Robust Neural Differential Solution for Time-Dependent Nonlinear Optimization With Noises Rejection

Model uncertainty, exogenous disturbance, and time-dependent computing are significant issues in constrained nonlinear optimization and related practical problems. This paper highlights two limitations of current methods for time-dependent nonlinear optimization, i.e., disturbance handling and restrictions on the objective function and constraints. First, a novel robust neural differential (RND) solution is proposed for solving equality-and inequality-constrained time-dependent nonlinear optimization (EIC-TDNO) problems involving the suppression of nonspecific disturbances. Second, the dynamic vector-valued error equation equivalent to the EIC-TDNO method is derived. Theoretical proofs ensure that the proposed RND solution converges to zero at an exponential rate, eliminates polynomial noise, and effectively reduces the time-varying disturbance. Finally, through simulation and applications to 2-D filter design and robot motion planning, the RND solution is verified to exhibit rapid convergence, high precision, strong robustness, and versatility in comparison with the existing models.
Note to Practitioners
—The primary impetus behind this paper is the disadvantage of current optimization methods when solving real-world application problems. First, most of the existing optimization methods consider time-invariant objective functions and constraints. Additionally, the latest time-varying optimization methods limit their application to unconstrained or equality-constrained situations and are capable of handling disturbances of only specific types and within specific numerical ranges. Note that the interplay between modules may be introduced if other components, such as filters, are used to handle disturbances. Therefore, in this paper, an optimization approach is intrinsically designed for time-dependent optimization with its own ability to stabilize the whole system while being perturbed by various types of time-varying disturbances, which could reduce large lagged errors and system complexity. Theoretical analysis provides performance guarantees as well as guidance on parameter tuning to ensure solution accuracy and robustness. This paper primarily investigates and verifies how the proposed robust neural differential solution solves constrained time-dependent nonlinear optimization problems with the presence of different time-varying disturbances; this approach can be applied to numerous practical engineering problems, such as robot control, energy dispatch, and image processing.