Convergence of a mountain pass type algorithm for strongly indefinite problems and systems
Provides theoretical convergence guarantees for a numerical method applied to indefinite variational problems, which is an incremental advance in optimization theory.
The paper proves convergence of a mountain pass algorithm for strongly indefinite problems, establishing whole-sequence convergence under a localization assumption, and illustrates results on a Schrödinger equation and system.
For a functional $\E$ and a peak selection that picks up a global maximum of $\E$ on varying cones, we study the convergence up to a subsequence to a critical point of the sequence generated by a mountain pass type algorithm. Moreover, by carefully choosing stepsizes, we establish the convergence of the whole sequence under a "localization" assumption on the critical point. We illustrate our results with two problems: an indefinite Schrödinger equation and a superlinear Schrödinger system.