Normalized Goldstein-type local minimax method for finding multiple unstable solutions of semilinear elliptic PDEs

Abstract

The local minimax method (LMM) proposed in [Y.

Nonmonotone local minimax methods for finding multiple saddle points

“…v are continuous at v. In view of Lemmas 2.2-2.4 and in accordance with the lines in the proof of Lemmas 3.6-3.7 in [26], one can obtain the following result, which is actually an improved version of Lemma 2.1 in [23] and Lemma 2.13 in [40] since the domain of the local peak selection is changed from S ∩ L ⊥ to S. Lemma 2.5 ([26]). Suppose E ∈ C 1 (X, R) and let p(v) = t v v + w L v be a local peak selection of E w.r.t.…”

“…The following two lemmas follow from direct calculations and their proofs are referred to those of Lemmas 3.2-3.4 in [26].…”

“…It is worthwhile to point out that step-size search rules used in traditional LMMs include the optimal/exact step-size search rule [23], normalized Armijo-type stepsize search rule [24,41,38], normalized Glodstein-type step-size search rule [26], and normalized Wolfe-Powell-type step-size search rules [27]. And up to now all stepsize search rules in traditional LMMs are monotone in the sense that the sequence {E(w k )} is monotonically decreasing.…”

“…And up to now all stepsize search rules in traditional LMMs are monotone in the sense that the sequence {E(w k )} is monotonically decreasing. Actually this feature is vital for the convergence analysis in traditional LMMs; see [24,48,38,26,27]. Since the work of this paper is closely related to the normalized Armijo-type step-size search rule, let us describe it a little more.…”

“…Thanks to this step-size rule, the global convergence result for the LMM was obtained in [24,48]. Then, a normalized Goldstein-type LMM was put forward in our work [26] by recommending a normalized Goldstein step-size search rule that guarantees the sufficient decrease of the energy functional and prevents the step-size from being too small simultaneously. The feasibility and global convergence analysis of this approach was also provided in it.…”