Robustness for uncertain multi-objective optimization: a survey and analysis of different concepts

In this paper, we discuss various concepts of robustness for uncertain multi-objective optimization problems. We extend the concepts of flimsily, highly, and lightly robust efficiency and we collect different versions of minmax robust efficiency and concepts based on set order relations from the literature. Altogether, we compare and analyze ten different concepts and point out their relations to each other. Furthermore, we present reduction results for the class of objective-wise uncertain multi-objective optimization problems.

[1]  Allen L. Soyster,et al.  Technical Note - Convex Programming with Set-Inclusive Constraints and Applications to Inexact Linear Programming , 1973, Oper. Res..

[2]  John R. Birge,et al.  Introduction to Stochastic Programming , 1997 .

[3]  Jürgen Branke,et al.  Creating Robust Solutions by Means of Evolutionary Algorithms , 1998, PPSN.

[4]  A Gerodimos,et al.  Robust Discrete Optimization and its Applications , 1996, J. Oper. Res. Soc..

[5]  Arkadi Nemirovski,et al.  Robust Convex Optimization , 1998, Math. Oper. Res..

[6]  Arkadi Nemirovski,et al.  Robust solutions of uncertain linear programs , 1999, Oper. Res. Lett..

[7]  Arkadi Nemirovski,et al.  Robust solutions of Linear Programming problems contaminated with uncertain data , 2000, Math. Program..

[8]  A. Ben-Tal,et al.  Adjustable robust solutions of uncertain linear programs , 2004, Math. Program..

[9]  Melvyn Sim,et al.  The Price of Robustness , 2004, Oper. Res..

[10]  S. Azarm,et al.  Multi-objective robust optimization using a sensitivity region concept , 2005 .

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

[12]  Serpil Sayin,et al.  The Multiobjective Discrete Optimization Problem: A Weighted Min-Max Two-Stage Optimization Approach and a Bicriteria Algorithm , 2005, Manag. Sci..

[13]  Yves De Smet,et al.  About the applicability of MCDA to some robustness problems , 2006, Eur. J. Oper. Res..

[14]  Kalyanmoy Deb,et al.  Introducing Robustness in Multi-Objective Optimization , 2006, Evolutionary Computation.

[15]  Carlos Henggeler Antunes,et al.  Robustness Analysis in Multi-Objective Optimization Using a Degree of Robustness Concept , 2006, 2006 IEEE International Conference on Evolutionary Computation.

[16]  Patrice Perny,et al.  A decision-theoretic approach to robust optimization in multivalued graphs , 2006, Ann. Oper. Res..

[17]  Jürgen Branke,et al.  Embedded evolutionary multi-objective optimization for worst case robustness , 2008, GECCO '08.

[18]  Matteo Fischetti,et al.  Light Robustness , 2009, Robust and Online Large-Scale Optimization.

[19]  Rolf H. Möhring,et al.  The Concept of Recoverable Robustness, Linear Programming Recovery, and Railway Applications , 2009, Robust and Online Large-Scale Optimization.

[20]  Martin W. P. Savelsbergh,et al.  Robust Optimization for Empty Repositioning Problems , 2009, Oper. Res..

[21]  Rolf H. Möhring,et al.  Robust and Online Large-Scale Optimization: Models and Techniques for Transportation Systems , 2009, Robust and Online Large-Scale Optimization.

[22]  Matthias Ehrgott,et al.  Trends in Multiple Criteria Decision Analysis , 2010 .

[23]  Dimitris Bertsimas,et al.  A Soft Robust Model for Optimization Under Ambiguity , 2010, Oper. Res..

[24]  John R. Birge,et al.  Introduction to Stochastic programming (2nd edition), Springer verlag, New York , 2011 .

[25]  Margaret M. Wiecek,et al.  A robust multiobjective optimization problem with application to Internet routing , 2012 .

[26]  Wei Chen,et al.  Including robustness in multi-criteria optimization for intensity-modulated proton therapy , 2011, Physics in medicine and biology.

[27]  K. Klamroth,et al.  A unified approach for different concepts of robustness and stochastic programming via non-linear scalarizing functionals , 2013 .

[28]  H. Yu,et al.  Robust Multiple Objective Game Theory , 2012, Journal of Optimization Theory and Applications.

[29]  Jonas Ide.  Concepts of Robustness for Uncertain Multi-Objective Optimization , 2014 .

[30]  Anita Schöbel,et al.  Recovery-to-optimality: A new two-stage approach to robustness with an application to aperiodic timetabling , 2014, Comput. Oper. Res..

[31]  Nikolaos Trichakis,et al.  Pareto Efficiency in Robust Optimization , 2014, Manag. Sci..

[32]  Matthias Ehrgott,et al.  Minmax robustness for multi-objective optimization problems , 2014, Eur. J. Oper. Res..

[33]  Elisabeth Köbis,et al.  Concepts of efficiency for uncertain multi-objective optimization problems based on set order relations , 2014, Math. Methods Oper. Res..

[34]  Anita Schöbel,et al.  Generalized light robustness and the trade-off between robustness and nominal quality , 2014, Math. Methods Oper. Res..

[35]  Jörg Fliege,et al.  Robust multiobjective optimization & applications in portfolio optimization , 2014, Eur. J. Oper. Res..

[36]  A. Schöbel,et al.  The relationship between multi-objective robustness concepts and set-valued optimization , 2014 .

[37]  Stephan Westphal,et al.  An application of deterministic and robust optimization in the wood cutting industry , 2015, 4OR.

[38]  Constantin Zalinescu,et al.  Set-valued Optimization - An Introduction with Applications , 2014, Vector Optimization.