发表机构
Center for Applied Mathematics of Fujian Province, School of Mathematics and Statistics, Fuzhou University; School of Mathematics and Statistics, Fuzhou University（福建省应用数学中心，福州大学数学与统计学院; 福州大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对带权等式约束的连续二次背包问题，提出动态阈值算法DTA及其无重置变体NDTA，证明了算法的有限终止性与复杂性，数值实验显示其在大规模变量下运行时间优于多种对比算法。
AI 中文摘要
Condat算法是一种高效的单纯形投影动态阈值方法，但它对带权等式约束的扩展以及重置、移除操作的算法作用分析有限。我们针对带权等式约束的连续二次背包问题提出了动态阈值算法（DTA），该算法通过添加、重置、移除三种操作维持阈值不变性，我们证明了其有限终止性与正确性。重置无法发生的充分条件启发了更简单的无重置变体NDTA。我们构造了算例，其中DTA运行时间为Θ(n)，而NDTA需要Θ(n²)时间，尽管两种算法的最坏情况复杂度均为二次。我们进一步证明，当权值比与每次移除过程的移除数量有界时，正阈值增量的线性次迭代要求输入值的最小非零间隙（经数据范围归一化）至多为exp[-Θ(n log n)]。变量数达10⁷的数值实验表明，DTA与NDTA实现了近似线性的经验缩放，在运行时间上优于Secant、WMVA、Variable Fixing、Newton、Median Search、Heap及Sort算法。
英文摘要
Condat's algorithm is an efficient dynamic-threshold method for projection onto the simplex, but its extension to weighted equality constraints and the algorithmic roles of resetting and removal have received limited analysis. We develop a dynamic-threshold algorithm (DTA) for a continuous quadratic knapsack problem with a weighted equality constraint. DTA maintains a threshold invariant through three operations--addition, reset, and removal--and we establish its finite termination and correctness. A sufficient condition under which reset cannot occur motivates a simpler no-reset variant, NDTA. We construct instances for which DTA runs in $Θ (n)$ time whereas NDTA requires $Θ(n^2)$ time, although both algorithms have quadratic worst-case complexity. We further show that, when the weight ratio and the number of deletions per removal pass are bounded, a linear number of passes with positive threshold increments requires the minimum nonzero gap between input values, normalized by the data range, to be at most $\exp [- Θ(n \log n)]$. Numerical experiments with up to 10^7 variables demonstrate that DTA and NDTA achieve approximately linear empirical scaling, and outperform Secant, WMVA, Variable Fixing, Newton, Median Search, Heap, and Sort in running time.
