applied mathematics Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical ...

and computer science, a local optimum of an

optimization problem In mathematics, computer science and economics, an optimization problem is the problem of finding the ''best'' solution from all feasible solutions. Optimization problems can be divided into two categories, depending on whether the variable ...

is a solution that is optimal (either maximal or minimal) within a neighboring set of candidate solutions. This is in contrast to a

global optimum In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given r ...

, which is the optimal solution among all possible solutions, not just those in a particular neighborhood of values.

Continuous domain

When the function to be optimized is

continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...

, it may be possible to employ

calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...

to find local optima. If the

first derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...

exists everywhere, it can be equated to zero; if the function has an unbounded

domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function ** Domain of holomorphy of a function * ...

, for a point to be a local optimum it is necessary that it satisfy this equation. Then the

second derivative test In calculus, a derivative test uses the derivatives of a function to locate the critical points of a function and determine whether each point is a local maximum, a local minimum, or a saddle point. Derivative tests can also give information abou ...

provides a

sufficient In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...

condition for the point to be a local maximum or local minimum.

Search techniques

Local search or

hill climbing numerical analysis, hill climbing is a mathematical optimization technique which belongs to the family of local search. It is an iterative algorithm that starts with an arbitrary solution to a problem, then attempts to find a better solution ...

methods for solving optimization problems start from an initial configuration and repeatedly move to an ''improving neighboring configuration''. A trajectory in search space is generated, which maps an initial point to a local optimum, where local search is stuck (no improving neighbors are available). The search space is therefore subdivided into basins of attraction, each consisting of all initial points which have a given local optimum as the final point of the local search trajectory. A local optimum can be isolated (surrounded by non-locally-optimal points) or part of a

plateau In geology and physical geography, a plateau (; ; ), also called a high plain or a tableland, is an area of a highland consisting of flat terrain that is raised sharply above the surrounding area on at least one side. Often one or more sides ha ...

, a locally optimal region with more than one point of equal value. If the problem to be solved has all locally optimal points with the same value of the function to be optimized, local search effectively solves the global problem: finding a local optimum delivers a globally optimal solution. The locality of the optimum is dependent on the neighborhood structure as defined by the local search method that is used for optimizing the function. In many cases, local optima deliver sub-optimal solutions to the global problem, and a local search method needs to be modified to continue the search beyond local optimality; see for example

iterated local search Iterated Local Search (ILS) is a term in applied mathematics and computer science defining a modification of local search or hill climbing methods for solving discrete optimization problems. Local search methods can get stuck in a local minimum, ...

tabu search Tabu search is a metaheuristic search method employing local search methods used for mathematical optimization. It was created by Fred W. Glover in 1986 and formalized in 1989. Local (neighborhood) searches take a potential solution to a prob ...

, reactive search optimization, and

simulated annealing Simulated annealing (SA) is a probabilistic technique for approximating the global optimum of a given function. Specifically, it is a metaheuristic to approximate global optimization in a large search space for an optimization problem. It ...

Continuous domain

Search techniques

See also