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.
Read more about this topic: Quadratic Programming
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The formulation was used as a motto by the English Nonconformist clergyman Richard Baxter (1615-1691)