  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 $\mathcal{O} \left({1}/k^{0.5-\epsilon}\right)$ in terms of
the lower level objective function for an arbitrary small $\epsilon>0$. We
present the performance of the algorithm on a binary text classification
problem.
