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 polyhedron, 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 function defined on this polyhedron. A linear programming algorithm finds a point in the polyhedron where this function has the smallest value if such a point exists.
Linear programs are problems that can be expressed in canonical form:
where x represents the vector of variables, c and b are vectors of coefficients, A is a matrix of coefficients, and is the matrix transpose. The expression to be maximized or minimized is called the objective function. The inequalities Ax ≤ b and x ≥ 0 are the constraints which specify a convex polytope over which the objective function is to be optimized.
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