Warm Start

• Also called: warm starting
• Category: Planning & manipulation › Trajectory optimisation
• Tags: optimization, model-predictive-control, trajectory-optimization, numerical-methods, real-time

Initializing an iterative solver with a good guess, often a previous or related solution, instead of from scratch, to cut solve time and steer it toward a useful local optimum.

Definition

A warm start initializes an iterative optimization or numerical solver with information from a related problem, rather than from a default or arbitrary starting point, which is called a cold start. The information can be the primal variables (the decision variables themselves, such as a trajectory), the dual variables or Lagrange multipliers, the active set of inequality constraints, or approximations such as a quasi-Newton Hessian. Because nonlinear robotics problems usually have many local optima, the starting point affects not only how fast a solver converges but also which solution it finds.

Model predictive control is the most common use. At each control cycle the controller solves an optimization over a horizon that overlaps almost entirely with the previous one, so the previous solution, shifted forward by one time step with the last step extended, is already close to the new optimum. Warm starting in this way often reduces the solver to a few iterations per cycle, and the real-time iteration scheme of Diehl and colleagues takes this further by performing only a single sequential quadratic programming iteration per cycle. Sampling-based controllers such as MPPI and the cross-entropy method warm start by reusing the previous mean control sequence.

How much a warm start helps depends on the solver. Active-set methods, such as the one in qpOASES, and first-order methods such as the ADMM solver OSQP benefit strongly, because a nearby primal and dual guess or a correct active set lets them finish quickly. Interior-point methods are harder to warm start, because they need iterates strictly inside the feasible region and away from the constraint boundaries where previous solutions usually lie. In offline trajectory optimization, a warm start can come from a simpler model, a coarser discretization, a relaxed version of the problem, a stored library of solutions or a neural network trained to predict good initial guesses. Physics engines similarly warm start iterative contact solvers with the impulses from the previous step. In machine learning the term is also used for initializing training from a previously trained model.

Why it matters

Real-time optimization-based controllers run at tens to thousands of hertz, and they usually meet those rates only because each solve starts close to its answer. In offline planning, a good warm start is often the difference between a solver that converges to a sensible motion and one that fails or finds a poor local minimum.

Key applications

• Real-time model predictive control on legged robots, drones and manipulators
• Contact-implicit and nonlinear trajectory optimization
• Sequences of related quadratic programs in whole-body control
• Iterative contact solvers in physics engines
• Learned initial guesses for fast motion planning

In practice

A legged robot runs a nonlinear MPC at 100 Hz. After each solve, the controller stores the optimal state and input trajectories and the multipliers. At the next cycle it shifts these by one step, copies the final step to fill the end of the horizon, replaces the first state with the newly measured state and passes the result to the solver as its initial guess. Typical solves then take two or three iterations instead of twenty. The engineers also handle the cases where warm starting fails: after a large push or a change of gait the shifted guess is far from the new optimum, so the controller falls back to a stored nominal trajectory for that gait and allows more iterations for that cycle.

Common misconceptions

• A warm start only speeds things up — for nonconvex problems it also decides which local optimum is found, so a poor warm start can lock the solver into a bad solution
• Any solver benefits equally — active-set and first-order methods gain the most; interior-point methods gain less and may need special handling
• A warm start only needs the previous decision variables — supplying dual variables or the active set as well often matters as much for fast convergence
• The previous solution is always a good guess — after large disturbances, mode changes or goal changes it can be far from the new optimum and needs a fallback

Who should know this

• Model predictive control engineers
• Trajectory optimization developers
• Legged locomotion and drone control engineers
• Numerical optimization practitioners in robotics

Related technologies

• OSQP
• qpOASES
• IPOPT
• acados
• CasADi
• Real-Time Iteration Scheme

Related terms

• Model Predictive Control
• Direct Collocation
• Differential dynamic programming
• Trajectory Planning
• Model Predictive Path Integral
• Complementarity Constraint

Related projects

• GENESIS: World's fastest physics engine: 43M FPS for embodied AI learning
• Cartographer: Google's real-time 2D and 3D SLAM system for multi-platform robotics
• FAST-LIVO2: Fast, direct LiDAR-inertial-visual odometry from HKU-MARS, published in IEEE T-RO
• Drake: MIT robotics toolbox for model-based dynamics, optimization, and control system design
• Dora AI: Dataflow middleware for AI robotics applications, built in Rust
• GTSAM: Factor-graph optimization library powering many modern SLAM backends