Quasi-Newton and Steepest Descent Methods for Objective Function Modeling

Both Quasi-Newton and Steepest Descent methods are gradient-based optimization algorithms that iteratively search for the minimum point of an objective function. Compared to the Steepest Descent method, the Quasi-Newton method significantly improves convergence speed and achieves superlinear convergence by constructing an approximate model of the objective function. This approach cleverly avoids the complexity of directly computing second derivatives while leveraging gradient information, making it more efficient and practical than the classical Newton method in many real-world applications. In code implementations, this typically involves maintaining an approximation of the inverse Hessian matrix using update formulas like BFGS or DFP.

The core concept of Quasi-Newton methods lies in constructing a local quadratic model of the objective function using gradient variation information. By measuring gradient differences between consecutive iteration points, the algorithm progressively updates its estimate of the Hessian matrix or its inverse, thereby more accurately characterizing the curvature properties of the objective function. This dynamic adjustment strategy enables Quasi-Newton methods to better adapt to the objective function's morphology in different regions, effectively avoiding the zigzag convergence path that Steepest Descent methods may exhibit in valley terrains. Implementation-wise, this involves algorithms that update matrix approximations using rank-2 updates while ensuring positive definiteness through careful line search procedures.

Modern optimization software widely adopts Quasi-Newton methods to solve various optimization problems, including unconstrained optimization, constrained problems, and large-scale optimization tasks. Their excellent performance characteristics and relatively low computational overhead make them essential tools in engineering optimization and scientific computing. Particularly when handling high-dimensional problems, Quasi-Newton methods often demonstrate superior practical value compared to traditional Newton methods. Code libraries typically implement these methods with efficient linear algebra operations and memory-efficient limited-memory variants (L-BFGS) for large-scale applications.