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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

     
     
     

where

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.


Examples

Quadratic Programming (QP) solver examples