On Lagrangian saddlepoints in vector optimization

Giancarlo Bigi and Massimo Pappalardo


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.

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