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

[2409.08948] Value-Function Root-Finding Algorithms for Composite Convex Simple Bilevel Optimization Skip to main content Search Submit Donate Log in Mathematics > Optimization and Control arXiv:2409.08948 (math) [Submitted on 13 Sep 2024 ( v1 ), last revised 22 Aug 2026 (this version, v3)] Title: Value-Function Root-Finding Algorithms for Composite Convex Simple Bilevel Optimization Authors: Rujun Jiang , Xu Shi , Weizheng Song , Jiulin Wang View PDF HTML (experimental) 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. Subjects: Optimization and Control (math.OC) Cite as: arXiv:2409.08948 [math.OC] (or arXiv:2409.08948v3 [math.OC] for this version) https://doi.org/10.48550/arXiv.2409.08948 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Xu Shi [ view email ] [v1] Fri, 13 Sep 2024 16:07:48 UTC (790 KB) [v2] Thu, 10 Jul 2025 13:12:25 UTC (149 KB) [v3] Sat, 22 Aug 2026 08:51:53 UTC (1,829 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.OC < prev | next > new | recent | 2024-09 Change to browse by: math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from (opens in new tab) (opens in new tab) (opens in new tab)

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