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Quadratic programming AI simulator
(@Quadratic programming_simulator)
Hub AI
Quadratic programming AI simulator
(@Quadratic programming_simulator)
Quadratic programming
Quadratic programming (QP) is the process of solving certain mathematical optimization problems involving quadratic functions. Specifically, one seeks to optimize (minimize or maximize) a multivariate quadratic function subject to linear constraints on the variables. Quadratic programming is a type of nonlinear programming.
"Programming" in this context refers to a formal procedure for solving mathematical problems. This usage dates to the 1940s and is not specifically tied to the more recent notion of "computer programming." To avoid confusion, some practitioners prefer the term "optimization" — e.g., "quadratic optimization."
The quadratic programming problem with n variables and m constraints can be formulated as follows. Given:
the objective of quadratic programming is to find an n-dimensional vector x, that will
where xT denotes the vector transpose of x, and the notation Ax ⪯ b means that every entry of the vector Ax is less than or equal to the corresponding entry of the vector b (component-wise inequality).
As a special case when Q is symmetric positive-definite, the cost function reduces to least squares:
where Q = RTR follows from the Cholesky decomposition of Q and c = −RT d. Conversely, any such constrained least squares program can be equivalently framed as a quadratic programming problem, even for a generic non-square R matrix.
When minimizing a function f in the neighborhood of some reference point x0, Q is set to its Hessian matrix H(f(x0)) and c is set to its gradient ∇f(x0). A related programming problem, quadratically constrained quadratic programming, can be posed by adding quadratic constraints on the variables.
Quadratic programming
Quadratic programming (QP) is the process of solving certain mathematical optimization problems involving quadratic functions. Specifically, one seeks to optimize (minimize or maximize) a multivariate quadratic function subject to linear constraints on the variables. Quadratic programming is a type of nonlinear programming.
"Programming" in this context refers to a formal procedure for solving mathematical problems. This usage dates to the 1940s and is not specifically tied to the more recent notion of "computer programming." To avoid confusion, some practitioners prefer the term "optimization" — e.g., "quadratic optimization."
The quadratic programming problem with n variables and m constraints can be formulated as follows. Given:
the objective of quadratic programming is to find an n-dimensional vector x, that will
where xT denotes the vector transpose of x, and the notation Ax ⪯ b means that every entry of the vector Ax is less than or equal to the corresponding entry of the vector b (component-wise inequality).
As a special case when Q is symmetric positive-definite, the cost function reduces to least squares:
where Q = RTR follows from the Cholesky decomposition of Q and c = −RT d. Conversely, any such constrained least squares program can be equivalently framed as a quadratic programming problem, even for a generic non-square R matrix.
When minimizing a function f in the neighborhood of some reference point x0, Q is set to its Hessian matrix H(f(x0)) and c is set to its gradient ∇f(x0). A related programming problem, quadratically constrained quadratic programming, can be posed by adding quadratic constraints on the variables.