擬牛頓法在非線性最小平方、對稱非線性方程組和最近似相關矩陣問題的應用

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黃德國(Huynh Duc Quoc) 查詢紙本館藏

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數學系

論文名稱

擬牛頓法在非線性最小平方、對稱非線性方程組和最近似相關矩陣問題的應用
(A family of quasi-Newton methods for solving nonlinear least-squares, symmetric nonlinear equations, and the nearest correlation matrix problems)

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至系統瀏覽論文 (2026-7-31以後開放)

摘要(中)

本論文探討無約束最佳化問題及其特殊形式，如非線性最小平方問題和非線性對稱方程求解，這些問題在數值最佳化中扮演重要角色，在現實世界中有許多應用。

為了解決這些最佳化挑戰，存在多種方法。本論文研究了一組擬牛頓法，這是一類用於解決這些問題的迭代技術，並展示了它們相較於傳統最佳化方法如最速下降法和牛頓法的優勢。擬牛頓法僅利用梯度評估來逼近包含目標函數二階信息的Hessian矩陣。這種逼近顯著減少了計算成本和複雜性，使擬牛頓法對於計算精確Hessian矩陣不可行的大規模問題尤為吸引。

使用類割線的對角矩陣近似，擬牛頓法為各種優化問題提供了高效的解決方案，展示了它們在不同情境下的有效性。這些方法在適當條件下還具有全局收斂性。

摘要(英)

This thesis aims to develop an efficient solution algorithm for the unconstrained optimization problem and its special cases, including nonlinear least-squares and nonlinear equations, which hold substantial importance in numerical optimization with numerous practical applications. Additionally, by exploring their use in addressing the nearest correlation matrix problems through numerical experiments, this study establishes a foundation for further research and practical implementations in real-world applications.

Various methods are available to handle these computational challenges. We focus on a family of quasi-Newton methods, a class of iterative techniques for solving these problems. We demonstrate their advantages over traditional optimization methods, such as the steepest descent and Newton′s method. Quasi-Newton methods only use gradient evaluations to approximate the Hessian matrix, which encodes second-order information about the objective function. This approximation significantly reduces computational cost and complexity, making quasi-Newton methods particularly appealing for large-sized problems where calculating the exact Hessian is impractical or impossible.

Using secant-like diagonal matrix approximations, quasi-Newton methods provide efficient solutions for various optimization problems, demonstrating their effectiveness across diverse scenarios. These methods also exhibit global convergence properties under suitable conditions.

關鍵字(中)

★ 非線
★ 問題
★ 方法
★ 牛頓法
★ 角色

關鍵字(英)

★ Newton method
★ nonlinear
★ least squares
★ matrix
★ correlation

論文目次

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction of the problems . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Contribution and organization of the thesis . . . . . . . . . . . . . . . . . 3
2 Review of unconstrained optimization and algorithms . . . . . . . . . . . . 6
2.1 Definition of unconstrained optimization problem . . . . . . . . . . . . . 6
2.2 Gradient, Hessian, and Jacobian . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Taylor’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4 Conditions for optimality . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.5 Algorithms for unconstrained optimization problem: Linesearch-type
methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.5.1 Search direction choices . . . . . . . . . . . . . . . . . . . . . . 10
2.5.2 Stepsize selections . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.5.3 Armijo-backtracking linesearch . . . . . . . . . . . . . . . . . . 12
2.5.4 Convergence of linesearch-type methods . . . . . . . . . . . . . 12
2.5.5 Rate of convergence . . . . . . . . . . . . . . . . . . . . . . . . 14
3 A new structured quasi-Newton method for nonlinear least squares problems 16
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2 A SQN method-based SLDA strategy . . . . . . . . . . . . . . . . . . . 18
3.3 Convergence analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.4 Numerical results and discussions . . . . . . . . . . . . . . . . . . . . . 25
3.4.1 Numerical experimental setup . . . . . . . . . . . . . . . . . . . 25
3.4.2 Comparison metric: Performance profiles . . . . . . . . . . . . . 27
3.4.3 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4 Nonmonotone quasi-Newton method for symmetric nonlinear equations . . 33
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2 A QN-SDAJ strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.3 Convergence analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.4 Numerical results and discussions . . . . . . . . . . . . . . . . . . . . . 44
4.4.1 Baseline methods . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.4.2 Test problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.4.3 Numerical experimental setup . . . . . . . . . . . . . . . . . . . 47
4.4.4 Nonmonotone vs. monotone linesearch for QN-SDAJ . . . . . . 48
4.4.5 Performance comparison of four methods . . . . . . . . . . . . . 48
5 A new quasi-Newton acceleration for the nearest correlation matrix problems 52
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.2 A review of dual problem-based approaches . . . . . . . . . . . . . . . . 55
5.2.1 Accelerated gradient descent method . . . . . . . . . . . . . . . 55
5.2.2 Inexact Jacobian-free semismooth inexact Newton methods . . . 56
5.2.3 A QN-SDAJ strategy finding the critical points for the nearest
correlation matrix problems . . . . . . . . . . . . . . . . . . . . 59
5.3 Proposed methods addressing dual problem: A combination of QN-SDAJ
and AGD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.4 Numerical experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.4.1 Solving the primal nearest correlation matrix problem by alternat-
ing projection method . . . . . . . . . . . . . . . . . . . . . . . 62
5.4.2 Numerical experimental setup . . . . . . . . . . . . . . . . . . . 63
5.4.3 Comparison of the primal and dual approaches in terms of the
norm of errors . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.4.4 AGD-SDAJ and the other dual-approach solvers . . . . . . . . . 64
5.4.5 Overall performance . . . . . . . . . . . . . . . . . . . . . . . . 65
6 Conclusions and future work . . . . . . . . . . . . . . . . . . . . . . . . . . 69

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指導教授

黃楓南(Hwang Feng Nan)

審核日期

2024-7-23

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