## 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:

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, it is easy to show that 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 type of models:

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

or as a consequence of physical laws, and so on. See Figure 9.1 for examples of convex, concave, and nonconvex objective functions.

Figure 9.1: Examples of Convex, Concave, and Nonconvex Objective Functions

The order of constraints is insignificant. Some or all components of or (lower and upper bounds, respectively) can be omitted.