Merit functions in vector optimization

We study the weak domination property and weakly efficient solutions in vector optimization problems. In particular scalarization of these problems is obtained by virtue of some suitable merit functions. Some natural conditions to ensure the existence of error bounds for merit functions are also given.

[1]  F. Di Guglielmo.  Nonconvex Duality in Multiobjective Optimization , 1977, Math. Oper. Res..

[2]  Phan Quoc Khanh.  Optimality conditions via norm scalarization in vector optimization , 1993 .

[3]  F. Facchinei,et al.  Finite-Dimensional Variational Inequalities and Complementarity Problems , 2003 .

[4]  A. Taa,et al.  Subdifferentials of multifunctions and Lagrange multipliers for multiobjective optimization , 2003 .

[5]  L. R. Huang,et al.  On First- and Second-Order Conditions for Error Bounds , 2004, SIAM J. Optim..

[6]  Alberto Zaffaroni,et al.  Degrees of Efficiency and Degrees of Minimality , 2003, SIAM J. Control. Optim..

[7]  Ivan Ginchev,et al.  Approximation of set-valued functions by single-valued one , 2002 .

[8]  Johannes Jahn,et al.  A Characterization of Properly Minimal Elements of a Set , 1985 .

[9]  J. Aubin.  Set-valued analysis , 1990 .

[10]  Constantin Zălinescu,et al.  A generalization of the Farkas lemma and applications to convex programming , 1978 .

[11]  J. Hiriart-Urruty.  New concepts in nondifferentiable programming , 1979 .

[12]  Wei Hong Yang,et al.  Regularities and their relations to error bounds , 2004, Math. Program..

[13]  R. Holmes.  Geometric Functional Analysis and Its Applications , 1975 .

[14]  JAHN Johannes.  Scalarization in vector optimization , 1984 .

[15]  Jean-Pierre Crouzeix,et al.  Condition numbers and error bounds in convex programming , 2008, Math. Program..

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

[17]  J. Aubin,et al.  Applied Nonlinear Analysis , 1984 .

[18]  J. B. Hiriart-Urruty,et al.  Tangent Cones, Generalized Gradients and Mathematical Programming in Banach Spaces , 1979, Math. Oper. Res..

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

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

[21]  Xi Yin Zheng,et al.  Scalarization of Henig Proper Efficient Points in a Normed Space , 2000 .

[22]  D. J. White,et al.  Fundamentals of decision theory , 1969 .

[23]  Ivan Ginchev,et al.  Second-order conditions in C1,1 constrained vector optimization , 2005, Math. Program..

[24]  O. Mangasarian.  Equivalence of the Complementarity Problem to a System of Nonlinear Equations , 1976 .

[25]  Xi Yin Zheng,et al.  Error Bounds for Lower Semicontinuous Functions in Normed Spaces , 2001, SIAM J. Optim..

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

[27]  Hans Jarchow,et al.  Topological Vector Spaces , 1981 .

[28]  Constantin Zalinescu,et al.  A Nonlinear Extension of Hoffman's Error Bounds for Linear Inequalities , 2003, Math. Oper. Res..

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

[30]  Johannes Jahn,et al.  Optimality conditions for set-valued optimization problems , 1998, Math. Methods Oper. Res..

[31]  X. Q. Yang,et al.  Weak Sharp Minima in Multicriteria Linear Programming , 2005, SIAM J. Optim..

[32]  J. Borwein,et al.  Super efficiency in vector optimization , 1993 .