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Introduction of Linear programming

The Linear programming or LP or linear optimization is a mathematical technique for determining a manner to achieve the finest outcome (such as maximum profit or lowest cost) in a given mathematical model for a few list of necessities represented as linear relationships. The Linear programming is a specific case of mathematical programming (i.e., mathematical optimization).

More officially, linear programming is a method for the optimization of a linear objective function, subject to linear inequality and the linear equality constraints. Its feasible area is a convex polyhedron which is a set defined as the intersection of finitely several half spaces, each of which is defined via a linear inequality. The objective function of it is a real-valued affine function defined on this polyhedron. The Linear programming algorithms find a point in the polyhedron where this function has the minimum (or maximum) value whenever such a point exists.

The problems which can be expressed in canonical form are known as linear programs.

The canonical form is:

Maximize cTy

Subject to Ay< d

And y > 0

Here, y represents the vector of variables, c and d are the vectors of (known) coefficients, A is a matrix of coefficients and T is the matrix transpose. Expression to be minimized or maximized is known as the objective function (cTy in this situation). The inequalities Ay ≤ b is the constraints that specify a convex polytope over which the objective function is to be optimized. In this situation the two vectors are comparable if they have similar dimensions. When every entry in the first is less-than or equal-to the corresponding entry in the second then we can say that the first vector is less-than or equal-to the second vector.

**Nonlinear programming**

In mathematics, nonlinear programming or NLP is the process of solving a system of equalities and inequalities, collectively termed constraints, over a set of not known real variables, along with an objective function to be maximized or minimized, where some of the constraints or the objective functions are nonlinear.

Non-linear optimization problem (NLO)

The problem can be stated simply as:

To maximize some variable like product throughput or to minimize a cost function where

f: R^{n} →R

X € R^{n}

s.t. (subject to)

h_{i} (x) = 0, I € I = 1,...,p

g_{j} (x) <= 0, j € J = 1,..., m

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