In this paper, we focus on simple bilevel optimization problems, where we minimize a convex smooth objective function over the optimal solution set of another convex smooth constrained optimization problem. We present a novel bilevel optimization method that locally approximates the solution set of the lower-level problem using a cutting plane approach and employs an accelerated gradient-based update to reduce the upper-level objective function over the approximated solution set. We measure the performance of our method in terms of suboptimality and infeasibility errors and provide non-asymptotic convergence guarantees for both error criteria. Specifically, when the feasible set is compact, we show that our method requires at most $\mathcal{O}(\max\{1/\sqrt{\epsilon_{f}}, 1/\epsilon_g\})$ iterations to find a solution that is $\epsilon_f$-suboptimal and $\epsilon_g$-infeasible. Moreover, under the additional assumption that the lower-level objective satisfies the $r$-th H\"olderian error bound, we show that our method achieves an iteration complexity of $\mathcal{O}(\max\{\epsilon_{f}^{-\frac{2r-1}{2r}},\epsilon_{g}^{-\frac{2r-1}{2r}}\})$, which matches the optimal complexity of single-level convex constrained optimization when $r=1$.

翻译：本文关注一类简单双层优化问题，旨在最小化关于另一个凸光滑约束优化问题最优解集的凸光滑目标函数。我们提出了一种新型双层优化方法，该方法利用切割平面技术局部近似下层问题解集，并在该近似解集上采用加速梯度更新以降低上层目标函数。通过次优性和不可行性误差衡量方法性能，并针对两类误差准则给出非渐近收敛性保证。具体而言，当可行域紧致时，证明该方法最多需要 $\mathcal{O}(\max\{1/\sqrt{\epsilon_{f}}, 1/\epsilon_g\})$ 次迭代即可找到 $\epsilon_f$-次优且 $\epsilon_g$-不可行的解。进一步，若假设下层目标满足 $r$ 阶Hölderian误差界，则所提方法迭代复杂度为 $\mathcal{O}(\max\{\epsilon_{f}^{-\frac{2r-1}{2r}},\epsilon_{g}^{-\frac{2r-1}{2r}}\})$，当 $r=1$ 时该复杂度与单层凸约束优化问题的最优复杂度一致。