Sufficient conditions for the existence of multipliers and Lagrangian duality in abstract optimization problems

We consider the following optimization problem: in an abstract set X, find and element x that minimizes a real function f subject to the constraints g(x)≤0 and h(x)=0, where g and h are functions from X into normed vector spaces. Assumptions concerning an overall convex structure for the problem in the image space, the existence of interior points in certain sets, and the normality of the constraints are formulated. A theorem of the alternative is proved for systems of equalities and inequalities, and an intrinsic multiplier rule and a Lagrangian saddle-point theorem (strong duality theorem) are obtained as consequences.