Convex Model Predictive Control (MPC)

• Convex MPC is a control strategy that reformulates receding-horizon optimal control problems into convex programs, ensuring unique global solutions.
• This approach leverages methods like QP, SOCP, and SDP, and utilizes techniques such as difference-of-convex decomposition to manage complex constraints.
• It enables efficient numerical solutions while providing robust guarantees on system stability and feasibility in dynamic environments.

Convex Model Predictive Control (MPC) is a class of model predictive control methodologies characterized by receding-horizon optimal control formulations that produce convex optimization problems at each time step. By construction, convex MPC frameworks guarantee unique global optima, efficient numerical solution methods, and strong theoretical guarantees on stability and feasibility. The convexity arises from appropriate selection of system models, cost functions, uncertainty sets, and constraint structures—often exploiting advanced representations (e.g., polytopes, convex neural networks, difference-of-convex decompositions) and duality theory.

1. Formulation and Theoretical Foundations

Convex MPC arises when the underlying optimal control problem, with all associated constraints and uncertainty models, can be cast as a convex program. The standard finite-horizon convex MPC problem for a discrete-time linear system is

$$\begin{aligned} \min_{x_0,\dots,x_N,u_0,\dots,u_{N-1}} &\quad \sum_{k=0}^{N-1} \ell(x_k,u_k) + \phi_N(x_N) \ \text{s.t.}\quad &x_{k+1} = A x_k + B u_k,\quad k = 0,\dots,N-1\ &x_k \in \mathcal{X},\quad u_k \in \mathcal{U}\ &x_0 = x_{\text{init}} \end{aligned}$$

where $\ell$ and $\phi_N$ are convex, and $\mathcal{X}$, $\mathcal{U}$ are convex sets (Hanks et al., 2023). Convexity can be preserved under more complex settings—e.g., risk-averse costs, polytopic uncertainty, and convex state/input transformations—provided all mappings remain convex in the decision variables (Chow et al., 2015).

Convex MPC exploits properties of convex analysis, duality, and LMI theory. Key structural results include:

• Time-consistent risk measures expressed as nested coherent risk mappings (e.g., polytopic risk envelopes) induce convex objectives under scenario tree dynamics (Chow et al., 2015).
• Convex relaxations of nonconvex sets (e.g., orbitopes for $SO(n)$) enable exact or approximate convex reformulations for nonlinear state representations, as in rotation dynamics (Huang et al., 2014).
• Difference-of-convex (DC) programming enables convexification of general smooth nonlinearities by successive linearization of the concave parts and explicit convex-concave decomposition (Doff-Sotta et al., 1 Feb 2026).

2. Algorithmic Structures and Solution Methods

The solution of a convex MPC problem at each control update involves the formulation and efficient solution of convex quadratic programs (QP), second-order cone programs (SOCP), or semidefinite programs (SDP), depending on the specific constraints and cost function structure.

First-order and primal-dual methods:

• Algorithms such as the Proportional-Integral Projected Gradient (PIPG) provide single-projection, primal-dual gradient methods tailored for convex MPC, achieving $O(1/k)$ or better convergence rates with one cheap projection per iteration (Yu et al., 2020).
• Split Bregman and operator-splitting approaches are effective for multi-convex decompositions, reducing complex trajectory optimizations to stacks of small QPs or LPs (Masnavi et al., 2021).

Data-driven architectures:

• Input-Convex Neural Networks (ICNNs) can be embedded as process models or cost-to-go surrogates, preserving problem convexity under weight and activation constraints (Bünning et al., 2020, Wang et al., 2024). Convex MPC can then be solved via QP, MIQP, or interior-point methods.
• Explicit MPC via multi-parametric QP decomposition