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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. Linear programming is a special case of mathematical programming (also known as mathematical optimization).
Given a transformation between input and output values, described by a mathematical function, optimization deals with generating and selecting the best solution from some set of available alternatives, by systematically choosing input values from within an allowed set, computing the output of the function and recording the best output values found during the process.
In the study of algorithms, an LP-type problem (also called a generalized linear program) is an optimization problem that shares certain properties with low-dimensional linear programs and that may be solved by similar algorithms.
Multi-objective linear programming is a subarea of mathematical optimization. A multiple objective linear program (MOLP) is a linear program with more than one objective function. An MOLP is a special case of a vector linear program .
Linear programming relaxation is a standard technique for designing approximation algorithms for hard optimization problems. In this application, an important concept is the integrality gap, the maximum ratio between the solution quality of the integer program and of its relaxation.
Mixed-integer linear programming (MILP) involves problems in which only some of the variables, , are constrained to be integers, while other variables are allowed to be non-integers. Zero–one linear programming (or binary integer programming ) involves problems in which the variables are restricted to be either 0 or 1.
Goal programming is a branch of multiobjective optimization, which in turn is a branch of multi-criteria decision analysis (MCDA). It can be thought of as an extension or generalisation of linear programming to handle multiple, normally conflicting objective measures.
While this formulation allows also fractional variable values, in this special case, the LP always has an optimal solution where the variables take integer values. This is because the constraint matrix of the fractional LP is totally unimodular – it satisfies the four conditions of Hoffman and Gale.