Davide Cherubini - PhD Thesis - UniCA Eprints

Chapter 6LP Optimization ModelsThis chapter introduces the basic concepts of Linear Programming Optimization,the flow optimization problems, and their extension to Multicommodity flow problemsin order to formulate three optimization models for the traffic flow in abackbone telecommunication network.6.1 IntroductionA specific class of mathematical problems, where a linear function has to beminimized (or maximized), and subject to given linear constraints, is called theclass of Linear Programming (LP) problems. A Linear Programming is a problemthat can be expressed, in its Standard Form, as follows:min cx (6.1)subject to Ax = b (6.2)x ≥ 0 (6.3)where x is the vector of unknown variables, A is a m × n matrix of known coefficients(with m representing the number of constraints and n being the numberof variables), and c and b are vectors of known coefficients.The expression cx represents the objective function, while the equations Ax =b are called the constraints. The matrix A is generally not square, but has morecolumns than rows (n > m, underdetermined), leaving great latitude (degrees of42