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Optimization problems arise in all quantitative disciplines from computer science and engineering [3] to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries.
Optimization problems can be divided into two categories, depending on whether the variables are continuous or discrete: 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 .
Optimality modeling is the modeling aspect of optimization theory. It allows for the calculation and visualization of the costs and benefits that influence the outcome of a decision, and contributes to an understanding of adaptations. The approach based on optimality models in biology is sometimes called optimality theory. [1]
In terms of mathematical optimization, dynamic programming usually refers to simplifying a decision by breaking it down into a sequence of decision steps over time. This is done by defining a sequence of value functions V 1 , V 2 , ..., V n taking y as an argument representing the state of the system at times i from 1 to n .
Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements and objective are represented by linear relationships.
Optimization comes at a price and it is important to be sure that the investment is worthwhile. An automatic optimizer (or optimizing compiler, a program that performs code optimization) may itself have to be optimized, either to further improve the efficiency of its target programs or else speed up its own operation. A compilation performed ...
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 ]
Stochastic optimization (SO) are optimization methods that generate and use random variables. For stochastic optimization problems, the objective functions or constraints are random. Stochastic optimization also include methods with random iterates .