A simplified single-parameter filled function method for unconstrained global optimization

Zengfu Chao; Zengfu Chao

doi:10.3934/math.20251076

AIMS Mathematics

2025, Volume 10, Issue 10: 24270-24293. doi: 10.3934/math.20251076

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Research article

A simplified single-parameter filled function method for unconstrained global optimization

Zengfu Chao ^,

Faculty of Mathematical and Physical Sciences, Yibin University, Yibin 644007, China

Received: 23 August 2025 Revised: 13 October 2025 Accepted: 16 October 2025 Published: 23 October 2025
MSC : 90C26, 90C30

To address the limitations of existing filled function methods—including complex multi-parameter tuning and ambiguous global-optimality verification—this paper proposes a simplified continuously differentiable filled function for unconstrained global optimization with only one interpretable parameter and no exponential or logarithmic terms. Under mild assumptions (continuous differentiability and coercivity of the objective function), we rigorously proved that the proposed function satisfies all essential axioms of a filled function, enabling explicit certification of global optimality. The resulting hybrid algorithm (SP-FFM) combines gradient-based local optimization with deterministic global search via grid sampling, requiring only a single grid-density parameter. This design eliminates the need for alternating between the original objective and auxiliary functions around local optima, significantly reducing computational effort and parameter-tuning complexity. Extensive numerical experiments on benchmark problems show that the algorithm converges to global minima within milliseconds, achieves a 100% success rate after grid refinement, and outperforms state-of-the-art methods in robustness and efficiency while maintaining insensitivity to initial points.
- global optimization,
- filled function,
- single-parameter,
- hybrid optimization,
- deterministic search,
- benchmark testing
Citation: Zengfu Chao. A simplified single-parameter filled function method for unconstrained global optimization[J]. AIMS Mathematics, 2025, 10(10): 24270-24293. doi: 10.3934/math.20251076

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Abstract

To address the limitations of existing filled function methods—including complex multi-parameter tuning and ambiguous global-optimality verification—this paper proposes a simplified continuously differentiable filled function for unconstrained global optimization with only one interpretable parameter and no exponential or logarithmic terms. Under mild assumptions (continuous differentiability and coercivity of the objective function), we rigorously proved that the proposed function satisfies all essential axioms of a filled function, enabling explicit certification of global optimality. The resulting hybrid algorithm (SP-FFM) combines gradient-based local optimization with deterministic global search via grid sampling, requiring only a single grid-density parameter. This design eliminates the need for alternating between the original objective and auxiliary functions around local optima, significantly reducing computational effort and parameter-tuning complexity. Extensive numerical experiments on benchmark problems show that the algorithm converges to global minima within milliseconds, achieves a 100% success rate after grid refinement, and outperforms state-of-the-art methods in robustness and efficiency while maintaining insensitivity to initial points.

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