Value-Function Root-Finding Algorithms for Composite Convex Simple Bilevel Optimization

Abstract

This paper considers composite convex simple bilevel optimization (SBO), which minimizes a composite convex function over the solution set of another composite convex minimization problem. We use a scalar value-function reformulation under which the bilevel optimal value is characterized as the leftmost root of a scalar equation. To evaluate this value function without level-set proximal access, we construct a first-order oracle based on Lagrangian duality. The resulting feasibility and value-error bounds, together with multiplier information, enable a bisection method and a safeguarded Newton-type method to obtain an $(\epsilon,\epsilon)$-solution. Under the stated assumptions and for fixed problem data and initialization tolerance, both methods achieve an operation complexity of $\widetilde{\mathcal{O}}(\epsilon^{-1/2})$, matching, up to logarithmic factors, the known near-optimal dependence for smooth convex SBO and the accelerated first-order rate for unconstrained composite convex optimization.