The Quadratic Programming Solver

Overview: QP Solver

The OPTMODEL procedure provides a framework for specifying and solving quadratic programs.

Mathematically, a quadratic programming (QP) problem can be stated as follows:

	$\text{[math]}$	$\text{[math]}$
	$\text{[math]}$	$\text{[math]}$
	$\text{[math]}$	$\text{[math]}$

where

$\text{[math]}$	$\text{[math]}$	$\text{[math]}$	is the quadratic (also known as Hessian) matrix
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$	is the constraints matrix
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$	is the vector of decision variables
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$	is the vector of linear objective function coefficients
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$	is the vector of constraints right-hand sides (RHS)
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$	is the vector of lower bounds on the decision variables
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$	is the vector of upper bounds on the decision variables

The quadratic matrix $\text{[math]}$ is assumed to be symmetric; that is,

$\text{[math]}$

Indeed, it is easy to show that even if $\text{[math]}$ , then the simple modification

$\text{[math]}$

produces an equivalent formulation $\text{[math]}$ hence symmetry is assumed. When you specify a quadratic matrix, it suffices to list only lower triangular coefficients.

In addition to being symmetric, $\text{[math]}$ is also required to be positive semidefinite for minimization type of models:

$\text{[math]}$

$\text{[math]}$ is required to be negative semidefinite for maximization type of models. Convexity can come as a result of a matrix-matrix multiplication

$\text{[math]}$

or as a consequence of physical laws, and so on. See Figure 8.1 for examples of convex, concave, and nonconvex objective functions.

Figure 8.1 Examples of Convex, Concave, and Nonconvex Objective Functions

The order of constraints is insignificant. Some or all components of $\text{[math]}$ or $\text{[math]}$ (lower and upper bounds, respectively) can be omitted.