector Optimization , 2014, Vector Optimization.

[25]  E. Polak,et al.  On Multicriteria Optimization , 1976 .

[26]  J. Dutta,et al.  ON APPROXIMATE MINIMA IN VECTOR OPTIMIZATION , 2001 .

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

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

[29]  Bastian Goldlücke,et al.  Variational Analysis , 2014, Computer Vision, A Reference Guide.

[30]  Kalyanmoy Deb,et al.  On Properly Pareto Optimal Solutions , 2005, Practical Approaches to Multi-Objective Optimization.

[31]  M. Wiecek.  Advances in Cone-Based Preference Modeling for Decision Making with Multiple Criteria , 2007 .

[32]  R. K. Ursem.  Multi-objective Optimization using Evolutionary Algorithms , 2009 .

[33]  Klaudia Beich,et al.  Theory Of Vector Optimization , 2016 .