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Saddle point with a given Morse index on a potential energy surface is animportant object related to energy landscape in physics and chemistry.Efficient numerical methods based on iterative minimization formulation havebeen proposed in the forms of the sequence of minimization subproblems or thecontinuous dynamics. We here present a differential game interpretation of thisformulation and theoretically investigate the Nash equilibrium of the proposedgame and the original saddle point on potential energy surface. To define thisdifferential game, a new proximal function growing faster than quadratic isintroduced to the cost function in the game and a robust IterativeProximal-Minimization algorithm (IPM) is then derived to compute the saddlepoints. We prove that the Nash equilibrium of the game is exactly the saddlepoint in concern and show that the new algorithm is more robust than theprevious iterative minimization algorithm without proximity, while the sameconvergence rate and the computational cost still hold. A two dimensionalproblem and the Cahn-Hillard problem are tested to demonstrate this numericaladvantage.

Towards Robust Calculation of index-$k$ Saddle Point: Iterative Proximal-Minimization and Differential Game Model