Global optimization is a branch of operations research, applied mathematics, and numerical analysis that attempts to find the global minima or maxima of a function or a set of functions on a given set.
Finding the global minimum of a function is far more difficult: analytical methods are frequently not applicable, and the use of numerical solution strategies often leads to very hard challenges.
Such procedures are popularly used to find integer solutions to mixed integer linear programming (MILP) problems, as well as to solve general, not necessarily differentiable convex optimization problems.
The use of cutting planes to solve MILP was introduced by Ralph E. Gomory and Václav Chvátal.
Branch and bound (BB or B&B) is an algorithm design paradigm for discrete and combinatorial optimization problems.
A branch-and-bound algorithm consists of a systematic enumeration of candidate solutions by means of state space search: the set of candidate solutions is thought of as forming a rooted tree with the full set at the root.
The algorithm explores branches of this tree, which represent subsets of the solution set.
Interval arithmetic helps find reliable and guaranteed solutions to equations and optimization problems.
It can be used in convex optimization Several exact or inexact Monte-Carlo-based algorithms exist: In this method, random simulations are used to find an approximate solution.
That is, all the facts (distances between each destination point) needed to determine the optimal path to follow are known with certainty and the goal is to run through the possible travel choices to come up with the one with the lowest total distance.
However, let's assume that instead of wanting to minimize the total distance traveled to visit each desired destination, we wanted to minimize the total time needed to reach each destination.
The replica exchange method was originally devised by Swendsen,[1] then extended by Geyer[2] and later developed, among others, by Giorgio Parisi.,[3][4] Sugita and Okamoto formulated a molecular dynamics version of parallel tempering:[5] this is usually known as replica-exchange molecular dynamics or REMD.
This results in a very robust ensemble which is able to sample both low and high energy configurations.
Other approaches include heuristic strategies to search the search space in a more or less intelligent way, including: Deterministic global optimization: For simulated annealing: For reactive search optimization: For stochastic methods: For parallel tempering: For continuation methods: For general considerations on the dimensionality of the domain of definition of the objective function: For strategies allowing one to compare deterministic and stochastic global optimization methods