The Quadratic Programming Solver (Experimental)

The QP solver in the OPTMODEL procedure implements an infeasible primal-dual predictor-corrector interior point algorithm that enables you to solve quadratic programming problems.

Mathematically, a quadratic programming problem can be stated as

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

where

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

$\text{[math]}$

for minimization models; it is required to be negative semidefinite for maximization models. Convexity can come as a result of a matrix-matrix multiplication

$\text{[math]}$

or as a consequence of physical laws, and so on.

For further details, see the SAS/OR^® User's Guide: Mathematical Programming: The Quadratic Programming Solver.
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The Quadratic Programming Solver (Experimental)

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