Characterizations of set relations with respect to variable domination structures via oriented distance function

ABSTRACT In this paper, we characterize several known set relations with respect to variable domination structures by defining scalarization functions in terms of the oriented distance function. The characterization of minimal elements of a family of sets and the properties of sets of minimal elements are discussed by means of scalarization functionals. An application of variable domination structures to uncertain vector optimization is also given.

[1]  Elisabeth Köbis,et al.  Set optimization by means of variable order relations , 2017 .

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

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

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

[5]  Shengjie Li,et al.  A new nonlinear scalarization function and applications , 2016 .

[6]  Jen-Chih Yao,et al.  Characterizations of set order relations and constrained set optimization problems via oriented distance function , 2017 .

[7]  Pascal Bouvry,et al.  Modelling, Computation and Optimization in Information Systems and Management Sciences, Second International Conference, MCO 2008, Metz, France - Luxembourg, September 8-10, 2008. Proceedings , 2008, MCO.

[8]  C. Gerth,et al.  Nonconvex separation theorems and some applications in vector optimization , 1990 .

[9]  Daishi Kuroiwa,et al.  On cone of convexity of set-valued maps , 1997 .

[10]  C. G. Liu,et al.  Merit functions in vector optimization , 2009, Math. Program..

[11]  Jen-Chih Yao,et al.  Vector Variational Inequalities and Vector Optimization , 2018 .

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

[13]  J. B. Hiriart-Urruty,et al.  Tangent Cones, Generalized Gradients and Mathematical Programming in Banach Spaces , 1979, Math. Oper. Res..

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

[15]  A. Taa,et al.  On Lagrance-Kuhn-Tucker Mulitipliers for Muliobjective Optimization Problems , 1997 .

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

[17]  Alberto Zaffaroni,et al.  Degrees of Efficiency and Degrees of Minimality , 2003, SIAM J. Control. Optim..

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

[19]  A. Wald.  Statistical Decision Functions Which Minimize the Maximum Risk , 1945 .

[20]  Johannes Jahn,et al.  Vector optimization - theory, applications, and extensions , 2004 .

[21]  R. Werner.  Robust Multiobjective Optimization , 2015 .

[22]  Gabriele Eichfelder,et al.  Set Approach for Set Optimization with Variable Ordering Structures Part I: Set Relations and Relationship to Vector Approach , 2016, J. Optim. Theory Appl..

[23]  E. Köbis,et al.  Treatment of set order relations by means of a nonlinear scalarization functional: a full characterization , 2016 .

[25]  Laurent El Ghaoui,et al.  Robust Optimization , 2021, ICORES.

[26]  E. Köbis,et al.  Variable Ordering Structures in Set Optimization , 2014 .

[27]  Jen-Chih Yao,et al.  Recent developments in vector optimization , 2012 .

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

[29]  Johannes Jahn,et al.  New Order Relations in Set Optimization , 2011, J. Optim. Theory Appl..

[30]  Gabriele Eichfelder,et al.  Set Approach for Set Optimization with Variable Ordering Structures Part II: Scalarization Approaches , 2014, J. Optim. Theory Appl..