High-Index Optimization-Based Shrinking Dimer Method for Finding High-Index Saddle Points

We present a high-index optimization-based shrinking dimer (HiOSD) method to compute index-k saddle points as a generalization of the optimization-based shrinking dimer method for index-1 saddle points [L. Zhang, Q. Du, and Z. Zheng, SIAM J. Sci. Comput., 38 (2016), pp. A528--A544]. We first formulate a minimax problem for an index-k saddle point that is a local maximum on a k-dimensional manifold and a local minimum on its orthogonal complement. The k-dimensional maximal subspace is spanned by the k eigenvectors corresponding to the smallest k eigenvalues of the Hessian, which can be constructed by the simultaneous Rayleigh-quotient minimization technique or the locally optimal block preconditioned conjugate gradient method. Under the minimax framework, we implement the Barzilai--Borwein gradient method to speed up the convergence. We demonstrate the efficiency of the HiOSD method for computing high-index saddle points by applying finite-dimensional examples and semilinear elliptic problems.