In many applications, gradient evaluations are inherently approximate, motivating the development of optimization methods that remain reliable under inexact first-order information. A common strategy in this context is adaptive evaluation, whereby coarse gradients are used in early iterations and refined near a minimizer. This is particularly relevant in differential equation-constrained optimization (DECO), where discrete adjoint gradients depend on iterative solvers. Motivated by DECO applications, we propose an inexact general descent framework and establish its global convergence theory under two step-size regimes. For bounded step sizes, the analysis assumes that the error tolerance in the computed gradient is proportional to its norm, whereas for diminishing step sizes, the tolerance sequence is required to be summable. The framework is implemented through inexact gradient descent and an inexact BFGS-like method, whose performance is demonstrated on a second-order ODE inverse problem and a two-dimensional Laplace inverse problem using discrete adjoint gradients with adaptive accuracy. Across these examples, adaptive inexact gradients consistently reduced optimization time relative to fixed tight tolerances, while incorporating curvature information further improved overall efficiency.

翻译：在许多应用中，梯度计算本质上是近似的，这推动了在非精确一阶信息下仍保持可靠的优化方法的发展。在此背景下，一种常见策略是自适应评估，即在早期迭代中使用粗略梯度，在接近极小点时进行细化。这在微分方程约束优化中尤为重要，其中离散伴随梯度依赖于迭代求解器。受DECO应用启发，我们提出了一种非精确通用下降框架，并在两种步长机制下建立了其全局收敛理论。对于有界步长，分析假设计算梯度的误差容限与其范数成正比；而对于递减步长，则要求容限序列是可求和的。该框架通过非精确梯度下降法和一种非精确类BFGS方法实现，其性能通过使用自适应精度离散伴随梯度的二阶ODE反问题和二维拉普拉斯反问题得到验证。在这些算例中，相较于固定严格容限，自适应非精确梯度持续减少了优化时间，而引入曲率信息进一步提升了整体效率。