A Correspondence-Driven Approach for Bilevel Decision-making with Nonconvex Lower-Level Problems

We consider bilevel optimization problems with general nonconvex lower-level objectives and show that the classical hyperfunction-based formulation is unsettled, since the global minimizer of the lower-level problem is generally unattainable. To address this issue, we propose a correspondence-driven hyperfunction φcd. In this formulation, the follower is modeled not as a rational agent always attaining a global minimizer, but as an algorithm-based bounded rational agent whose decisions are produced by a fixed algorithm with initialization and step size. Since φcd is generally discontinuous, we apply Gaussian smoothing to obtain a smooth approximation φcd_xi, then show that its value and gradient converge to those of φcd. In the nonconvex setting, we identify that bifurcation phenomena, which arise when g(x,·) has a degenerate stationary point, pose a key challenge for hyperfunction-based methods. This is especially the case when φcd_xi is solved using gradient methods. To overcome this challenge, we analyze the geometric structure of the bifurcation set under some weak assumptions. Building on these results, we design a biased projected SGD-based algorithm SCiNBiO to solve φcd_xi with a cubic-regularized Newton lower-level solver. We also provide convergence guarantees and oracle complexity bounds for the upper level. Finally, we connect bifurcation theory from dynamical systems to the bilevel setting and define the notion of fold bifurcation points in this setting. Under the assumption that all degenerate stationary points are fold bifurcation points, we establish the oracle complexity of SCiNBiO for the lower-level problem.