Study on "Weak" Efficiencies in Vector Optimization

Abstract

It is a well-known fact that efficient points enjoy special attention in vector optimization given that numerous applications in areas as mathematical economics, game theory, geometry of normed spaces. Following the various types of efficient points introduced and studied overtime by many authors (see  [1-7], [8])  we propose an unified approach concerning the "weak" efficient points (efficient points defined by respect to cones that have some interiority properties). These generalized weak efficient points will be called -efficient points and we will present conditions of existence, domination properties and comparative results for them. As application, we will study a generalized vector optimization problem defined with the -efficient points. The definition of the notion of solution generated some difficulties given that the usual notion of solution would have reduced the study of the generalized problem to a classical problem, the MIN problem. Finally, the solution for this problem will be a net of approximate efficient points which is closely related with the notion of asymptotically weakly Pareto optimizing sequence used in [9]. In order to obtain conditions on existence and properties of these solutions, we introduce the INFSUP problem, a generalization of MINMAX problem. Following the studies for the MINMAX problem (see for example [10-14], [15,16], [17-19], some saddle points theorems and duality results using a suitable lagrangian adapted for the INFSUP problem, a generalization of the MINMAX problem are obtained. Also, we’ll present the links between our problems and two special problems, the scalar and the linear approximate problems.