Degrees of efficiency and degrees of minimality. (English) Zbl 1046.90084

Zaffaroni, Alberto

SIAM J. Control Optimization 42, No. 3, 1071-1086 (2003).

Summary: We characterize different types of solutions of a vector optimization problem by means of a scalarization procedure. Usually different scalarizing functions are used in order to obtain the various solutions of the vector problem. Here we consider different kinds of solutions of the same scalarized problem. Our results allow us to establish a parallelism between the solutions of the scalarized problem and the various efficient frontiers: stronger solution concepts of the scalar problem correspond to more restrictive notions of efficiency. Besides the usual notions of weakly efficient and efficient points, which are characterized as global and strict global solutions of the scalarized problem, we also consider some restricted notions of efficiency, such as strict and proper efficiency, which are characterized as Tikhonov well-posed minima and sharp minima for the scalarized problem.

MSC:

90C29	Multi-objective and goal programming
90C31	Sensitivity, stability, parametric optimization
90C48	Programming in abstract spaces
49K40	Sensitivity, stability, well-posedness

Keywords:

vector optimization; scalarization; proper efficiency; strict efficiency; sharp minima; well-posed minima