{"ID":2856613,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.11990","arxiv_id":"2510.11990","title":"Linear Convergence of a Unified Primal--Dual Algorithm for Convex--Concave Saddle Point Problems with Quadratic Growth","abstract":"In this paper, we study saddle point (SP) problems, focusing on convex-concave optimization involving functions that satisfy either two-sided quadratic functional growth (QFG) or two-sided quadratic gradient growth (QGG)--novel conditions tailored specifically for SP problems as extensions of quadratic growth conditions in minimization. These conditions relax the traditional requirement of strong convexity-strong concavity, thereby encompassing a broader class of problems. We propose a generalized accelerated primal-dual (GAPD) algorithm to solve SP problems with non-bilinear objective functions, unifying and extending existing methods. We prove that our method achieves a linear convergence rate under these relaxed conditions. Additionally, we provide examples of structured SP problems that satisfy either two-sided QFG or QGG, demonstrating the practical applicability and relevance of our approach.","short_abstract":"In this paper, we study saddle point (SP) problems, focusing on convex-concave optimization involving functions that satisfy either two-sided quadratic functional growth (QFG) or two-sided quadratic gradient growth (QGG)--novel conditions tailored specifically for SP problems as extensions of quadratic growth condition...","url_abs":"https://arxiv.org/abs/2510.11990","url_pdf":"https://arxiv.org/pdf/2510.11990v1","authors":"[\"Cody Melcher\",\"Afrooz Jalilzadeh\",\"Erfan Yazdandoost Hamedani\"]","published":"2025-10-13T22:31:20Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[]","has_code":false}