| This dissertation proposes a reconstructed optimization model of the Scalable Probabilistic Approximation(SPA)that includes both oblique and linear manifold constraints,and optimizes the multiplicative manifold constraint matrix.This improved SPA model facilitates simultaneous data-driven optimal discretization,feature selection,and prediction,resulting in reliable discrete approximations for complex systems.The dissertation introduces various Riemannian first-order optimization methods based on the geometric properties of the multiplicative manifold to solve this reconstructed model.Chapter 2 of the dissertation presents the geometrical properties of the product manifold,along with explicit expressions of the Riemannian gradient and Riemannian Hessian for the objective function.In Chapter 3,the dissertation proposes a novel Riemannian nonlinear conjugate gradient method that combins the geometrical properties of the product manifold with the Zhang-Hager technique and a classical Armijo-type monotone line search to solve the reconstruction model.Additionally,a new non-monotone line search criterion is introduced to enhance the efficiency of the reconstruction model.Iterative initial values are selected using improved BB steps,resulting in trial steps with consistent upper and lower bounds leading to a complete iteration format and global convergence analysis.In Chapter 4,the Dai nonlinear conjugate gradient algorithm is extended to the Riemannian product manifold,and the proposed Riemannian nonlinear conjugate gradient method is based on the Dai non-monotone line search and is utilized for solving the reconstructed model.The method utilizes the geometric features of the product manifold and includes a global convergence analysis.Chapter 5 combines the geometric features of the product manifold with the variational Fletcher-Reeves conjugate gradient method on Euclidean space.This leads to the development of the Riemannian variational FR nonlinear conjugate gradient method,which is utilized for solving the reconstruction model.Additionally,the chapter includes an analysis of the global convergence of the algorithm.Finally,Chapter 6 presents numerical experiments and comparisons that verify the feasibility and efficiency of the proposed algorithms for solving the reconstructed model.The three Riemannian nonlinear conjugate gradient methods proposed in this study have unique advantages in terms of iteration efficiency.The non-monotone line search criterion proposed in Chapter 3 outperforms the two mainstream non-monotone line search techniques(Max-type and Zhang-Hager-type)in terms of iteration time,and the Riemannian nonlinear conjugate gradient method based on this criterion(Algorithm 1)outperforms other optimization algorithms in terms of iteration efficiency. |
