The Quadratic Programming (QP) Solver

Procedures in Online Documentation

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



is the quadratic (also known as Hessian) matrix

is the constraints matrix

is the vector of decision variables

is the vector of linear objective function coefficients

is the vector of constraints right-hand sides (RHS)

is the vector of lower bounds on the decision variables

is the vector of upper bounds on the decision variables

The quadratic matrix is assumed to be symmetric; that is,


Indeed, even if , then the simple modification


produces an equivalent formulation hence symmetry is assumed. When you specify a quadratic matrix, it suffices to list only lower triangular coefficients.

In addition to being symmetric, is also required to be positive semidefinite


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


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


Quadratic Programming (QP) solver examples