Vertical halfspaces as solutions of dual extremal problems

This paper is a contribution to the duality theory of extremal problems and prepares for a unified approach to this theory and to Lagrange’s principle of elimination of constraints in the convex case, to be given in a forthcoming paper. We introduce additional – so-called vertical – solutions of dual extremal problems using a compactification construction. The new problem – called the complete dual problem – is solvable if it has a finite value. We introduce a set which is relatively easy to handle, the set of promising vertical slopes. This set contains the set of vertical solutions; moreover these two sets are equal if the complete dual problem is solvable. Finally, we give a geometric interpretation of the notions vertical solutions and promising vertical slopes in terms of pointed halfspaces.