Multiobjective Quadratic Problem: Characterization of the Efficient Points

Abstract

Here we consider the multiobjective quadratic problem with convex objective functions: (MQP)

$$ \begin{array}{*{20}{c}} {Min\left( {{f_1}\left( x \right), \ldots ,{f_m}\left( x \right)} \right)} \\ {x \in \mathbb{R}} \end{array} $$

(1)

where ƒ _i (x)=1/2 x ^t A _i x+b ^t _i x, i = 1,…,m are convex functions, x,b _i ∈ ℝⁿ, and A _i ∈ M _n×n

We present the result that characterizes weakly efficient points for the problem (MQP). The purpose of this work is to find the conditions under which the previous points are also efficient points for the problem (MQP). For this, we give a procedure based on the iterative application of Kuhn-Tucker conditions on gradients of objective functions.

Finally, we will show some extensions of obtained results to find the efficient points in the multiobjective programming problem with generalized convex objective functions.