We give a damped proximal augmented Lagrangian method (DPALM) for solving problems with a weakly-convex objective and convex linear/nonlinear constraints. Instead of taking a full stepsize, DPALM adopts a damped dual stepsize to ensure the boundedness of dual iterates. We show that DPALM can produce a (near) $\vareps$-KKT point within $O(\vareps^{-2})$ outer iterations if each DPALM subproblem is solved to a proper accuracy. In addition, we establish overall iteration complexity of DPALM when the objective is either a regularized smooth function or in a regularized compositional form. For the former case, DPALM achieves the complexity of $\widetilde{\mathcal{O}}\left(\varepsilon^{-2.5} \right)$ to produce an $\varepsilon$-KKT point by applying an accelerated proximal gradient (APG) method to each DPALM subproblem. For the latter case, the complexity of DPALM is $\widetilde{\mathcal{O}}\left(\varepsilon^{-3} \right)$ to produce a near $\varepsilon$-KKT point by using an APG to solve a Moreau-envelope smoothed version of each subproblem. Our outer iteration complexity and the overall complexity either generalize existing best ones from unconstrained or linear-constrained problems to convex-constrained ones, or improve over the best-known results on solving the same-structured problems. Furthermore, numerical experiments on linearly/quadratically constrained non-convex quadratic programs and linear-constrained robust nonlinear least squares are conducted to demonstrate the empirical efficiency of the proposed DPALM over several state-of-the art methods.

翻译：本文提出一种阻尼近端增广拉格朗日方法（DPALM），用于求解具有弱凸目标函数和凸线性/非线性约束的问题。DPALM不采用全步长，而是使用阻尼对偶步长来保证对偶迭代的有界性。我们证明，若每个DPALM子问题以适当精度求解，则该方法可在$O(\vareps^{-2})$次外迭代内生成（近似）$\vareps$-KKT点。此外，我们建立了DPALM在目标函数为正则化光滑函数或正则化复合形式时的整体迭代复杂度。对于前者，通过将加速近端梯度（APG）方法应用于每个DPALM子问题，DPALM以$\widetilde{\mathcal{O}}\left(\varepsilon^{-2.5} \right)$的复杂度生成$\varepsilon$-KKT点。对于后者，通过使用APG求解每个子问题的莫罗包络平滑版本，DPALM以$\widetilde{\mathcal{O}}\left(\varepsilon^{-3} \right)$的复杂度生成近似$\varepsilon$-KKT点。我们的外迭代复杂度与整体复杂度要么将无约束或线性约束问题的最优结果推广至凸约束问题，要么改进了同类结构问题的最佳已知结果。此外，通过对线性/二次约束非凸二次规划以及线性约束鲁棒非线性最小二乘问题的数值实验，验证了所提出的DPALM在实证效率上优于多种最先进方法。