Sequential Convex Programming (SCP)

Author: Tianjiang Shuo
Website: https://cislunarspace.cn

Definition

Sequential convex programming (SCP, also successive convexification / SCvx) is a direct method tailored to nonconvex optimal control problems. At a reference trajectory $\mathbf{x}^{*}(t)$ , the nonlinear dynamics are replaced by a first-order Taylor expansion; combined with a trust-region constraint and lossless convexification of nonconvex control constraints, the original problem becomes a sequence of second-order cone programs (SOCP). Each iteration solves the SOCP and updates the reference trajectory until $\|\mathbf{x}^{(n)}-\mathbf{x}^{(n-1)}\|$ converges (Acikmese & Ploen 2007; Szmuk et al. 2017; Wang & Grant 2018).

Unlike general direct collocation NLPs, SCP subproblems are convex:

Interior-point methods solve each to a given accuracy in polynomial time with a deterministic stopping criterion.
User-supplied initial guesses do not affect interior-point startup (self-dual embedding constructs a feasible start).
Subproblem-level global optimality is guaranteed (though the original nonconvex problem still admits only local optimality).

These properties make SCP well suited to real-time / on-board applications — the core distinction from general NLP direct methods (Acikmese & Ploen 2007).

Mathematical form

For the nonconvex OCP

\min_{\mathbf{x},\mathbf{u}} J,\ \ \dot{\mathbf{x}}=\mathbf{f}(\mathbf{x},\mathbf{u}),\ \ \mathbf{x}(t_0)=\mathbf{x}_0,\ \ \text{boundary/path constraints},

SCP iteration $n$ linearizes the dynamics at $\mathbf{x}^{(n-1)}(t)$ :

\dot{\mathbf{x}} \approx A(t)\,\mathbf{x}+B(t)\,\mathbf{u}+\mathbf{q}(t),

A(t)=\frac{\partial\mathbf{f}}{\partial\mathbf{x}}\bigg|_{\mathbf{x}^{(n-1)}},\quad B(t)=\frac{\partial\mathbf{f}}{\partial\mathbf{u}}\bigg|_{\mathbf{x}^{(n-1)}},\quad \mathbf{q}(t)=\mathbf{f}(\mathbf{x}^{(n-1)},\mathbf{u}^{(n-1)})-A\,\mathbf{x}^{(n-1)}-B\,\mathbf{u}^{(n-1)}.

A trust-region constraint $\|\mathbf{x}-\mathbf{x}^{(n-1)}\|\leq \delta_n$ limits linearization error. The subproblem is now linear-time-varying (LTV) dynamics with convex constraints. After pseudospectral/Radau discretization it becomes a finite-dimensional SOCP, solved by ECOS, Mosek, or CVX.

Lossless convexification

Many aerospace problems have nonconvex control constraints, not nonconvex dynamics. The canonical example is powered descent: $\rho_2\leq\|\mathbf{T}_c\|\leq\rho_1$ — the lower bound (nonzero) makes the feasible set an annulus, nonconvex. Introduce a slack $\Gamma\geq\|\mathbf{T}_c\|$ and replace the control constraint by the convex cone $\|\mathbf{T}_c\|\leq\Gamma,\ \rho_2\leq\Gamma\leq\rho_1$ , giving a relaxed problem.

The key Lemma of Acikmese & Ploen (2007) shows that, under mild conditions, the optimal solution of the relaxed problem lies on the boundary of the original nonconvex feasible set — so relaxation costs nothing in optimality (lossless). The result was later generalized to control-affine problems (Wang & Grant 2018).

Virtual control

Linearization may produce artificial infeasibility — subproblems that are infeasible even though the original is feasible. An unconstrained virtual control $\boldsymbol{\nu}$ is added to the linearized dynamics to guarantee subproblem feasibility; $\|\boldsymbol{\nu}\|$ is heavily penalized in the cost to force $\boldsymbol{\nu}\to\mathbf{0}$ at convergence. If $\boldsymbol{\nu}=\mathbf{0}$ at convergence, the subproblem solution satisfies the nonlinear dynamics exactly (Szmuk et al. 2017).

Trust region and convergence

The trust-region radius $\delta_n$ is adapted: if $r_n=\Delta J_{\text{act}}/\Delta J_{\text{pred}}$ is close to 1 (linearization accurate), $\delta_{n+1}$ grows; if $r_n$ is close to 0, $\delta_{n+1}$ shrinks. Hofmann & Topputo (2021) use a $\rho_i,\alpha_i,\beta_i$ three-segment rule.

Convergence criteria usually combine: dynamics violation $\epsilon_c\to 0$ , cost change $\epsilon_\phi\to 0$ , virtual control $\|\boldsymbol{\nu}\|\to 0$ .

Variants

SCvx (Szmuk et al.) — classical trust region + virtual control, with the most complete convergence proof.
GUSTO (Mao, Szmuk, Acikmese) — a success/failure-step strategy; implementation-friendly.
ECIPS / cvxgen — embedded code generation for on-board real-time solves.
PC-SCoP (phase-constrained sequential cone program) — multiphase extension with linearized inter-phase event constraints.
Successive convex optimization + switching-time extraction (Hofmann-Topputo 2021) — FRPM discretization with bang-off-bang mesh refinement, accurately capturing switching structure.

Comparison with direct collocation / pseudospectral

Feature	SCP	Collocation/Pseudospectral
Subproblem	SOCP (convex)	General NLP (nonconvex)
Convergence	Polynomial-time, deterministic	Local, depends on initial guess
Real-time	Strong (ms–s)	Weak (s–min)
Global optimality	Subproblem-level guarantee	None
Application	Guidance, on-board	Mission design, ground optimization
Control-structure accuracy	Needs switching-time extraction	Captured directly

Applications

Powered descent guidance. Mars / Moon pinpoint landing is SCP's killer app — Acikmese & Ploen's lossless convexification was tailor-made for this; derivatives are used for SpaceX Starship on-board real-time planning (Acikmese & Ploen 2007; Song et al. 2021).
Low-thrust transfers. Wang & Grant first applied SCP to time- and fuel-optimal low-thrust orbit transfers; Hofmann & Topputo extended it to interplanetary transfers with hp-Radau + bang-off-bang refinement (Wang & Grant 2018; Hofmann & Topputo 2021).
Earth-Moon libration transfers. Halo ↔ NRHO and other highly nonlinear regimes: Kayama et al. (2022) show SCP still converges in the CR3BP neighborhood, but a thrust-continuation method (gradually lowering thrust acceleration from a large value to the mission value) is required at low acceleration levels to avoid divergence.
Collision avoidance and rendezvous. Near-field RPO and passively safe rendezvous with convexified keep-out constraints (Elango et al. 2025).

References

Acikmese, B., & Ploen, S. R. (2007). Convex programming approach to powered descent guidance for Mars landing. J. Guidance, Control, and Dynamics, 30(5), 1353–1366.
Szmuk, M., Reynolds, T. P., & Acikmese, B. (2017). Successive convexification for real-time six-degree-of-freedom powered descent guidance. J. Guidance, Control, and Dynamics.
Wang, Z., & Grant, M. J. (2018). Minimum-fuel low-thrust transfers for spacecraft: a convex approach. IEEE Trans. Aerospace and Electronic Systems.
Hofmann, C., & Topputo, F. (2021). Rapid low-thrust trajectory optimization in deep space based on convex programming. J. Guidance, Control, and Dynamics.
Kayama, H., et al. (2022). Low-thrust trajectory design with successive convex optimization for libration point orbits.
Song, Z., et al. (2021). Adaptive powered descent guidance based on multi-phase pseudospectral convex optimization.
Tang, S., & Conway, B. A. (2018). Fuel-optimal low-thrust trajectory optimization using indirect method and successive convex programming.
Elango, A., et al. (2025). Successive convexification for passively-safe spacecraft rendezvous on near rectilinear halo orbit.
Mao, Y., Szmuk, M., & Acikmese, B. (2016). Successive convexification of non-convex optimal control problems and its convergence properties. IEEE CDC.