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# 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

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.

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