Variational convergence for vector-valued functions and its applications to convex multiobjective optimization

The aim of this work is to study a notion of variational convergence for vector-valued functions. We show that it is suitable for obtaining existence and stability results in convex multiobjective optimization. We obtain various of properties of the variational convergence. We characterize it via the set convergence of epigraphs, coepigraphs, level sets, and some infima. We also characterize it by means of two metrics. We compare it with other notions of convergence for vector-valued functions from the literature and we show that it is more general than most of them. For obtaining the existence and stability results we employ an asymptotic method that has shown to be very useful in optimization theory. In this method we couple the variational convergence with notions of asymptotic analysis (asymptotic cones and functions).

[1]  P. Loridan.  ε-solutions in vector minimization problems , 1984 .

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

[3]  X. X. Huang.  Stability in vector-valued and set-valued optimization , 2000, Math. Methods Oper. Res..

[4]  Fabián Flores-Bazán.  Ideal, weakly efficient solutions for vector optimization problems , 2002 .

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

[6]  Shu-Wen Xiang,et al.  On essential sets and essential components of efficient solutions for vector optimization problems , 2006 .

[7]  Sien Deng.  On Efficient Solutions in Vector Optimization , 1998 .

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

[9]  Xiaoqi Yang,et al.  Vector Optimization: Set-Valued and Variational Analysis , 2005 .

[10]  Roger J.-B. Wets,et al.  A characterization of epi-convergence in terms of convergence of level sets , 1992 .

[11]  Felipe Lara,et al.  Inner and outer estimates for the solution sets and their asymptotic cones in vector optimization , 2012, Optim. Lett..

[12]  C. Combari,et al.  Sous-différentiels de fonctions convexes composées , 1994 .

[13]  Variational Convergence of Composed Convex Functions , 2005 .

[14]  J. Dutta,et al.  ON APPROXIMATE MINIMA IN VECTOR OPTIMIZATION , 2001 .

[15]  Sien Deng.  On Approximate Solutions in Convex Vector Optimization , 1997 .

[16]  Sien Deng.  Boundedness and Nonemptiness of the Efficient Solution Sets in Multiobjective Optimization , 2009 .

[17]  Roger J.-B. Wets,et al.  Variational convergence of bivariate functions: lopsided convergence , 2008, Math. Program..

[18]  Rubén López,et al.  Approximations of Equilibrium Problems , 2012, SIAM J. Control. Optim..

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

[20]  M. Teboulle,et al.  Asymptotic cones and functions in optimization and variational inequalities , 2002 .

[21]  Hédy Attouch,et al.  Stability Results for Ekeland's ε-Variational Principle and Cone Extremal Solutions , 1993, Math. Oper. Res..

[22]  Bernard Lemaire,et al.  Approximation in multiobjective optimization , 1992, J. Glob. Optim..

[23]  A. Iusem,et al.  Set-valued mappings and enlargements of monotone operators , 2008 .

[24]  P. Oppezzi,et al.  A convergence for vector valued functions , 2008 .

[25]  Fabián Flores-Bazán.  Radial Epiderivatives and Asymptotic Functions in Nonconvex Vector Optimization , 2003 .

[26]  Pirro Oppezzi,et al.  A convergence for infinite dimensional vector valued functions , 2008, J. Glob. Optim..

[27]  Matthias Ehrgott,et al.  Multicriteria Optimization (2. ed.) , 2005 .

[28]  S. Xiang,et al.  Stability Results for Efficient Solutions of Vector Optimization Problems , 2007 .

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

[30]  E. Miglierina,et al.  Stability for convex vector optimization problems , 2004 .

[31]  S. J. Li,et al.  Stability results for convex vector-valued optimization problems , 2011 .

[32]  Yonghui Zhou,et al.  Continuity properties of solutions of vector optimization , 2006 .

[33]  Rubén López,et al.  On the set of weakly efficient minimizers for convex multiobjective programming , 2008, Oper. Res. Lett..

[34]  H. Attouch.  Variational convergence for functions and operators , 1984 .