On Lagrangian saddlepoints in vector optimization
Giancarlo Bigi and Massimo Pappalardo
Summary
In Vector Optimization, duality theorems and Lagrangian functions have
been known for a long time. One of the first works was that of Gale,
Kuhn and Tucker appeared in 1951, which treated linear problems
and matrix optimization. Since then, there have been several
developments for the linear and the nonlinear case. In this last one,
which is the main interest of our paper, Tanino and Sawaragi
in 1979 reported very interesting results, using a
vector-valued Lagrangian function with vector multipliers. These results have
been generalized in several directions in these last two
decades. Another different approach, the one we follow in this paper,
consists in treating duality theory based on scalarization, instead of
using vector-valued Lagrangian function. Two of the first pioneering works in
this field were published by Jahn in 1983 and by Luc
in 1984. The main purpose of our paper is not devoted to
the duality theory but to the connections between saddlepoints of the
scalarized Lagrangian function and efficient points of the given
problem. In this context we underline that different definitions of
saddlepoints can be given and they are strictly related to different
definitions of efficient solution. The role of regularity and
constraint qualification for vector optimization problems is emphasized.
If you are interested in this paper, feel free to contact me.
My Home Page