Quadratic Programming Models

Abstract

Quadratic programming (QP) deals with a special class of mathematical programs in which a quadratic function of the decision variables is required to be optimized (i.e., either minimized or maximized) subject to linear equality and/or inequality constraints.

Let x = (x ₁, …, x _n )^T denote the column vector of decision variables. In mathematical programming, it is standard practice to handle a problem requiring the maximization of a function f(x) subject to some constraints by minimizing − f(x) subject to the same constraints. Both problems have the same set of optimum solutions. Because of this, we restrict our discussion to minimization problems.

A quadratic function of decision variables x is a function of the form

$$Q(x) = \sum \limits _{i=1}^{n} \sum \limits _{j=i}^{n}{q}_{ ij}{x}_{i}{x}_{j}\ \ + \sum \limits _{j=1}^{n}{c}_{ j}{x}_{j}\ \ + {c}_{0}.$$