寻找一般Morse指标的鞍点的水平集方法
Level set methods for finding saddle points of general Morse index
AI总结：
本文基于多维山脊定理，提出并分析了几种水平集方法以寻找一般Morse指标的鞍点，证明了非光滑情形下的收敛性和光滑有限维情形下的局部超线性收敛。
摘要：
对于实值函数，若其导数为零则该点为临界点，若临界点不是局部极值点则为鞍点。本文研究寻找一般Morse指标的鞍点的算法。我们的方法受多维山脊定理启发，并扩展了我们早期基于水平集研究的方法以寻找山脊型鞍点。我们证明了算法在非光滑情形下的收敛性，以及另一种算法在光滑有限维情形下的局部超线性收敛。
英文摘要：
For a real valued function, a point is critical if its derivatives are zero, and a critical point is a saddle point if it is not a local extrema. In this paper, we study algorithms to find saddle points of general Morse index. Our approach is motivated by the multidimensional mountain pass theorem, and extends our earlier work on methods (based on studying the level sets) to find saddle points of mountain pass type. We prove the convergence of our algorithms in the nonsmooth case, and the local superlinear convergence of another algorithm in the smooth finite dimensional case.
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