Based on the hyperbolic smoothing function and truncation strategy, this paper proposes a truncated hyperbolic smoothing Newton method for finite minimax problems with a large number of component functions. Gaps between the original and truncated hyperbolic smoothing functions are established in terms of the function values, gradients, and Hessian matrices, respectively, which in turn lead to a rule for selecting the truncation parameter. Based on this rule, the global convergence and the local quadratic convergence rate of the method are established. Numerical experiments demonstrate that, under the same accuracy requirement, the proposed method substantially reduces CPU time and is generally more efficient than both the exponential smoothing method and the untruncated smoothing method.
Citation: Wenzhe Zhao, Liping Tang. A truncated hyperbolic smoothing Newton method for finite minimax problems[J]. Journal of Industrial and Management Optimization, 2026, 22(9): 4547-4575. doi: 10.3934/jimo.2026158
Based on the hyperbolic smoothing function and truncation strategy, this paper proposes a truncated hyperbolic smoothing Newton method for finite minimax problems with a large number of component functions. Gaps between the original and truncated hyperbolic smoothing functions are established in terms of the function values, gradients, and Hessian matrices, respectively, which in turn lead to a rule for selecting the truncation parameter. Based on this rule, the global convergence and the local quadratic convergence rate of the method are established. Numerical experiments demonstrate that, under the same accuracy requirement, the proposed method substantially reduces CPU time and is generally more efficient than both the exponential smoothing method and the untruncated smoothing method.
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Wenzhe Zhao, Liping Tang. A truncated hyperbolic smoothing Newton method for finite minimax problems[J]. Journal of Industrial and Management Optimization, 2026, 22(9): 4547-4575. doi: 10.3934/jimo.2026158
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