Operator Splitting for Optimal Control

A Julia package for solving optimal control problems efficiently.

Based on: Operator Splitting for Optimal Control by Brendan O'Donoghue, George Stathopoulos and Stephen Boyd.

Freely inspired by the existing C implementation of the same algorithm.

What does this solve ?

$$ \begin{aligned} \min_{\substack{x_t, u_t \ t=0,\ldots,T}} \quad & \sum_{t=0}^{T} \left( \phi_t(x_t, u_t) + \psi_t(x_t, u_t) \right) \\ \text{subject to} \quad & x_{t+1} = A_t x_t + B_t u_t + c_t,\quad t=0,\ldots,T-1. \end{aligned} $$

Where the $\phi_t$ terms are quadratic and the $\psi_t$ terms are arbitrary, supplied by the user in the form of a proximal operator.

Usage

Problem Data

data = all_data(A, B, c, Q, S, R, q, r, x_init;
    rho=50.0,
    alpha=1.8,
    eps_abs=1e-3,
    eps_rel=1e-3,
    reg=1e-6,
)

See src/types.jl for guidelines on variable types.

Proximal operator

The proximal operator must have the signature:

prox!(x_tilde, u_tilde, v, w, rho)

and update x_tilde and u_tilde in place.

Cache

Before solving, one must build a cache from the problem data.

cache = setup_cache(data)

Repeated calls to the solver with the same cache will reuse the internal problem state, thus resulting in a "warm start" of the resolution, yielding a noticeable performance increase.

You can reuse the same cache while these stay fixed:

• A, B, Q, S, R
• rho, reg

Equivalently, you can freely change these variables without rebuilding the cache:

• q, r, c, x_init

Solver

x, u, tt = solve(cache, prox!; max_iters=3000)

The tt output is a Timings struct which contains the timing logs of the solver run.

# Compact output
println(tt)

# Detailed output
display(tt)

Examples

Runnable examples (from Operator Splitting for Optimal Control) live in examples/. You can run them by specifying the problem size (small, medium, large).