Design and Convergence Analysis of Optimization Algorithms Constrained by Nonlinear Partial Differential Equations

Optimization problems constrained by nonlinear partial differential equations (PDEs) are fundamentally critical to advanced engineering applications, including aerodynamic shape design and steady-state thermal control systems. These intricate tasks require minimizing specific cost functionals while strictly adhering to complex nonlinear physical laws. Existing first-order adjoint methods frequently exhibit sub-linear convergence rates when navigating highly ill-conditioned optimization landscapes. Conversely, standard second-order Newton-type methods often incur prohibitive computational costs due to the well-known curse of dimensionality inherent in full Hessian matrix construction. To overcome these substantial limitations, this paper proposes a novel Adaptive Second-order Adjoint-based (ASA) algorithm. The core methodological innovation lies in formulating a localized Hessian approximation, intricately coupled with an adaptive damping mechanism. This mechanism strategically utilizes first-order adjoint information to accurately capture manifold curvature without the overhead of exact second-order derivatives. Furthermore, an Augmented Lagrangian framework is rigorously employed to ensure robust constraint satisfaction throughout the optimization process. Comprehensive numerical experiments conducted on semilinear elliptic control problems demonstrate that the proposed ASA algorithm achieves a remarkable 1.75x computational speedup compared to the widely used L-BFGS method. The algorithm successfully reaches a stringent convergence tolerance of 10−7 in 38±3 iterations, yielding a mean relative error of 0.48%±0.03%, and maintains a flawless success rate of 100% under optimal penalty configurations. Global convergence to a stationary point is mathematically established under mild regularity assumptions. This research provides a highly scalable framework that effectively bridges the gap between low-cost gradient methods and high-precision second-order accuracy for large-scale industrial optimization.