Quadratic Programming - Problem Formulation

Problem Formulation

The quadratic programming problem can be formulated as:

Assume x belongs to space. Both x and c are column vectors with n elements (n×1 matrices), and Q is a symmetric n×n matrix.

Minimize (with respect to x)

Subject to one or more constraints of the form:

(inequality constraint)

(equality constraint)

where indicates the vector transpose of . The notation means that every entry of the vector is less than or equal to the corresponding entry of the vector .

If the matrix is positive semidefinite, then is a convex function: In this case the quadratic program has a global minimizer if there exists some feasible vector (satisfying the constraints) and if is bounded below on the feasible region. If the matrix is positive definite and the problem has a feasible solution, then the global minimizer is unique.

If is zero, then the problem becomes a linear program.

A related programming problem, quadratically constrained quadratic programming, can be posed by adding quadratic constraints on the variables.