梯度不确定下约束优化的任意时刻可行梯度下降
Anytime-Feasible Gradient Descent for Constrained Optimization Under Gradient Uncertainty
- Johns Hopkins University（约翰斯·霍普金斯大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对梯度不确定下的非线性约束优化，提出任意时刻可行的一阶方法，通过二阶锥规划和带保护回溯保证严格可行性与目标下降，并在多智能体导航中验证了噪声下的无碰撞性能。
AI中文摘要：
约束优化是许多工程系统的核心，在这些系统中，决策必须满足严格的安全和运行要求，尤其是在计算预算有限的实时环境中。在此类场景下，优化算法通常在完全收敛之前就被终止，这使得任意时刻可行性对于安全部署至关重要。现有的保证每次迭代可行性的方法通常依赖于精确的梯度信息，而由于测量噪声、随机逼近或模型失配，这一假设在实践中经常被违反。我们针对目标函数和约束梯度存在范数有界误差的非线性约束优化，开发了一种任意时刻可行的一阶方法。该方法通过求解二阶锥规划来计算鲁棒搜索方向，并通过带保护的回溯法选择步长。假设函数求值精确且初始点严格可行，该方法保持严格可行性，并保证在计算出的搜索方向非零时目标函数有足够的下降。我们建立了接受的步长的统一正下界，平均平方搜索方向范数的O(1/K)界，以及搜索方向收敛到零的性质。我们还证明了在严格可行点处零搜索方向可证明近似一阶平稳性。我们在杂乱环境中的多智能体导航任务上验证了所提出的方法，并表明尽管梯度信息有噪声，它仍能保持无碰撞轨迹。
英文摘要：
Constrained optimization is central to many engineering systems in which decisions must satisfy strict safety and operational requirements, especially in real-time settings with limited computational budgets. In such scenarios, optimization algorithms are often terminated before full convergence, making *anytime feasibility* essential for safe deployment. Existing methods that guarantee feasibility at every iterate typically rely on exact gradient information, an assumption that is often violated in practice due to measurement noise, stochastic approximations, or model mismatch. We develop an anytime-feasible first-order method for nonlinear constrained optimization under norm-bounded errors in the objective and constraint gradients. The method computes a robust search direction by solving a second-order cone program and selects a step size through safeguarded backtracking. Assuming exact function evaluations and a strictly feasible initialization, the method preserves strict feasibility and guarantees sufficient objective decrease whenever the computed search direction is nonzero. We establish a uniform positive lower bound on the accepted step sizes, an O(1/K) bound on the average squared search direction norm, and convergence of the search directions to zero. We also show that a zero search direction at a strictly feasible point certifies approximate first-order stationarity. We validate the proposed method on a multi-agent navigation task in cluttered environments and show that it maintains collision-free trajectories despite noisy gradient information.
