发表机构
School of Science, Xihua University（西华大学理学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对带非线性不等式约束的凸优化，提出首个Nesterov型原始-对偶乘子框架，构造带Nesterov型消失阻尼的连续时间动力学并导出不精确加速算法，证明其可达𝒪(t⁻²)及𝒪(k⁻²)的收敛速率。
AI 中文摘要
我们研究带非线性不等式约束的凸优化问题，提出一种在连续时间与离散时间中均保持一致性的原始-对偶乘子框架。首先，我们构造带有Nesterov型消失阻尼α/t的连续时间动力学，同时结合对偶变量与非线性约束映射的合适外推。在凸性假设及α≥3的条件下，我们证明非线性可行性与目标残差均达到𝒪(t⁻²)的收敛速率。随后，我们通过对该动力学的扰动版本进行相容离散化，得到一种不精确加速原始-对偶算法。对于复合凸目标，原始不精确性满足加权可和性条件时，可行性与目标残差可达到𝒪(k⁻²)的收敛速率，与连续时间对应项的加速速率相匹配。据我们所知，这是首个针对带非线性不等式约束的凸优化的Nesterov型原始-对偶乘子框架。
英文摘要
We consider convex optimization with nonlinear inequality constraints and develop a primal-dual multiplier framework that is consistent in continuous and discrete time. We first propose continuous-time dynamics with Nesterov-type vanishing damping $α/t$, together with compatible extrapolations of the dual variable and the nonlinear constraint mapping. Under convexity assumptions and $α\geq3$, we establish $\mathcal O(t^{-2})$ convergence rates for both nonlinear feasibility and the objective residual. In the noncritical regime $α>3$, with an admissible choice of the extrapolation parameter, we further prove that the entire primal-dual trajectory converges to a KKT pair and sharpen both continuous-time estimates to $o(t^{-2})$. We then derive an inexact accelerated primal-dual algorithm through a compatible discretization of a perturbed version of the dynamic. For composite convex objectives, a weighted summability condition on the primal inexactness yields $\mathcal O(k^{-2})$ rates for feasibility and the objective residual. In the corresponding noncritical regime, the discrete primal-dual sequence converges to a KKT pair and both residual estimates improve to $o(k^{-2})$. Thus the continuous and discrete results exhibit matching accelerated rates and matching asymptotic improvements. To the best of our knowledge, this is the first Nesterov-type primal-dual multiplier framework for convex optimization with nonlinear inequality constraints.
