A new MM algorithm for root‐finding problems

• Published in: Statistica Neerlandica Vol. 78; no. 4; pp. 692 - 701
• Main Authors: Li, Xun‐Jian, Li, Shuang, Tian, Guo‐Liang
• Format: Journal Article
• Language: English
• Published: Oxford Blackwell Publishing Ltd 01.11.2024
• Subjects: Algorithms; Convergence; integral version of Taylor expansion; MM algorithm; Newton methods; Newton's method; Nonlinear equations; Optimization; Statistical analysis; univariate nonlinear equation
• ISSN: 0039-0402; 1467-9574
• DOI: 10.1111/stan.12345

The minorization–maximization (MM) algorithm is an optimization technique for iteratively calculating the maximizer of a concave target function rather than a root–finding tool. In this paper, we in the first time develop the MM algorithm as a new method for seeking the root x∗$$ {x}^{\ast } $$ of a univariate nonlinear equation g(x)=0$$ g(x)=0 $$. The key idea is to transfer the root–finding issue to iteratively calculate the maximizer of a concave target function by designing a new MM algorithm. According to the ascent property of the MM algorithm, we know that the proposed algorithm converges to the root x∗$$ {x}^{\ast } $$ and does not depend on any initial values, in contrast to Newton's method. Several statistical examples are provided to demonstrate the proposed algorithm.