Solution Concepts in Vector Optimization

Many applications require the optimization of multiple conflicting goals at the same time. Such a problem can be modeled as a vector optimization problem. Vector optimization deals with the problem of finding efficient elements of a vector-valued function. In that sense, vector optimization generalizes the concept of scalar optimization. In scalar optimization, there is only one concept for efficiency which characterizes efficient elements, namely the solution which generates the smallest function value. But, due to the lack of a total order in general spaces, order relations that are defined within the optimality concept need to be chosen. In this chapter, we discuss several solution concepts for a vector optimization problem. In particular, solution concepts for vector optimization problem equipped with a variable domination structure are studied. Moreover, we present some existence results for solutions of vector optimization problems.

[1]  Hirotaka Nakayama,et al.  Theory of Multiobjective Optimization , 1985 .

[2]  J. Zowe.  A duality theorem for a convex programming problem in order complete vector lattices , 1975 .

[3]  C. Tammer,et al.  Theory of Vector Optimization , 2003 .

[4]  Andreas H. Hamel,et al.  Closing the Duality Gap in Linear Vector Optimization , 2004 .

[5]  Michel Théra,et al.  Semicontinuity of vector-valued mappings , 2007 .

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

[7]  Tetsuzo Tanino,et al.  Conjugate Duality in Vector Optimization , 1992 .

[8]  Yacov Y. Haimes,et al.  Multiobjective Decision Making: Theory and Methodology , 1983 .

[9]  Lizhen Shao,et al.  A dual variant of Benson’s “outer approximation algorithm” for multiple objective linear programming , 2012, J. Glob. Optim..

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

[11]  Johannes Jahn.  A generalization of a theorem of Arrow, Barankin, and Blackwell , 1988 .

[12]  D. Blackwell,et al.  5. Admissible Points of Convex Sets , 1953 .

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

[14]  Christiane Tammer,et al.  A new approach to duality in vector optimization , 2007 .

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

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

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

[18]  Adriaan C. Zaanen,et al.  Introduction to Operator Theory in Riesz Spaces , 1997 .

[19]  J. Nieuwenhuis.  Supremal points and generalized duality , 1980 .

[20]  Christiane Tammer,et al.  Set-valued duality theory for multiple objective linear programs and application to mathematical finance , 2009, Math. Methods Oper. Res..

[21]  F. Ferro.  A New Abb Theorem In Banach Spaces , 1999 .

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

[23]  Andreas Löhne,et al.  Geometric Duality in Multiple Objective Linear Programming , 2008, SIAM J. Optim..

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

[25]  Diethard Pallaschke,et al.  Foundations of mathematical optimization : convex analysis without linearity , 1997 .