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abstract: 'We study a class of bilevel convex optimization problems where the goal is to find the minimizer of [[an objective function]{}]{} in the upper level, among the set of all optimal solutions of [[an optimization]{}]{} problem in the lower level. A wide range of problems in convex optimization can be formulated [[using]{}]{} this class. [[An important example is the case where an optimization problem is ill-posed.]{}]{} [[In this paper, our interest lies in addressing the bilevel problems, where the lower level objective is given as a finite sum of separate nondifferentiable convex component functions. This is the case]{}]{} in a variety of applications in distributed optimization, such as large-scale data processing in machine learning and neural networks. [[To the best of our knowledge, this class of bilevel problems, with a finite sum in the lower level, has not been addressed before. Motivated by this gap, we]{}]{} develop an iterative regularized incremental subgradient method[[,]{}]{} [[where the agents update their iterates in a cyclic manner using a regularized subgradient]{}]{}. Under a suitable choice of [[the]{}]{} regularization [[parameter sequence]{}]{}, we establish the convergence of the proposed algorithm and [[derive a]{}]{} rate of $\mathcal{\cal O} \left({1}/k^{0.5-\epsilon}\right)$ [[in terms of]{}]{} the lower level 