DifferentiableOSQP

This package provides a thin wrapper of OSQP.jl, but also provides the ability to differentiate through the quadratic progam, based on the equations in OptNet.

The package exports 2 commands: solve and solve_and_jac.

Installation

Activate your environment and simply add DifferentialOSQP:

] add https://github.com/dev10110/DifferentiableOSQP.jl

Interface 1: Solve a QP

   x = solve(P, q, A, u; kwargs...)

Solves the quadratic program:

$$\begin{aligned} \min_x &\quad \frac{1}{2} x^T P x + q^T x \\ s.t. &\quad A x \leq u \end{aligned}$$

where kwargs are keyword arguments passed into OSQP.setup!

To include equality constraints, for example G x = h, modify A, u matrices as:

A = [A ; G ; -G]
u = [u ; h; -h]

This introduces the constraints G x \leq h and G x \geq h, allowing equalities to be handled. Yes, this is rather inefficient, but the easiest way to solve the problem I think.

Interface 2: Solve a Parameteric QP

  x = solve(θ, P_fn, q_fn, A_fn, u_fn; kwargs)

Solves the quadratic program:

$$\begin{aligned} \min_x &\quad \frac{1}{2} x^T P(\theta) x + q(\theta)^T x \\ s.t. &\quad A(\theta) x \leq u(\theta) \end{aligned}$$

i.e., assumes the P_fn, q_fn, A_fn, u_fn are functions of θ. Note, the shape and size of each output must be correct - for example, A_fn(θ) must return a matrix, and q_fn(θ) must return a vector.

Interface 3: Jacobians of a QP

Thinking about interface 2, notice that a QP solver is essentially a function

$$QP : \mathbb{R}^p \to \mathbb{R}^n\\\ x = QP(θ)$$

Therefore, the Jacobian of the QP

$$J = \frac{\partial x}{\partial \theta}$$

If we want the jacobian of QP, we can call it as the following:

x, J = solve_and_jac(θ, P_fn, q_fn, A_fn, u_fn; kwargs)

which gives the optimal solution x, and the jacobian J

Interaface 4: ForwardDiff

For convenience, we overloaded solve to handle Dual numbers. This means we can directly use ForwardDiff.jacobian, as in the following example:

J = ForwardDiff.jacobian(θ -> solve(θ, P_fn, q_fn, A_fn, u_fn), θ0)

or more explictly

function parameteric_qp(θ)
  return solve(θ, P_fn, q_fn, A_fn, u_fn)
end

J = ForwardDiff.jacobian(parametric_qp, θ0)

Warning

Naturally, not all QPs are differentiable. This library will always return a derivative, but doesnt check/warn if the derivative doesnt exist. I want to add this functionality in the future. See this or this paper for some results on existence of derivatives/Lipschitz continuity of QPs.