<p>We introduce and study a safeguard strategy for the search directions of iterative methods in the vector setting. A general algorithm with safeguarded directions for solving vector optimization problems is proposed, and, with the (vector) standard Wolfe line search, its global convergence is presented. Based on it, focusing on nonlinear conjugate gradient (CG) method, we explore the conjugate parameters with built-in safeguard features and investigate the technique of redefinition related to the conjugate parameters, which inspire the creation of a novel redefined formula. Moreover, an efficient hybrid conjugate parameter is also suggested. These lead to the development of a nonlinear CG-type algorithm with safeguarded directions, which, in particular, falls into the aforementioned general algorithm. Under standard assumptions, we then establish its global convergence. Numerical experiments illustrate that the proposed algorithms are encouraging.</p>

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The convergence properties of nonlinear conjugate gradient methods with safeguarded directions for vector optimization problems

  • Qing-Rui He,
  • Sheng-Jie Li,
  • Chun-Rong Chen

摘要

We introduce and study a safeguard strategy for the search directions of iterative methods in the vector setting. A general algorithm with safeguarded directions for solving vector optimization problems is proposed, and, with the (vector) standard Wolfe line search, its global convergence is presented. Based on it, focusing on nonlinear conjugate gradient (CG) method, we explore the conjugate parameters with built-in safeguard features and investigate the technique of redefinition related to the conjugate parameters, which inspire the creation of a novel redefined formula. Moreover, an efficient hybrid conjugate parameter is also suggested. These lead to the development of a nonlinear CG-type algorithm with safeguarded directions, which, in particular, falls into the aforementioned general algorithm. Under standard assumptions, we then establish its global convergence. Numerical experiments illustrate that the proposed algorithms are encouraging.