Ekeland’s variational principle with weighted set order relations

The main results of the paper are a minimal element theorem and an Ekeland-type variational principle for set-valued maps whose values are compared by means of a weighted set order relation. This relation is a mixture of a lower and an upper set relation which form the building block for modern approaches to set-valued optimization. The proofs rely on nonlinear scalarization functions which admit to apply the extended Brézis–Browder theorem. Moreover, Caristi’s fixed point theorem and Takahashi’s minimization theorem for set-valued maps based on the weighted set order relation are obtained and the equivalences among all these results is verified. An application to generalized intervals is given which leads to a clear interpretation of the weighted set order relation and versions of Ekeland’s principle which might be useful in (computational) interval mathematics.

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

[2]  C. Baiocchi,et al.  Variational and quasivariational inequalities: Applications to free boundary problems , 1983 .

[3]  Karl Nickel,et al.  Verbandstheoreitsche Grundlagen der Intervall-Mathematik , 1975, Interval Mathematics.

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

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

[6]  Truong Xuan Duc Ha.  Some Variants of the Ekeland Variational Principle for a Set-Valued Map , 2005 .

[7]  L. Thibault,et al.  Strict approximate solutions in set-valued optimization with applications to the approximate Ekeland variational principle☆ , 2010 .

[8]  F. Browder,et al.  A general principle on ordered sets in nonlinear functional analysis , 1976 .

[9]  C. Zălinescu,et al.  Vector variational principles for set-valued functions , 2011 .

[10]  Yousuke Araya,et al.  Ekeland's variational principle and its equivalent theorems in vector optimization , 2008 .

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

[12]  A. Rubinov.  SUBLINEAR OPERATORS AND THEIR APPLICATIONS , 1977 .

[13]  R. Branzei,et al.  Set-valued solution concepts using interval-type payoffs for interval games , 2011 .

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

[15]  Qamrul Hasan Ansari,et al.  EKELAND'S VARIATIONAL PRINCIPLE FOR SET-VALUED MAPS WITH APPLICATIONS TO VECTOR OPTIMIZATION IN UNIFORM SPACES , 2014 .

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

[17]  Yousuke Araya,et al.  Nonlinear scalarizations and some applications in vector optimization , 2010 .

[18]  Yousuke Araya,et al.  Four types of nonlinear scalarizations and some applications in set optimization , 2012 .

[19]  Daishi Kuroiwa,et al.  On set-valued optimization , 2001 .

[20]  Enrico Miglierina,et al.  Scalarization in set optimization with solid and nonsolid ordering cones , 2015, J. Glob. Optim..

[21]  X. X. Huang,et al.  Ekeland's ε-variational principle for set-valued mappings , 1998, Math. Methods Oper. Res..

[22]  I. Ekeland.  On the variational principle , 1974 .

[23]  S. H. Hou,et al.  General Ekeland's Variational Principle for Set-Valued Mappings , 2000 .

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

[25]  X. X. Huang.  A New Variant of Ekeland's Variational Principle for Set-Valued Maps , 2003 .

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

[27]  Andreas H. Hamel,et al.  Set Optimization and Applications - The State of the Art , 2015 .