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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 If there exists an optimal solution, then there exists an optimal bfs. Linear programming is a special case of mathematical programming (also known as mathematical optimization).

Fundamental theorem of linear programming in mathematical optimization, the fundamental theorem of linear programming states, in a weak formulation, that the maxima and minima of a linear function over a convex polygonal region occur at the region's corners. Geometrically, each bfs corresponds to a vertex of the polyhedron of feasible solutions Karmarkar's paper created a surge of interest in interior point methods.

The theory of linear programming dictates that under mild assumptions (if the linear program has an optimal solution, and if the feasible region does not contain a line), one can always find an extreme point or a corner point that is optimal

The obtained optimum is tested for being an integer solution. 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. The graph illustrates the simplex algorithm solving a linear programming problem with two variables

In mathematical optimization, dantzig 's simplex algorithm (or simplex method) is an algorithm for linear programming [1] the name of the algorithm is derived from the concept of a simplex and was suggested by t [2] simplices are not actually used in the method, but one. In mathematics, farkas' lemma is a solvability theorem for a finite system of linear inequalities

It was originally proven by the hungarian mathematician gyula farkas

[1] farkas' lemma is the key result underpinning the linear programming duality and has played a central role in the development of mathematical optimization (alternatively, mathematical programming) In linear programming, a discipline within applied mathematics, a basic solution is any solution of a linear programming problem satisfying certain specified technical conditions For a polyhedron and a vector , is a basic solution if All the equality constraints defining p {\displaystyle p} are active at x ∗ {\displaystyle \mathbf {x} ^ {*}} of all the constraints that are active at that.

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