| 基于双曲正切函数的一类新的精确罚函数算法 |
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| 引用本文：唐加会.基于双曲正切函数的一类新的精确罚函数算法[J].上海第二工业大学(中文版),2026,43(2):227-231 |
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| 中文摘要:罚函数法是求解约束优化问题的一类重要方法, 其核心思想是将约束问题转化为一系列无约束问题进行求解。传统罚函数法(如二次罚函数(quadratic penalty, QP) 和精确罚函数) 存在“维数灾难”、不可微、罚因子需趋于无穷等问题。本文提出一类基于双曲正切(hyperbolic tangent, tanh) 函数的新型光滑精确罚函数。该罚函数在任意有限罚因子下均能保持光滑性, 且具有精确性, 即当罚因子足够大时, 无约束罚问题的局部极小点与原约束问题的局部极小点一致。本文建立了新罚函数的精确性定理并给出了严格证明。通过经典测试算例的数值实验, 将新算法与传统QP 法进行比较。结果表明, 新算法在求解精度、稳定性和收敛速度方面均表现出显著优势, 有效避免了因罚因子过大而引起的数值困难, 为求解约束优化问题提供了一种新的有效途径。 |
| 中文关键词:约束优化问题 精确罚函数 光滑化 双曲正切函数 局部最优解 |
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| A New Class of Exact Penalty Function Algorithm Based on Hyperbolic Tangent Function |
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| Abstract:The penalty function method is an important class of algorithms for solving constrained optimization problems, whose core idea is to transform the constrained problem into a series of unconstrained problems. Traditional penalty function methods (such as quadratic penalty (QP) and exact penalty methods) suffer from issues such as the “curse of dimensionality”, non-differentiability, and the requirement for a penalty parameter that tends to infinity. This paper proposes a new class of smooth exact penalty functions based on the hyperbolic tangent (tanh) function. The penalty function maintains smoothness for any finite penalty parameter and possesses exactness, meaning that when the penalty parameter is sufficiently large, the local minimizers of the unconstrained penalty problem coincide with those of the original constrained problem. The paper establishes an exactness theorem for the new penalty function and provides a rigorous proof. Numerical experiments on classical test problems are used to compare the new algorithm with the traditional QP method. The results demonstrate that the new algorithm exhibits significant advantages in terms of solution accuracy, stability, and convergence speed, effectively avoiding numerical difficulties caused by excessively large penalty parameters, thus providing a new and effective approach for solving constrained optimization problems. |
| keywords:constrained optimization problem exact penalty function smoothing hyperbolic tangent function local optimal solution |
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