optimisations

Optimisation, or optimization, is the process of making something as effective or functional as possible. In mathematics and computer science, optimization involves selecting input values, sometimes from a specified domain, to maximize or minimize an objective function, possibly subject to constraints. The result is an optimal solution, though in practice one often seeks good approximations due to complexity or uncertainty.

Problems may be continuous, discrete, or mixed. Continuous optimization deals with variables that can take any value in a range, as in linear programming, nonlinear programming, or convex optimization. Discrete optimization restricts variables to integers or categorical choices and includes combinatorial optimization, graph problems, and assignment problems. Mixed-integer problems combine both. Common frameworks include linear programming (LP), quadratic programming (QP), and more general nonlinear or stochastic formulations.

Solution methods fall into analytical, numerical, and heuristic categories. Analytical approaches use calculus and duality theory; numerical methods include gradient-based algorithms like gradient descent, Newton’s method, or quasi-Newton methods. Derivative-free methods, such as Nelder–Mead, are used when derivatives are unavailable. For discrete problems, exact methods include branch-and-bound, dynamic programming, and integer programming; heuristic and metaheuristic methods include genetic algorithms, simulated annealing, and swarm optimization.

In practice, optimisation also concerns performance engineering. Code and system optimisations aim to reduce time or memory usage, using profiling, data-structure choices, caching, parallelization, vectorization, and compiler optimisations. The field emphasizes the trade-offs among accuracy, robustness, and computational cost. Applications span scheduling, routing, resource allocation, finance, and machine learning, including hyperparameter tuning.