This paper deals with the characterization of the efficient set of a set \(Z\subseteq {\mathbb{R}}^ n\), further constrained by constraints \(g_ k(x)\leq 0\), \(1\leq k\leq p\), with respect to a multiple objective vector function \(f: {\mathbb{R}}^ n\to {\mathbb{R}}^ m\). This characterization is obtained by extending the usual scalar Lagrangean function to a vector Lagrangean function \(f_{\mu}=f-\mu g\), with \(\mu \in {\mathbb{R}}^ p\), \(\mu\geq 0\). If \(X=\{x\in Z:\) \(g_ k(x)\leq 0\), \(1\leq k\leq p\}\), the paper looks at the relationships between E(X,f) and \(E(Z,f_{\mu})\), where these sets are, respectively, the efficient sets of X with respect to f and of Z with respect to \(f_{\mu}.\) The first part of the paper deals with some elementary vector saddle point results. The middle part of the paper deals with four theorems producing similar results but under different analytic conditions on the functions f, g. Corollary 3 for example derives the following complete characterization of E(X,f). \(E(x,f)=\cup \{E^ 0(Z,f_{\mu}):\) \(\mu \in {\mathbb{R}}^ p\), \(\mu\geq 0\}\) where \(E^ 0(Z,f_{\mu})=\{x\in E(Z,f_{\mu}):\) \(\mu g(x)=0\}^ i.\)e. those members of \(E(Z,f_{\mu})\) which satisfy the complementary slackness condition. (Note that in Lemma 3, h should have been defined as \(h=(f,-g)).\) The final part deals with vector Lagrangean relaxation, and extends Everett's scalar Lagrangrean relaxation method to give the following result: \(E(Z,f_{\mu})=\cup \{E(X_ y,f):\) \(y\in E(Z,f_{\mu})\) for all \(\mu \in {\mathbb{R}}^ P\) and \(\mu\geq 0\}\) where \(X_ y=\{x\in Z:\) g(x)\(\leq g(y)\}\).