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2025, Volume 10, Issue 4: 10163-10186. doi: 10.3934/math.2025463
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Research article

The steepest descent method for fuzzy optimization problems under granular differentiability

  • School of Science, Lanzhou University of Technology, Lanzhou 730050, China
  • Received: 26 January 2025 Revised: 17 April 2025 Accepted: 24 April 2025 Published: 28 April 2025

  • MSC : 03E72

  • This paper aimed to establish optimality conditions and develop an efficient solution method for unconstrained fuzzy optimization problems using the steepest descent method, based on the granular differentiability and granular convexity of fuzzy mappings. First, building upon the gr-difference for fuzzy numbers, the granular gradient, gr-differentiable and twice gr-differentiable of fuzzy mappings were researched. Furthermore, the efficient solution of fuzzy optimization was introduced under the granular convexity of fuzzy mappings. Finally, we proposed a steepest descent method for fuzzy optimization problems by employing the characterization function of fuzzy mappings. Our analysis demonstrated that the proposed method converged linearly under standard convexity conditions.

    Citation: Shexiang Hai, Liang He. The steepest descent method for fuzzy optimization problems under granular differentiability[J]. AIMS Mathematics, 2025, 10(4): 10163-10186. doi: 10.3934/math.2025463

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  • This paper aimed to establish optimality conditions and develop an efficient solution method for unconstrained fuzzy optimization problems using the steepest descent method, based on the granular differentiability and granular convexity of fuzzy mappings. First, building upon the gr-difference for fuzzy numbers, the granular gradient, gr-differentiable and twice gr-differentiable of fuzzy mappings were researched. Furthermore, the efficient solution of fuzzy optimization was introduced under the granular convexity of fuzzy mappings. Finally, we proposed a steepest descent method for fuzzy optimization problems by employing the characterization function of fuzzy mappings. Our analysis demonstrated that the proposed method converged linearly under standard convexity conditions.





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