The OPTQP Procedure

The OPTQP procedure solves quadratic programs—problems with quadratic objective function and a collection of linear constraints, including lower or upper bounds (or both) on the decision variables.

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 , 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 1 for 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.

Figure 1: Examples of Convex, Concave, and Nonconvex Objective Functions