---
title: Successive Convexification
url: https://www.emergentmind.com/topics/successive-convexification
type: topic
---

# Successive Convexification

Successive convexification is a trajectory optimization methodology that transforms a nonconvex optimal control problem (OCP)—where nonlinearity and nonconvex constraints preclude direct global solution—into a rapidly convergent sequence of tractable convex subproblems. At each iteration, the original dynamics and nonconvex constraints are locally linearized about a reference trajectory, and feasibility is safeguarded through the inclusion of artificial constructs such as virtual controls and trust regions. Successive convexification techniques have demonstrated strong theoretical guarantees for convergence, practical real-time performance in a range of guidance and control applications, and the facility to handle challenging constraints such as logical obstacle avoidance via state-triggered constructs [2010.05201, 1701.00558, 1804.06539, 2505.17251, 2606.06896]. The framework supports both discrete- and continuous-time problem settings, generalizes readily to uncertainty and robustness, and can be realized using a variety of convex optimization back ends (LP, QP, SOCP).

## 1. Core Methodology and Theoretical Foundations

Successive convexification (often abbreviated SCvx) is an iterative framework for solving nonconvex OCPs by repeatedly constructing and solving convex approximations around the current iterate. Nonlinear and/or nonconvex components—typically in the system dynamics, path constraints, and boundary conditions—are locally linearized (most commonly via first-order Taylor expansion, though intrinsic collocation and difference-of-convex representations are used in related variants [2302.07744, 2512.09551]). The resulting subproblem at each iteration is a convex finite-dimensional program.

A typical SCvx iteration includes the following steps, as formalized in [2010.05201, 1701.00558, 1804.06539]:

1. **Linearization:** Dynamics, constraints, and sometimes the cost functional are linearized about the most recent solution trajectory (state, control, and possibly time).
2. **Virtual Control/Slack Augmentation:** To guarantee subproblem feasibility even far from the true dynamics (thereby preventing artificial infeasibility), “virtual control” (slack variables) are introduced into the dynamics and penalized in the cost (exact-penalty style) [1608.05133, 1804.06539].
3. **Trust Region Enforcement:** To prevent “artificial unboundedness” due to the local validity of the linear model, deviations from the linearization point are bounded, either via explicit norm constraints or quadratic penalty terms in the cost [2010.05201].
4. **Convex Subproblem Solution:** The resulting convex problem (frequently QP/SOCP) is solved to (near-)optimality.
5. **Step Acceptance and Trust Region Update:** The actual improvement (cost reduction under nonlinear OCP) and predicted improvement (under the convex surrogate) are compared; the solution is accepted if agreement is sufficient, and trust regions are adaptively adjusted [1804.06539, 2010.05201].
6. **Iterate Update:** The new solution

Source: https://www.emergentmind.com/topics/successive-convexification