Exact-Penalty Prox-Linear Methods for Bilevel Optimization with $\ell_1$ Lower-Level Gradient Penalty

Abstract: Bilevel optimization is a fundamental framework for hierarchical decision-making, but its solution is challenging due to the implicit and typically set-valued nature of the lower-level optimality condition. In this paper, we study bilevel optimization problems through an exact-penalty reformulation based on the $\ell_1$-norm of the lower-level gradient. Under suitable regularity assumptions, we show that this penalty defines a distance-bound function and yields an exact penalty property for sufficiently large penalty parameters. To solve the resulting nonsmooth penalized problem, we propose an exact-penalty prox-linear (EPPL) method and establish a stationarity-oriented convergence guarantee. We further specialize the method to the simple bilevel setting, where the subproblem admits an explicit dual reformulation as a box-constrained quadratic program. This structure leads to a dual spectral projected gradient method with closed-form primal recovery, for which convergence of the inner dual iterates is proved. Numerical experiments on a minimum-norm least-squares bilevel model show that the proposed method is effective in reducing both the lower-level and upper-level gaps to high accuracy. Compared with several existing methods, the proposed approach attains the best final solution accuracy on the tested instance.