Nonconvex constrained optimization problems can be used to model a number of machine learning problems, such as multi-class Neyman-Pearson classification and constrained Markov decision processes. However, such kinds of problems are challenging because both the objective and constraints are possibly nonconvex, so it is difficult to balance the reduction of the loss value and reduction of constraint violation. Although there are a few methods that solve this class of problems, all of them are double-loop or triple-loop algorithms, and they require oracles to solve some subproblems up to certain accuracy by tuning multiple hyperparameters at each iteration. In this paper, we propose a novel gradient descent and perturbed ascent (GDPA) algorithm to solve a class of smooth nonconvex inequality constrained problems. The GDPA is a primal-dual algorithm, which only exploits the first-order information of both the objective and constraint functions to update the primal and dual variables in an alternating way. The key feature of the proposed algorithm is that it is a single-loop algorithm, where only two step-sizes need to be tuned. We show that under a mild regularity condition GDPA is able to find Karush-Kuhn-Tucker (KKT) points of nonconvex functional constrained problems with convergence rate guarantees. To the best of our knowledge, it is the first single-loop algorithm that can solve the general nonconvex smooth problems with nonconvex inequality constraints. Numerical results also showcase the superiority of GDPA compared with the best-known algorithms (in terms of both stationarity measure and feasibility of the obtained solutions).

翻译：非凸约束优化问题可用于建模多种机器学习问题，例如多类奈曼-皮尔逊分类和约束马尔可夫决策过程。然而，此类问题具有挑战性，因为目标函数和约束条件均可能为非凸函数，难以平衡损失值的降低与约束违反程度的减少。尽管已有若干方法能求解此类问题，但它们均为双循环或三循环算法，且需要在每次迭代时通过调节多个超参数来调用求解器，以将子问题求解至特定精度。本文提出一种新颖的梯度下降与扰动上升（GDPA）算法，用于求解一类光滑非凸不等式约束问题。GDPA是一种原始-对偶算法，仅利用目标函数与约束函数的一阶信息，以交替方式更新原始变量与对偶变量。该算法的关键特征在于其为单循环结构，仅需调节两个步长参数。我们证明在温和的正则性条件下，GDPA能够以可证明的收敛速率找到非凸函数约束问题的卡鲁什-库恩-塔克（KKT）点。据我们所知，这是首个能求解具有非凸不等式约束的通用光滑非凸问题的单循环算法。数值实验结果也表明，相较于当前最优算法（就所得解的平稳性度量与可行性而言），GDPA具有显著优越性。