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| Hlavní autoři: | , , , |
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| Médium: | Preprint |
| Vydáno: |
2024
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| Témata: | |
| On-line přístup: | https://arxiv.org/abs/2402.08097 |
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- 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 $\mathcal{O}(\max\{1/\sqrt{ε_{f}}, 1/ε_g\})$ iterations to find a solution that is $ε_f$-suboptimal and $ε_g$-infeasible. Moreover, under the additional assumption that the lower-level objective satisfies the $r$-th Hölderian error bound, we show that our method achieves an iteration complexity of $\mathcal{O}(\max\{ε_{f}^{-\frac{2r-1}{2r}},ε_{g}^{-\frac{2r-1}{2r}}\})$, which matches the optimal complexity of single-level convex constrained optimization when $r=1$.