Applying set optimization to weak efficiency

Set-valued extensions of vector-valued functions are used to investigate the relations between weak efficiency and variational inequalities (both Stampacchia and Minty type) which allows to apply the complete lattice framework from set optimization. Since the seminal work of Giannessi, it has been a challenge to generalize scalar results to the vector case. In this effort, some notions of generalized derivatives for vector-valued functions have been introduced, either in the form of set-valued functions or introducing appropriate notions of infinite elements in vector spaces. Switching the focus to set optimization in conlinear spaces, we propose a Dini-type derivative, that keeps the same set-valued form of the optimization problem.

[1]  Carola Schrage.  Scalar representation and conjugation of set-valued functions , 2010, 1011.5860.

[2]  Kok Lay Teo,et al.  Some Remarks on the Minty Vector Variational Inequality , 2004 .

[3]  G. Crespi,et al.  Set Optimization Meets Variational Inequalities , 2013, 1303.4212.

[4]  Peter Jipsen,et al.  Residuated lattices: An algebraic glimpse at sub-structural logics , 2007 .

[5]  G. Crespi,et al.  Some remarks on the Minty vector variational principle , 2008 .

[6]  Andreas H. Hamel,et al.  A Duality Theory for Set-Valued Functions I: Fenchel Conjugation Theory , 2009 .

[7]  Semen S. Kutateladze,et al.  MINKOWSKI DUALITY AND ITS APPLICATIONS , 1972 .

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

[9]  Kim C. Border,et al.  Infinite Dimensional Analysis: A Hitchhiker’s Guide , 1994 .

[10]  A. Hamel,et al.  Set Optimization—A Rather Short Introduction , 2014, 1404.5928.

[11]  G. Crespi,et al.  A Minty variational principle for set optimization , 2014, 1403.2898.

[12]  Qamrul Hasan Ansari,et al.  Nonsmooth Vector Optimization Problems and Minty Vector Variational Inequalities , 2010 .

[13]  Juan Enrique Martínez-Legaz,et al.  Dualities Associated To Binary Operations On R , 1995 .

[14]  Matteo Rocca,et al.  Variational inequalities in vector optimization , 2005 .

[15]  H. Riahi,et al.  Variational Methods in Partially Ordered Spaces , 2003 .

[16]  丸山 徹.  Convex Analysisの二,三の進展について , 1977 .

[17]  Matteo Rocca,et al.  Variational Inequalities Characterizing Weak Minimality in Set Optimization , 2014, J. Optim. Theory Appl..

[18]  Carola Schrage,et al.  Continuity concepts for set-valued functions and a fundamental duality formula for set-valued optimization , 2013 .

[19]  L. Rodríguez-Marín,et al.  Nonconvex scalarization in set optimization with set-valued maps , 2007 .

[20]  Andreas Löhne,et al.  Vector Optimization with Infimum and Supremum , 2011, Vector Optimization.

[21]  Giorgio Giorgi,et al.  Dini derivatives in optimization — Part I , 1992 .

[22]  Ivan Ginchev,et al.  First-order optimality conditions in set-valued optimization , 2006, Math. Methods Oper. Res..

[23]  László Fuchs,et al.  Teilweise geordnete algebraische Strukturen , 1966 .

[24]  A. Hamel,et al.  Variational principles on metric and uniform spaces , 2005 .

[25]  Andreas Löhne,et al.  Solution concepts in vector optimization: a fresh look at an old story , 2011 .

[26]  Ivan Ginchev,et al.  Vector optimization problems with quasiconvex constraints , 2009, J. Glob. Optim..

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

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

[29]  Jen-Chih Yao,et al.  Solution Concepts in Vector Optimization , 2018 .

[30]  F. Giannessi.  On Minty Variational Principle , 1998 .

[31]  C. Zălinescu.  Convex analysis in general vector spaces , 2002 .

[32]  G. Crespi,et al.  Minty Variational Inequalities, Increase-Along-Rays Property and Optimization1 , 2004 .

[33]  A. Hamel,et al.  Notes about extended real- and set-valued functions , 2010, 1011.3179.