We investigate frugal splitting operators for finite sum monotone inclusion problems. These operators utilize exactly one direct or resolvent evaluation of each operator of the sum, and the splitting operator's output is dictated by linear combinations of these evaluations' inputs and outputs. To facilitate analysis, we introduce a novel representation of frugal splitting operators via a generalized primal-dual resolvent. The representation is characterized by an index and four matrices, and we provide conditions on these that ensure equivalence between the classes of frugal splitting operators and generalized primal-dual resolvents. Our representation paves the way for new results regarding lifting numbers and the development of a unified convergence analysis for frugal splitting operator methods, contingent on the directly evaluated operators being cocoercive. The minimal lifting number is $n-1-f$ where $n$ is the number of monotone operators and $f$ is the number of direct evaluations in the splitting. Notably, this lifting number is achievable only if the first and last operator evaluations are resolvent evaluations. These results generalize the minimal lifting results by Ryu and Malitsky--Tam that consider frugal resolvent splittings. Building on our representation, we delineate a constructive method to design frugal splitting operators, exemplified in the design of a novel, convergent, and parallelizable frugal splitting operator with minimal lifting.

翻译：我们研究有限和单调包含问题的节俭分裂算子。此类算子利用求和过程中每个算子恰好一次直接求值或预解式求值，且分裂算子的输出由这些求值的输入与输出的线性组合决定。为便于分析，我们通过广义原始-对偶预解式引入节俭分裂算子的新型表示。该表示由一个指标和四个矩阵表征，我们给出确保节俭分裂算子类与广义原始-对偶预解式类之间等价性的条件。基于直接求值算子满足共强制性，该表示为提升数的新结果和节俭分裂算子方法的统一收敛性分析奠定基础。最小提升数为$n-1-f$，其中$n$为单调算子个数，$f$为分裂中的直接求值次数。值得注意的是，该提升数仅当第一个和最后一个算子求值为预解式求值时才可实现。这些结果推广了Ryu及Malitsky--Tam关于节俭预解式分裂的最小提升结论。基于该表示，我们提出设计节俭分裂算子的构造性方法，并通过设计一种新颖、收敛且可并行的具有最小提升的节俭分裂算子加以示例。