Numerical Optimization Algorithms: From Classical Methods to Heuristic Approaches

Numerical optimization is a crucial field in mathematics and computer science that focuses on developing algorithms to find minimum or maximum values of functions. Different optimization algorithms suit various problem scenarios, ranging from classical first-order and second-order methods to heuristic approaches, each with distinct advantages and limitations.

Newton's Method Newton's method is a second-order optimization algorithm that utilizes second derivative information (Hessian matrix) for iterative optimization, achieving rapid convergence. The core concept involves approximating the objective function via Taylor expansion at the current point, then updating the iteration point by solving the optimal solution of the quadratic approximation. In implementation, developers typically use automatic differentiation or numerical differentiation to compute the Hessian matrix. Newton's method requires the objective function to be twice differentiable with a positive definite Hessian matrix, but becomes computationally expensive for high-dimensional problems due to Hessian matrix calculation and inversion.

Quasi-Newton Methods Quasi-Newton methods improve upon Newton's method by avoiding direct Hessian matrix computation. These methods progressively approximate the inverse Hessian matrix using algorithms like BFGS and L-BFGS. In code implementation, BFGS maintains an approximation of the inverse Hessian through rank-two updates, while L-BFGS uses limited memory storage for large-scale problems. These approaches maintain fast convergence rates while reducing computational complexity, making them particularly suitable for large-scale optimization problems.

Genetic Algorithms Genetic algorithms represent a heuristic optimization approach that simulates natural selection and genetic mechanisms from biological evolution. Through operations like selection, crossover, and mutation, they perform global search in the solution space, making them suitable for non-convex, non-differentiable, or discrete optimization problems. In practical implementation, genetic algorithms require careful parameter tuning including population size, mutation rate, and crossover probability. Although typically slower in convergence, they demonstrate strong robustness for complex, multi-modal optimization problems.

Additionally, other significant algorithms include gradient descent, conjugate gradient method, and particle swarm optimization (PSO), each suited for different problem characteristics such as convex optimization, non-convex optimization, and constrained optimization. Gradient descent implementations often involve learning rate scheduling and momentum terms, while PSO maintains particle positions and velocities for swarm intelligence. Selecting appropriate numerical optimization algorithms requires comprehensive consideration of problem scale, differentiability, computational efficiency, and other relevant factors.