Linear Programming and Network Optimization

Here, C denotes a column vector of cost coefficients, X a column vector of variables, A = [ai,j ] is a coefficient matrix, B is a column vector representing the bounds in the inequalities and the multiplication operation is matrix multiplication. Linear programs arise in a wide range of applications and can be solved efficiently. The classical simplex algorithm has been very successful in practice, although its worst-case running time is exponential in the size of the problem instance. So-called interior point methods can be used to solve linear programs in polynomial time, although they are often slower than the simplex algorithm, in practice. It’s importat to note that the variables xj are defined as real-valued. If some or all of the variables are required to be integers we get an Integer Linear Program and in general, these problems are NP -hard.