Accelerated Gradient Descent With Quasi‐Newton Preconditioning for Nearest Correlation Matrix Problems

ABSTRACT The nearest correlation matrix (NCM) problem poses computational difficulties with critical implications in finance and actuarial science. Accurate correlation matrices are essential for effective risk management and investment strategies. The NCM problem is mathematically formulated as a constrained optimization problem, often solved using the alternating projection method (APM). However, the slow linear convergence of APM, requiring many iterations for convergence and complete eigenvalue decomposition at each iteration, makes it computationally expensive and impractical for large‐scale problems. Alternatively, as our target problem, the NCM problem can be reformulated as an unconstrained optimization formulation in its dual form, taking advantage of its convex structure. It allows for a focus on minimizing the dual objective function by solving for the zeros of its gradient to satisfy first‐order optimality conditions. Our proposed method belongs to the framework of the generalized linesearch iterative methods, incorporating two essential elements: search direction and stepsize determination. Additionally, we introduce two practical mechanisms, including an accelerated step and nonlinear preconditioning, to improve the robustness and efficiency of the method. We propose the quasi‐Newton method with diagonal Jacobian approximation (QN‐SDAJ) as a nonlinear preconditioner to accelerate gradient descent for the NCM dual problem, significantly reducing the number of iterations and computational time. The numerical results show that our proposed method outperforms existing alternatives, including accelerated gradient descent, semismooth inexact Newton, and APM, particularly for large‐scale problems, demonstrating superior efficiency.