Week 7–8: Linear Programming

A linear program (LP for short) is an optimization problem in which the constraints are linear inequalities and equalities, and the objective function is also linear. There are many equivalent standard forms for LPs. We will use the following form for a maximization problem: max cx s.t. Ax ≤ b x ≥ 0 Let us explain the notation a bit. Here A is an m× n matrix (i.e. m constraints and n variables) and b is an m× 1 column vector; c is an n× 1 column vector which encodes the objective function and cT is its transpose; x is an n× 1 column vector which contains the variables we are optimizing over. The inequalities between vectors mean that the inequality should hold in all coordinates simultaneously. The value of this LP is the minimum value of the objective cTx achieved subject to x satisfying the constraints. Any value of x that satisfies the constraints x ≥ 0 and Ax ≤ b is called feasible. The set of feasible x is called the feasible set or feasible region of the LP. When the feasible set is empty, the LP is called infeasible. The maximum value of the objective cTx over feasible x is the optimal value of the LP. If this maximum is infinity, i.e. for any t ∈ R there exists a feasible x s.t. cTx ≥ t, then the LP is called unbounded.