Title: Convergence rates for an inexact linearized ADMM for nonsmooth nonconvex optimization with nonlinear equality constraints Show Chinese title

Title: 非光滑非凸优化问题中带非线性等式约束的不精确线性化ADMM的收敛速率

Abstract: In this paper, we consider nonconvex optimization problems with nonsmooth nonconvex objective function and nonlinear equality constraints. We assume that both the objective function and the functional constraints can be separated into 2 blocks. To solve this problem, we introduce a new inexact linearized alternating direction method of multipliers (ADMM) algorithm. Specifically, at each iteration, we linearize the smooth part of the objective function and the nonlinear part of the functional constraints within the augmented Lagrangian and add a dynamic quadratic regularization. We then compute the new iterate of the block associated with nonlinear constraints inexactly. This strategy yields subproblems that are easily solvable and their (inexact) solutions become the next iterates. Using Lyapunov arguments, we establish convergence guarantees for the iterates of our method toward an $\epsilon$-first-order solution within $\mathcal{O}(\epsilon^{-2})$ iterations. Moreover, we demonstrate that in cases where the problem data exhibit e.g., semi-algebraic properties or more general the KL condition, the entire sequence generated by our algorithm converges, and we provide convergence rates. To validate both the theory and the performance of our algorithm, we conduct numerical simulations for several nonlinear model predictive control and matrix factorization problems. Show Chinese abstract

Abstract: 本文考虑了具有非光滑非凸目标函数和非线性等式约束的非凸优化问题。我们假设目标函数和函数约束都可以分成两块。为了解决这个问题，我们引入了一种新的不精确线性化交替方向乘子法（ADMM）算法。具体来说，在每次迭代中，我们将目标函数的光滑部分和函数约束的非线性部分在线性增广拉格朗日中线性化，并添加一个动态二次正则化。然后，我们近似地计算与非线性约束相关的块的新迭代点。这一策略产生的子问题易于求解，其（不精确）解成为下一次迭代点。通过Lyapunov论证，我们证明了我们的方法的迭代点在$\mathcal{O}(\epsilon^{-2})$次迭代内收敛到一个$\epsilon$-一阶解。此外，我们还表明，在问题数据具有半代数性质或其他更一般的KL条件的情况下，由我们的算法生成的整个序列都收敛，并且提供了收敛率。为了验证理论和算法的性能，我们在几个非线性模型预测控制和矩阵分解问题上进行了数值模拟。