globalOptimization

Global optimization is the field of mathematical optimization focused on finding the best possible value of an objective function over a given domain, particularly when the function is nonconvex and may have many local optima. The term is sometimes encountered in code or informal references as globalOptimization. It contrasts with local optimization, which guarantees optimality only within a neighborhood of the starting point.

A typical problem formulation is to minimize or maximize f(x) subject to x belonging to a feasible set D, where f can be continuous, differentiable, or a black-box function. Global optimization seeks the global optimum, not just a local one, and is frequently challenged by nonconvex landscapes, expensive evaluations, and complex constraints. Problems may be unconstrained, bound-constrained, or involve equalities and inequalities.

Methods for global optimization fall into several categories. Deterministic approaches, such as branch-and-bound, interval analysis, and reformulations, aim to prove global optimality or provide guaranteed bounds. Stochastic and heuristic methods—such as simulated annealing, genetic algorithms, differential evolution, and particle swarm optimization—explore the search space without guarantees but often perform well on large, complex problems. Bayesian optimization and surrogate-based methods build cheaper approximations of f to guide search, especially when evaluations are costly. Hybrid methods combine global search with local refinement.

Applications span engineering design, energy systems, finance, machine learning hyperparameter tuning, chemistry, and materials science. Common challenges include handling noisy or expensive evaluations, high-dimensional spaces, and multi-objective formulations where trade-offs between simultaneous goals must be managed.