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Discrete optimization is a branch of optimization in applied mathematics and computer science. As opposed to continuous optimization , some or all of the variables used in a discrete optimization problem are restricted to be discrete variables —that is, to assume only a discrete set of values, such as the integers .
Mathematical optimization (alternatively spelled optimisation) or mathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives. [ 1 ] [ 2 ] It is generally divided into two subfields: discrete optimization and continuous optimization .
A minimum spanning tree of a weighted planar graph.Finding a minimum spanning tree is a common problem involving combinatorial optimization. Combinatorial optimization is a subfield of mathematical optimization that consists of finding an optimal object from a finite set of objects, [1] where the set of feasible solutions is discrete or can be reduced to a discrete set.
Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuous functions).
An optimization problem with discrete variables is known as a discrete optimization, in which an object such as an integer, permutation or graph must be found from a countable set. A problem with continuous variables is known as a continuous optimization, in which an optimal value from a continuous function must be found.
In practice, this generally requires numerical techniques for some discrete approximation to the exact optimization relationship. Alternatively, the continuous process can be approximated by a discrete system, which leads to a following recurrence relation analog to the Hamilton–Jacobi–Bellman equation:
Optimization algorithms and methods (10 C, ... Pages in category "Mathematical optimization" ... Discrete optimization;
Optimal control is an extension of the calculus of variations, and is a mathematical optimization method for deriving control policies. [6] The method is largely due to the work of Lev Pontryagin and Richard Bellman in the 1950s, after contributions to calculus of variations by Edward J. McShane . [ 7 ]