# Linear Program Solving Task

A linear program solving task is an continuous optimization task that requires a solution to a linear program (with a linear cost function).

**AKA:**Linear Optimization (LO).**Context:**- It can be solved by a Linear Programming System (that applies a linear programming algorithm, such as the simplex algorithm).
- It can range from
- being an Integer Programming Task/General Integer Linear Programming Task the Variables are Integer Variables.
- to being a Numeric Optimization Task,
- to being a Combinatorial Optimization Task, if the solution space is the choice of which variables to make basic.

**Example(s):****Counter-Example(s):****See:**System of Equations Solving, Mixed Integer Programming, Assignment Problem, Feasible Region, Convex Polytope, Intersection (Mathematics).

## References

### 2016

- (Wikipedia, 2016) ⇒ https://en.wikipedia.org/wiki/Linear_programming Retrieved:2016-12-17.
**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 are represented by linear relationships. Linear programming is a special case of mathematical programming (mathematical optimization).More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints. Its feasible region is a convex polytope, which is a set defined as the intersection of finitely many half spaces, each of which is defined by a linear inequality . Its objective function is a real-valued affine (linear) function defined on this polyhedron. A linear programming algorithm finds a point in the polyhedron where this function has the smallest (or largest) value if such a point exists.

Linear programs are problems that can be expressed in canonical form as : [math] \begin{align} & \text{maximize} && \mathbf{c}^\mathrm{T} \mathbf{x}\\ & \text{subject to} && A \mathbf{x} \leq \mathbf{b} \\ & \text{and} && \mathbf{x} \ge \mathbf{0} \end{align} [/math] where

**x**represents the vector of variables (to be determined),**c**and**b**are vectors of (known) coefficients,*A*is a (known) matrix of coefficients, and [math] (\cdot)^\mathrm{T} [/math] is the matrix transpose. The expression to be maximized or minimized is called the*objective function*(**c**^{T}**x**in this case). The inequalities*A***x**≤**b**and**x**≥**0**are the constraints which specify a convex polytope over which the objective function is to be optimized. In this context, two vectors are comparable when they have the same dimensions. If every entry in the first is less-than or equal-to the corresponding entry in the second then we can say the first vector is less-than or equal-to the second vector.Linear programming can be applied to various fields of study. It is widely used in business and economics, and is also utilized for some engineering problems. Industries that use linear programming models include transportation, energy, telecommunications, and manufacturing. It has proved useful in modeling diverse types of problems in planning, routing, scheduling, assignment, and design.

### 2012

- Introduction to Linear Optimization - AMS Class Notes http://ams.org/open-math-notes/files/course-material/OMN-201611-110636-1-Course_notes-v1-pdf