Local Second-Order Limit Dynamics of the Alternating Direction Method of Multipliers for Semidefinite Programming
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908861072408576 |
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| author | Kang, Shucheng Yang, Heng |
| author_facet | Kang, Shucheng Yang, Heng |
| contents | The alternating direction method of multipliers (ADMM) is widely used for solving large-scale semidefinite programs (SDPs), yet on instances with multiple primal-dual optimal solution pairs, it often enters prolonged slow-convergence regions where the Karush-Kuhn-Tucker (KKT) residuals nearly stall. To explain and predict the fine-grained dynamical behavior inside these regions, we develop a local second-order limit dynamics framework for ADMM near an arbitrary KKT point -- not necessarily the eventual limit point of the iterates. Assuming the existence of a strictly complementary primal-dual solution pair, we derive a second-order local expansion of the ADMM dynamics by leveraging a refined and simplified variational characterization of the (parabolic) second-order directional derivative of the PSD projection operator. This expansion reveals a closed convex cone of directions along which the local first-order update vanishes, and it induces a second-order limit map that governs the persistent drift after transient effects are filtered out. We characterize fundamental properties of this mapping, including its kernel, range, and continuity. A primal-dual decoupling further yields a clean scaling law for the effect of the penalty parameter in ADMM. We connect these properties to second-order dynamical features of ADMM, including fixed points, almost-invariant sets, and microscopic phases. Three empirical phenomena in slow-convergence regions are then explained or predicted: (i) angles between consecutive iterate differences are small yet nonzero, except for sparse spikes; (ii) primal and dual infeasibilities are insensitive to penalty-parameter updates; and (iii) iterates can be transiently trapped in a low-dimensional subspace for an extended period. Extensive numerical experiments on the Mittelmann dataset corroborate our theoretical predictions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_20103 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Local Second-Order Limit Dynamics of the Alternating Direction Method of Multipliers for Semidefinite Programming Kang, Shucheng Yang, Heng Optimization and Control The alternating direction method of multipliers (ADMM) is widely used for solving large-scale semidefinite programs (SDPs), yet on instances with multiple primal-dual optimal solution pairs, it often enters prolonged slow-convergence regions where the Karush-Kuhn-Tucker (KKT) residuals nearly stall. To explain and predict the fine-grained dynamical behavior inside these regions, we develop a local second-order limit dynamics framework for ADMM near an arbitrary KKT point -- not necessarily the eventual limit point of the iterates. Assuming the existence of a strictly complementary primal-dual solution pair, we derive a second-order local expansion of the ADMM dynamics by leveraging a refined and simplified variational characterization of the (parabolic) second-order directional derivative of the PSD projection operator. This expansion reveals a closed convex cone of directions along which the local first-order update vanishes, and it induces a second-order limit map that governs the persistent drift after transient effects are filtered out. We characterize fundamental properties of this mapping, including its kernel, range, and continuity. A primal-dual decoupling further yields a clean scaling law for the effect of the penalty parameter in ADMM. We connect these properties to second-order dynamical features of ADMM, including fixed points, almost-invariant sets, and microscopic phases. Three empirical phenomena in slow-convergence regions are then explained or predicted: (i) angles between consecutive iterate differences are small yet nonzero, except for sparse spikes; (ii) primal and dual infeasibilities are insensitive to penalty-parameter updates; and (iii) iterates can be transiently trapped in a low-dimensional subspace for an extended period. Extensive numerical experiments on the Mittelmann dataset corroborate our theoretical predictions. |
| title | Local Second-Order Limit Dynamics of the Alternating Direction Method of Multipliers for Semidefinite Programming |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2602.20103 |