Solving Extremum and Extremum Point Problems for Convex Functions

In optimization design, solving extremum and extremum point problems for convex functions represents a core computational challenge. Newton's method and quadratic interpolation are two widely adopted efficient algorithms for this purpose.

Newton's method is an iterative optimization technique based on second-order Taylor expansion. It leverages second-derivative information (the Hessian matrix) at the current point to iteratively approach the extremum point. The algorithm's strength lies in its quadratic convergence rate, particularly suitable for smooth functions where second derivatives are obtainable. Implementation typically involves calculating the Hessian matrix and solving the Newton system equation: x_{k+1} = x_k - H^{-1}(x_k)∇f(x_k). However, this method requires Hessian matrix computation, which can be computationally expensive for high-dimensional problems. Additionally, improper initial point selection may lead to convergence failures in certain scenarios.

Quadratic interpolation employs a function approximation approach through point selection and quadratic polynomial construction. The algorithm selects multiple points to build a quadratic model approximating the objective function, then minimizes this polynomial to estimate extremum points. Key advantages include avoiding second-derivative calculations and reduced computational overhead, making it suitable for functions with complex or unavailable derivatives. Code implementation typically involves three-point pattern selection and parabolic fitting. The trade-off is slower convergence compared to Newton's method and higher sensitivity to initial point selection.

In practical applications, Newton's method excels for smooth convex functions where higher-order derivatives are easily computable. Quadratic interpolation proves advantageous when computational resources are limited or derivative calculations are impractical. The choice between these methods requires balancing computational efficiency and convergence performance based on specific problem characteristics.