Constrained Convex Generators (CCGs)

• Constrained Convex Generators (CCGs) are a parametric affine image representation of convex sets with equality constraints, unifying traditional set classes like zonotopes and ellipsoids.
• They support exact closed-form operations—such as affine mapping, Minkowski sum, and convex hull—with minimal growth in generators and constraints.
• CCGs reduce conservatism in state estimation and safety-critical control, improving performance in applications like autonomous navigation and obstacle avoidance.

A Constrained Convex Generator (CCG) is a set representation in $\mathbb{R}^n$ characterized by a parametric affine image of a product of convex sets, subject to affine equality constraints on the parameterization. This formulation unifies and generalizes classical convex set classes, enabling exact and efficient manipulation of convex sets in state estimation, reachability analysis, and safety-critical control. CCGs support closure under affine maps, Minkowski sums, and generalized intersections, with minimal descriptional complexity and provably minimal variable growth for set operations such as convex hulls. Their indirect representation permits reduced conservatism compared to polytopic or interval approaches, especially in systems with coupling between uncertainties and dynamics.

1. Definition and Structural Properties

A CCG in $\mathbb{R}^n$ is given by

$$\mathcal{Z} = \{ G \xi + c : A \xi = b,\ \xi \in \mathcal{C}_1 \times \cdots \times \mathcal{C}_{n_p} \}$$

where $G \in \mathbb{R}^{n \times n_g}$ is the generator matrix, $c \in \mathbb{R}^n$ is the offset (center), $A \in \mathbb{R}^{n_c \times n_g}$ and $b \in \mathbb{R}^{n_c}$ impose affine equality constraints, and each $\mathcal{C}_i$ is a convex "base" set (e.g., $\ell_p$ balls, intervals, cones), so that the parameter vector $\xi = [\xi^{(1)}; \cdots; \xi^{(n_p)}]$ with $\xi^{(i)} \in \mathcal{C}_i$ (Silvestre, 2023, Matias et al., 12 Jan 2026). This indirect representation allows CCGs to subsume intervals, zonotopes, ellipsoids, constrained zonotopes, cones, ellipsotopes, affine-hull polytopes, and related classes.

Key properties include:

• Convexity by construction: The combination of an affine image and convex generator sets yields a convex set $\mathcal{Z}$.
• Expressivity: Special choices recover known set classes, e.g., zonotopes ($\|\xi\|_\infty \leq 1$ and diagonal $G$), ellipsoids ($\|\xi\|_2 \leq 1$), cones ($\xi \geq 0$).
• Parameter-space representation: Avoids wrapping effects seen in direct vertex/hyperplane descriptions, facilitating support function and analytic operations.

2. Algebraic Operations and Closure

CCGs are closed under major set operations essential for estimation and control:

• Affine Maps: $R \mathcal{Z} + t$ is a CCG with parameters $(R G, R c + t, A, b, \mathfrak{C})$.
• Minkowski Sum: $\mathcal{Z} \oplus \mathcal{W}$ gives $$([G_Z\; G_W], c_Z + c_W, \mathrm{blockdiag}(A_Z, A_W), [b_Z; b_W], \mathfrak{C}_Z \times \mathfrak{C}_W)$$.
• Generalized Intersection: For affine alignment $R$, the intersection is described with augmented parameters and constraints (Matias et al., 12 Jan 2026).

Notably, the parameter tuples $(G, c, A, b, \mathfrak{C})$ can be updated in closed-form after each operation, with the number of generators and constraints scaling additively.

3. Convex Hulls and Minimal Variable Growth

The convex hull of two CCGs $X, Y$ in $\mathbb{R}^n$ admits a closed-form CCG representation:

• A new generator variable $\xi_\lambda = \lambda - 1/2 \in [-1/2,1/2]$ is introduced.
• The resulting generator, center, constraints, and lifted generator sets are given by:

$$G_h = [G_x\;\;G_y\;\;c_x - c_y],\quad c_h = \tfrac{1}{2}(c_x + c_y)$$

$$A_h = \begin{bmatrix} A_x & 0 & -b_x \ 0 & A_y & b_y \end{bmatrix},\quad b_h = \begin{bmatrix} \tfrac{1}{2}b_x \ \tfrac{1}{2}b_y \end{bmatrix}$$

$$\mathfrak{C}_h = \left\{ \mathfrak{C}_x^{(\tau_x+1)}(\xi_x, \xi_\lambda;-1,0.5),\, \mathfrak{C}_y^{(\tau_y+1)}(\xi_y, \xi_\lambda;1,0.5) \right\}$$

This yields exactly $n_g^x + n_g^y + 1$ generators and $n_c^x + n_c^y$ constraints, which is minimal by results of Conforti–Di Summa–Faenza and Balas’ theorem on polytopal unions. Volume-minimality and absence of conservatism are ensured (Silvestre, 2023).

4. Complexity, Reduction, and Practical Computation

While CCGs permit exact, closed-form operations, the order (number of generators) can grow linearly with the number of convex hull or Minkowski sum operations and exponentially in linear parameter-varying (LPV) settings. To address this, ray-shooting order-reduction is employed:

• Directions $\{v_i\}_{i=1}^\gamma$ are sampled.
• For each direction, the support point $p_i = \arg\max_{x \in X} v_i^T x$ is computed.
• The outer-approximating polytope is constructed as $\{x: v_i^T x \leq v_i^T p_i\}$.
• Parameters of a new, smaller CCG are obtained by wrapping these support points (Silvestre, 2023).
• In benchmarks, for $\gamma \approx 50$, reduction runs in tens to hundreds of milliseconds per reduction.

This heuristic produces outer-approximations with volume errors comparable to established constrained zonotope (CZ) order-reduction methods, as implemented in CORA.

5. Applications in Estimation and Safety

CCGs have been deployed for exact set-valued state estimation under model and measurement uncertainty, such as in autonomous vehicle localization in GPS-denied environments:

• In set-valued finite-horizon estimation for uncertain linear systems, sequences of CCGs track the reachable set of the system state. Operations such as affine propagation, Minkowski addition (for process noise), and intersection with measurement sets are computed in closed form (Matias et al., 12 Jan 2026, Silvestre, 2023).
• In navigation and obstacle avoidance, CCGs are used to represent both agent and obstacle sets. The support for arbitrary convex body inflation, affine geometry, and generalized intersection is critical for representing complex agents and uncertain obstacles (Matias et al., 12 Jan 2026).
• Simulation evidence demonstrates up to 30% smaller volume in state estimates compared to CZ approaches for figure-8 maneuvers and 10–20% lower volume during spiral maneuvers. Computation per step typically ranges from 0.6–1.5 s in MATLAB + YALMIP + Mosek environments. The presence of many $\ell_2$ constraints can slow computation by 20–50% in later steps (Silvestre, 2023).

6. Integration with Control Barrier Functions

The indirect (parameter-space) description of CCGs enables their use in formal safety synthesis via Control Barrier Functions (CBF):

• For a CCG-valued set $\mathcal{O}(t)$, a corresponding CBF $h(p,t)$ is constructed as the minimum of a convex function $f(\eta)$ subject to $G\eta + c = p$, with $f(\eta)$ a smooth approximation of the tightest generator constraint (Matias et al., 12 Jan 2026). The structure is established via the Implicit Function Theorem, ensuring $h$ is well-defined and $C^1$ in $(p,t)$.
• This permits synthesis of QP-based safety filters for rigid-body and point-mass agents, in both single- and second-order dynamic regimes. Simulation benchmarks indicate typical CBF computation times of 3–8 ms per obstacle per step, supporting real-time deployment.

7. Advantages, Limitations, and Extensions

Advantages:

• Unification: CCGs subsume a wide class of convex set representations.
• Exactness: Operations such as convex hull are exact, non-conservative, and optimally minimal in terms of parameter growth.
• Expressivity for Uncertainty: Parameter-space coupling sharply reduces conservatism in state estimation under parametric and structural uncertainty.

Limitations:

• Complexity: Introduction of curved (norm-cone) constraints increases solver burden, especially for high-rate control tasks.
• Order Growth: Without reduction, the number of generators grows unbounded with repeated operations.
• Reduction Suboptimality: Ray-shooting reduction produces guaranteed outer-approximations but is not volume-optimal. There is no theoretical guarantee on minimality of the reduced order.

Extensions:

• Roundabout Constrained Convex Generators (RCGs) extend CCGs to represent singly-holed (multiply-connected) sets by subtracting an inner CCG from an outer CCG, with analogous closed-form algorithms for set operations, solver integration, and applications in safety-critical navigation (Xie et al., 10 Nov 2025).

A plausible implication is that future CCG research may target more efficient reductions, smarter order-management, and broader integration with reachability and formal verification tools across hybrid dynamic and non-affine systems.