On mountain pass type algorithms

Abstract

We consider constructive proofs of the mountain pass lemma, the saddle point theorem and a linking type theorem. In each, an initial “path” is deformed by pushing it downhill using a (pseudo) gradient flow, and, at each step, a high point on the deformed path is selected. Using these high points, a Palais–Smale sequence is constructed, and the classical minimax theorems are recovered. Because the sequence of high points is more accessible from a numerical point of view, we investigate the behavior of this sequence in the final two sections. We show that if the functional satisfies the Palais–Smale condition and has isolated critical points, then the high points form a Palais–Smale sequence, and—passing to a subsequence—the high points will in fact converge to a critical point.