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For more information, see Integer Linear Programming and its related pages.The importance of linear programming derives both from its many applications and from the existence of effective general purpose techniques for finding optimal solutions.
These techniques are general purpose in that they take as input an LP and determine a solution without reference to any information concerning the origin of the LP or any special structure of the LP.
They are fast and reliable over a substantial range of problem sizes and applications.
For the purposes of describing and analyzing algorithms, the problem is often stated in standard form as \[ \begin \min & c^T x & & \\ \mbox & A x & = & b \\ & x & \geq & 0 \end \] where \(x\) is the vector of unknown variables, \(c\) is the cost vector, and \(A\) is the constraint matrix.
The matrix \(A\) is generally not square; therefore, solving the LP is not as simple as just inverting the \(A\) matrix.
In these lessons, we will learn about linear programming and how to use linear programming to solve word problems.
Many problems in real life are concerned with obtaining the best result within given constraints.Although all linear programs can be converted to standard form, it is not usually necessary to do so to solve them.Most LP solvers can handle other forms such as There are two families of techniques in wide use today, simplex methods and barrier or interior point methods.Both techniques generate an improving sequence of trial solutions until a solution is reached that satisfies the conditions for an optimal solution.Simplex methods were introduced by George Dantzig in the 1940s.An optimal solution is a feasible solution that has the smallest value of the objective function for a minimization problem.An LP may have one, more than one or no optimal solutions.These methods derive from techniques for nonlinear programming that were developed and popularized in the 1960s by Anthony Fiacco and Garth Mc Cormick.Their application to linear programming dates back to Narendra Karmarkar's innovative analysis in 1984. Due to advances in solution techniques and in computing power over the past two decades, linear programming problems with tens or hundreds of thousands of continuous variables are routinely solved.Simplex methods visit basic solutions computed by fixing enough of the variables at their bounds to reduced the constraints \(Ax = b\) to a square system, which can be solved for unique values of the remaining variables.Basic solutions represent extreme boundary points of the feasible region defined by \(Ax = b\), \(x = 0\), and the simplex method can be viewed as moving from one such point to another along the edges of the boundary. Barrier or interior-point methods by contrast visit points within the interior of the feasible region.