Pointwise well-posedness in set optimization with cone proper sets

Abstract This paper deals with the well-posedness property in the setting of set optimization problems. By using a notion of well-posed set optimization problem due to Zhang et al. (2009)  [18] and a scalarization process, we characterize this property through the well-posedness, in the Tykhonov sense, of a family of scalar optimization problems and we show that certain quasiconvex set optimization problems are well-posed. Our approach is based just on a weak boundedness assumption, called cone properness, that is unavoidable to obtain a meaningful set optimization problem.

[1]  Enrico Miglierina,et al.  A Mountain Pass-type Theorem for Vector-valued Functions , 2011 .

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

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

[4]  Marius Durea Scalarization for pointwise well-posed vectorial problems , 2007, Math. Methods Oper. Res..

[5]  Roberto Lucchetti,et al.  Minima of quasi-convex functions , 1989 .

[6]  Andreas H. Hamel,et al.  Minimal element theorems and Ekeland's principle with set relations , 2006 .

[7]  Andreas H. Hamel,et al.  Duality for Set-Valued Measures of Risk , 2010, SIAM J. Financial Math..

[8]  Bienvenido Jiménez,et al.  A Set-Valued Ekeland's Variational Principle in Vector Optimization , 2008, SIAM J. Control. Optim..

[9]  César Gutiérrez,et al.  A Brézis–Browder principle on partially ordered spaces and related ordering theorems , 2011 .

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

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

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

[13]  Luis Rodríguez-Marín,et al.  Existence theorems for set optimization problems , 2007 .

[14]  C. Zălinescu,et al.  Recession cones and asymptotically compact sets , 1993 .

[15]  César Gutiérrez,et al.  Generalized ε-quasi-solutions in multiobjective optimization problems: Existence results and optimality conditions , 2010 .

[16]  Enrico Miglierina,et al.  Well-posedness and convexity in vector optimization , 2003, Math. Methods Oper. Res..

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

[18]  Kok Lay Teo,et al.  Well-posedness for set optimization problems , 2009 .

[19]  Darinka Dentcheva,et al.  On variational principles, level sets, well-posedness, and ∈-solutions in vector optimization , 1996 .

[20]  A. Göpfert Variational methods in partially ordered spaces , 2003 .

[21]  Daishi Kuroiwa,et al.  Convexity for set-valued maps , 1996 .

[22]  R. Young The algebra of many-valued quantities , 1931 .

[23]  Nicolae Popovici,et al.  Characterizations of convex and quasiconvex set-valued maps , 2003, Math. Methods Oper. Res..

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

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

[26]  Bienvenido Jiménez,et al.  Strict Efficiency in Set-Valued Optimization , 2009, SIAM J. Control. Optim..

[27]  H. W. Corley,et al.  Existence and Lagrangian duality for maximizations of set-valued functions , 1987 .

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

[29]  Enrico Miglierina,et al.  Well-Posedness and Scalarization in Vector Optimization , 2005 .