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A Fast and Simple Modification of Newton’s Method Avoiding Saddle Points

Author

Listed:
  • Tuyen Trung Truong

    (University of Oslo)

  • Tat Dat To

    (Ecole Nationale de l’Aviation Civile
    Sorbonne University)

  • Hang-Tuan Nguyen

    (Axon AI Research)

  • Thu Hang Nguyen

    (Torus Actions SAS)

  • Hoang Phuong Nguyen

    (Torus Actions SAS)

  • Maged Helmy

    (University of Oslo
    ODI Medical AS)

Abstract

We propose in this paper New Q-Newton’s method. The update rule is conceptually very simple, using the projections to the vector subspaces generated by eigenvectors of positive (correspondingly negative) eigenvalues of the Hessian. The main result of this paper roughly says that if a sequence $$\{x_n\}$$ { x n } constructed by the method from a random initial point $$x_0$$ x 0 converges, then the limit point is a critical point and not a saddle point, and the convergence rate is the same as that of Newton’s method. A subsequent work has recently been successful incorporating Backtracking line search to New Q-Newton’s method, thus resolving the global convergence issue observed for some (non-smooth) functions. An application to quickly find zeros of a univariate meromorphic function is discussed, accompanied with an illustration on basins of attraction.

Suggested Citation

  • Tuyen Trung Truong & Tat Dat To & Hang-Tuan Nguyen & Thu Hang Nguyen & Hoang Phuong Nguyen & Maged Helmy, 2023. "A Fast and Simple Modification of Newton’s Method Avoiding Saddle Points," Journal of Optimization Theory and Applications, Springer, vol. 199(2), pages 805-830, November.
    • Handle: RePEc:spr:joptap:v:199:y:2023:i:2:d:10.1007_s10957-023-02270-9
    DOI: 10.1007/s10957-023-02270-9
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