Nonconvex Minimization Problems

I. The central result. The grandfather of it all is the celebrated 1961 theorem of Bishop and Phelps (see [7], [8]) that the set of continuous linear functionals on a Banach space E which attain their maximum on a prescribed closed convex bounded subset X c E is norm-dense in £*. The crux of the proof lies in introducing a certain convex cone in E, associating with it a partial ordering, and applying to the latter a transfinite induction argument (Zorn's lemma). This argument was later used in different settings by Brondsted and Rockafellar (see [9]) and by F. Browder (see [11]). The various situations can be adequately summarized in a diagram: