Recent Advances in the Application of Generalized Lagrangian Methods for Second-Order Cone Optimization with Emphasis on Quadratic Competence

In recent decades, second-order cone optimization (SOCO) has garnered significant attention due to its wide-ranging applications in engineering, finance, machine learning, and other fields. One of the innovative approaches within this domain is the use of generalized Lagrangian methods, particularly the augmented Lagrangian method (ALM), which effectively simplifies and stabilizes optimization problems with complex constraints. This paper explores the recent advancements in these methods, emphasizing the importance of “quadratic competence” and its impact on convergence rates and solution accuracy in solving SOCO problems. The primary aim of this paper is to investigate novel methodologies that leverage second-order sufficiency conditions to reduce reliance on rigid assumptions, such as linear independence constraint qualifications or strict complementarity conditions, thereby guaranteeing local stability and convergence. Additionally, the research focuses on the detailed formulation of augmented Lagrangian methods and their use in solving complex constrained optimization problems, both in exact and approximate forms. Another key objective is to examine the performance and applicability of generalized Lagrangian methods in practical scenarios such as engineering design, machine learning models, and financial optimization tasks, all characterized by large-scale structures and sophisticated constraints. This study also highlights the role of second-order growth conditions in broadening the theoretical scope of SOCO while providing robustness under relaxed criteria. By doing so, these frameworks contribute to greater flexibility in handling complex constraints, even in cases where non-unique Lagrange multipliers exist. Furthermore, the paper elaborates on the incorporation of advanced differential analysis tools, which ensure precise model formulations while achieving linear and robust convergence guarantees for exact and inexact ALM formulations. The scalability of the methods is reviewed in depth, addressing their effectiveness for large-scale optimization problems where computational resources and accuracy are critical. Applications ranging from engineering and urban planning to machine learning and finance are studied to demonstrate the practical versatility of these methods. The results underscore the potential of generalized Lagrangian methods to handle real-world decision-making tasks more efficiently.

کلیدواژه ها:

Second-Order Cone Optimization ، Augmented Lagrangian Method ، Quadratic Competence ، Second-Order Sufficiency Conditions ، Convergence Analysis