Multiobjective programming and penalty functions

Abstract

This paper generalizes the penalty function method of Zang-will for scalar problems to vector problems. The vector penalty function takes the form

$$g(x,\lambda ) = f(x) + \lambda ^{ - 1} P(x)e,$$

wheree ⊀R ^m, with each component equal to unity;f:R ⁿ →R ^m, represents them objective functions {f ⁱ} defined onX $\subseteq$ R ⁿ; λ ∈R ¹, λ>0;P:R ⁿ →R ¹ X $\subseteq$ Z $\subseteq$ R ⁿ,P(x)≦0, ∨x ∈R ⁿ,P(x) = 0 ⇄x ∈X.

The paper studies properties of {E (Z, λ _r )} for a sequence of positive {λ _r } converging to 0 in relationship toE(X), whereE(Z, λ _r ) is the efficient set ofZ with respect tog(·, λ_r) andE(X) is the efficient set ofX with respect tof. It is seen that some of Zangwill's results do not hold for the vector problem. In addition, some new results are given.