We consider rather a general class of multi-level optimization problems, where a convex objective function is to be minimized subject to constraints of optimality of nested convex optimization problems. As a special case, we consider a trilevel optimization problem, where the objective of the two lower layers consists of a sum of a smooth and a non-smooth term.~Based on fixed-point theory and related arguments, we present a natural first-order algorithm and analyze its convergence and rates of convergence in several regimes of parameters.

翻译：本文考虑一类相当一般的多层优化问题，其中在嵌套凸优化问题的最优性约束下，需最小化一个凸目标函数。作为一个特例，我们研究了一个三层优化问题，其中两个较低层的目标函数由光滑项与非光滑项之和构成。基于不动点理论及相关论证，我们提出了一种自然的一阶算法，并在若干参数设置下分析了其收敛性及收敛速率。