On O(n) algorithms for projection onto the top-k-sum sublevel set

• Published in: Mathematical programming computation Vol. 17; no. 2; pp. 307 - 348
• Main Authors: Roth, Jake, Cui, Ying
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2025; Springer Nature B.V
• Subjects: Algorithms; Complexity; Floating point arithmetic; Full Length Paper; Iterative methods; Mathematics; Mathematics and Statistics; Mathematics of Computing; Methods; Operations Research/Decision Theory; Optimization; Search methods; Solvers; Theory of Computation; 90C20; 90-04; Top; 90C25; 90C33; 90-08; 90C06; Projection; sum; matrix; Superquantile; Z-matrix; Top-k-sum
• ISSN: 1867-2949; 1867-2957
• DOI: 10.1007/s12532-024-00273-9

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 ≤ 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.