Linear Programming

Abstract

Solving linear programs is one of the best investigated and most important problems in mathematical optimization. Formally, it is the problem of minimizing a linear objective function

$$ {{\upsilon }_{1}}{{x}_{1}} + {{\upsilon }_{2}}{{x}_{2}} + \cdots + {{\upsilon }_{d}}{{x}_{d}} $$

subject to a collection of constraints, where each constraint is a linear inequality

$$ {{\omega }_{{i,1}}}{{x}_{1}} + {{\omega }_{{i,2}}}{{x}_{2}} + ... + {{\omega }_{i}}{{,}_{d}}{{x}_{d}} \leqslant {{\omega }_{{i,d + 1,}}}$$

for real numbers υ₁ through υ_d and ω _i,1 through ω_{i ,d+1},. If the linear program involves d variables, x₁,x₂,...,x_d, then each inequality can be interpreted as a closed half-space in E^d, and the collection of constraints becomes a convex polyhedron Pthat is the intersection of all half-spaces. The problem is then to find a point x =(ξ₁,ξ₂,..., ξ_d) in P if it exists, that minimizes

$$ \left\langle {u,x} \right\rangle = {{\upsilon }_{1}}{{\xi }_{1}} + {{\upsilon }_{2}}{{\xi }_{2}} + \cdots + {{\upsilon }_{d}}{{\xi }_{d}}, $$

where u =(v ₁,v₂,...,v _d ). is the vector which defines the objective function. Intuitively, x; is a point of P which lies furthest in the direction determined by the vector −u = (−v₁, −v₂,..., −v _d ). However, such a point does not always exist. We will return to this issue in Section 10.1, where we discuss the different types of linear programs existing and where we specify what we require from a solution to a linear program.