Mathematical Programming Computation最新文献

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On O ( n ) algorithms for projection onto the top- k -sum sublevel set. 关于O (n)算法在top- k和子水平集上的投影。

IF 3.6 1区数学 Q1 COMPUTER SCIENCE, SOFTWARE ENGINEERING

Mathematical Programming Computation

Pub Date : 2025-06-01 Epub Date: 2025-01-08 DOI: 10.1007/s12532-024-00273-9

Jake Roth, Ying Cui

The top-k-sum operator computes the sum of the largest $k$ components of a given vector. The Euclidean projection onto the top- $k$ -sum sublevel set serves as a crucial subroutine in iterative methods to solve composite superquantile optimization problems. In this paper, we introduce a solver that implements two finite-termination algorithms to compute this projection. Both algorithms have $O (n)$ complexity of floating point operations when applied to a sorted $n$ -dimensional input vector, where the absorbed constant is independent of $k$ . This stands in contrast to an existing grid-search-inspired method that has $O (k (n - k))$ complexity, a partition-based method with $O (n + D log D)$ complexity, where $D \leq n$ is the number of distinct elements in the input vector, and a semismooth Newton method with a finite termination property but unspecified floating point complexity. The improvement of our methods over the first method is significant when $k$ is linearly dependent on $n$ , which is frequently encountered in practical superquantile optimization applications. In instances where the input vector is unsorted, an additional cost is incurred to (partially) sort the vector, whereas a full sort of the input vector seems unavoidable for the other two methods. To reduce this cost, we further derive a rigorous procedure that leverages approximate sorting to compute the projection, which is particularly useful when solving a sequence of similar projection problems. Numerical results show that our methods solve problems of scale $n = 10^{7}$ and $k = 10^{4}$ within 0.05 s, whereas the most competitive alternative, the semismooth Newton-based method, takes about 1 s. The existing grid-search method and Gurobi's QP solver can take from minutes to hours.

top-k-sum算子计算给定向量的最大k个分量的和。在求解复合超分位数优化问题的迭代方法中，顶k和子水平集上的欧几里得投影是一个至关重要的子程序。在本文中，我们引入了一个求解器，该求解器实现了两种有限终止算法来计算该投影。当应用于排序的n维输入向量时，这两种算法的浮点运算复杂度都为O (n)，其中吸收常数与k无关。这与现有的网格搜索启发的方法形成对比，该方法具有O （k (n- k)）复杂度，基于分区的方法具有O （n + D log D）复杂度，其中D≤n是输入向量中不同元素的数量，以及具有有限终止性质但未指定浮点复杂度的半光滑牛顿方法。当k线性依赖于n时，我们的方法对第一种方法的改进是显著的，这在实际的超分位数优化应用中经常遇到。在输入向量未排序的情况下，对向量进行（部分）排序会产生额外的开销，而对于其他两种方法来说，对输入向量进行完整排序似乎是不可避免的。为了降低这个成本，我们进一步推导了一个严格的程序，利用近似排序来计算投影，这在解决一系列类似的投影问题时特别有用。数值结果表明，我们的方法在0.05 s内解决了尺度n = 10.7和k = 10.4的问题，而最具竞争力的替代方法，基于半光滑牛顿的方法，大约需要1 s。现有的网格搜索方法和Gurobi的QP求解器可能需要几分钟到几个小时。

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