Hybrid Zonotopes: Compact Nonconvex Sets

• Hybrid zonotopes are mixed-integer set representations that extend zonotopes by incorporating binary generators and affine constraints to capture nonconvex unions of convex sets.
• They support closed-form operations such as linear maps, Minkowski sums, intersections, and projections, which are critical in reachability analysis and verification of hybrid systems.
• Their compact design offers a practical balance between modeling complexity and precision, benefiting applications in motion planning, temporal logic, neural network verification, and set-valued estimation.

Searching arXiv for relevant hybrid zonotope papers to ground the article. Hybrid zonotopes are mixed-integer set representations that extend zonotopes and constrained zonotopes by adding binary generators and affine coupling constraints on continuous and binary factors. Their defining role in the literature is to represent nonconvex, and often disjoint, unions of convex sets within a single algebraic object while preserving closed-form formulas for fundamental set operations. Introduced as a reachability representation for mixed logical dynamical systems, they have since been used for exact reachable-set computation in linear hybrid and ReLU-network settings, and for tight outer-approximation in nonlinear dynamics, smooth neural activations, temporal-logic planning, and set-valued estimation (Bird et al., 2021, Ortiz et al., 2023, Siefert et al., 2022).

1. Formal definition and representational semantics

A standard hybrid zonotope in hybrid constrained-generator form is written as

$$\mathcal Z_h = \Bigl\{ G^c\xi^c + G^b\xi^b + c \;\Big|\; \xi^c\in[-1,1]^{n_g},\; \xi^b\in\{-1,1\}^{n_b},\; A^c\xi^c + A^b\xi^b = b \Bigr\},$$

or compactly

$\mathcal Z_h=\langle G^c,G^b,c,A^c,A^b,b\rangle.$

Here $c$ is the center, $G^c$ the continuous-generator matrix, $G^b$ the binary-generator matrix, and $A^c,A^b,b$ define affine equality constraints on the factors (Bird et al., 2021).

A central structural property is that a hybrid zonotope compactly encodes an implicit union of up to $2^{n_b}$ constrained zonotopes. For each binary assignment $\xi^b\in\{-1,1\}^{n_b}$, one obtains a constrained-zonotope slice; the hybrid zonotope is the union of all nonempty such slices. This is the basic mechanism by which nonconvexity enters the representation without explicit enumeration of all branches (Bird et al., 2021).

Several special cases are immediate. If $n_b=0$, the representation reduces to a constrained zonotope. If, in addition, the affine constraints are absent, it reduces to an ordinary zonotope. Conversely, works on motion planning and sensor fusion also use equivalent $0/1$-valued conventions, with continuous factors in $\mathcal Z_h=\langle G^c,G^b,c,A^c,A^b,b\rangle.$0 and binary factors in $\mathcal Z_h=\langle G^c,G^b,c,A^c,A^b,b\rangle.$1, while preserving the same center-generator-constraint architecture (Thompson et al., 30 Jan 2026, Narri et al., 12 Dec 2025).

This representational flexibility explains why hybrid zonotopes appear in settings where convexity is too restrictive but full polyhedral branching is too expensive. They can encode unions of polytopes, piecewise-affine mode logic, graphs of ReLU activations, and temporal-logic region memberships in one object.

2. Algebraic structure and closure properties

The operational appeal of hybrid zonotopes is that the basic set operations needed in reachability and verification admit explicit algebraic updates. In particular, they are closed under linear maps, Minkowski sums, generalized intersections, half-space intersections, Cartesian products, and projections (Bird et al., 2021, Siefert et al., 2022).

For a linear map $\mathcal Z_h=\langle G^c,G^b,c,A^c,A^b,b\rangle.$2,

$\mathcal Z_h=\langle G^c,G^b,c,A^c,A^b,b\rangle.$3

For two hybrid zonotopes $\mathcal Z_h=\langle G^c,G^b,c,A^c,A^b,b\rangle.$4 and $\mathcal Z_h=\langle G^c,G^b,c,A^c,A^b,b\rangle.$5, the Minkowski sum is obtained by concatenating generators and block-diagonalizing the constraint matrices: $\mathcal Z_h=\langle G^c,G^b,c,A^c,A^b,b\rangle.$6 (Bird et al., 2021).

Intersection is equally important. Generalized intersection enforces compatibility between two HZ parameterizations through additional affine constraints, while half-space intersection augments the factor-space description with slack-related continuous generators and new constraints. Projection onto selected coordinates is performed by deleting rows of $\mathcal Z_h=\langle G^c,G^b,c,A^c,A^b,b\rangle.$7, $\mathcal Z_h=\langle G^c,G^b,c,A^c,A^b,b\rangle.$8, and $\mathcal Z_h=\langle G^c,G^b,c,A^c,A^b,b\rangle.$9 corresponding to discarded state components (Thompson et al., 30 Jan 2026).

Operation	Closed-form update	Representative cost
Linear image	Map $c$0 by $c$1	$c$2
Minkowski sum	Concatenate generators, stack constraints	$c$3
Half-space intersection	Add one continuous generator and one constraint per inequality	$c$4
Projection	Remove rows	$c$5

For these four operations, the motion-planning literature emphasizes a notable point: no new binary variables are introduced by affine image, Minkowski sum, half-space intersection, or projection; the binary variables remain associated with region membership or logical truth values rather than with the mechanics of set propagation itself (Thompson et al., 30 Jan 2026). By contrast, explicit union constructions do add binary generators and constraints, as in mode-union constructions for piecewise-affine reachability (Xie et al., 6 Apr 2025).

A common misconception is that hybrid zonotopes merely append integer variables to a zonotope. The literature is more specific: the essential feature is the joint use of binary generators and affine coupling constraints, which turns the representation into an implicit union-of-constrained-zonotopes calculus rather than a purely continuous convex set model (Bird et al., 2021).

3. Reachability analysis for hybrid, nonlinear, and data-driven systems

The original reachability result concerns discrete-time mixed logical dynamical systems. If the current state set and admissible input sets are hybrid zonotopes, then the exact one-step reachable set is again a hybrid zonotope, and the representation complexity grows linearly with time: $c$6 (Bird et al., 2021). This replaces explicit branch growth in the number of convex pieces by linear growth in HZ description size.

For nonlinear systems, the dominant pattern is different. Hybrid zonotopes are used with state-update sets, special ordered set approximations, and functional decomposition to obtain tight outer-approximations of open-loop and closed-loop successor sets. The central identities compute a successor by intersecting an HZ representation of the state-update graph with the current state-input set and then projecting onto next-state coordinates. If the state-update set is over-approximated, the resulting successor set is also an over-approximation (Siefert et al., 2022, Siefert et al., 2023).

The data-driven literature extends this approach to piecewise-affine systems identified from noisy measurements. For each mode, a model-set is built from input-state data, the current reachable hybrid zonotope is intersected with the polyhedral cell for that mode, the data-driven model set is applied, and the resulting modewise reachable sets are unified into a single hybrid zonotope. Under the full-row-rank hypothesis on data, the true reachable set is contained in the data-driven reachable set at each step (Xie et al., 6 Apr 2025). An important motivation is boundary handling: hybrid zonotopes can jump across mode boundaries without splitting into separate zonotopes for each mode sequence, which reduces conservatism at piecewise-affine guard surfaces (Xie et al., 6 Apr 2025).

The same article develops three mathematically equivalent measurement-update schemes for noisy input-output data: Reverse-Mapping, Implicit Intersection, and Generalized Intersection. Empirically, Reverse-Mapping was fastest, with median $c$7 s per step, followed by Generalized Intersection at $c$8 s and Implicit Intersection at $c$9 s; all three were equivalent in the error bounds established in the paper (Xie et al., 6 Apr 2025).

These results delimit the exactness frontier. Exact HZ propagation is established for linear hybrid and certain piecewise-linear settings; nonlinear settings typically rely on graph over-approximations, SOS envelopes, or functional decomposition, with soundness but not exactness (Siefert et al., 2022, Siefert et al., 2023).

4. Neural networks, piecewise-linear controllers, and closed-loop verification

One of the most consequential later developments is the exact encoding of feed-forward fully connected ReLU networks as hybrid zonotopes. Over a bounded pre-activation interval $G^c$0, the graph of a single ReLU unit $G^c$1 with $G^c$2 can be represented exactly as a hybrid zonotope with

$G^c$3

Stacking across all neurons yields an exact graph representation of the network with

$G^c$4

(Ortiz et al., 2023).

This result underlies exact forward and backward reachability for neural feedback systems with ReLU controllers. In the backward-reachability formulation, the graph of the controller and the target set are assembled with the plant dynamics into a single HZ; the resulting one-step predecessor is again a hybrid zonotope in closed form (Zhang et al., 2023). Safety verification reduces to emptiness of an HZ intersection and can be posed as a mixed-integer linear program (Zhang et al., 2023, Zhang et al., 2022).

For feedforward neural controllers, the reachability literature also stresses convex-relaxation quality. Relaxing binary factors from $G^c$5 to $G^c$6 yields a constrained zonotope equal to the convex hull of the hybrid zonotope, and the resulting formulation is described as the tightest convex relaxation for the reachable sets of the neural feedback system (Zhang et al., 2022).

The framework has since been extended to recurrent networks. For closed-loop ReLU-RNNs, state-pair sets $G^c$7 are computed as hybrid zonotopes without unrolling. To trade exactness for scalability, unstable ReLUs are ranked by a triangle-area score

$G^c$8

and only a fixed number of high-score units are preserved exactly; the remainder use triangle relaxations. Exact reachability is recovered as a special case when the binary budget is large enough (Zhang et al., 12 Mar 2026).

Hybrid zonotopes also exactly represent continuous piecewise-linear explicit MPC laws and the graphs of ReLU networks in software form. The zonoLAB toolbox exposes these constructions through the classes zono, conZono, and hybZono, with operations such as plus, mtimes, and, union, projection, and cartProd, and a default GUROBI-based MILP interface (Koeln et al., 2023).

5. Temporal logic, motion planning, and estimation

In temporal-logic planning, hybrid zonotopes serve as a geometric substrate for encoding atomic propositions and their evolution over time. In motion planning with Metric Temporal Logic, the method of (Thompson et al., 30 Jan 2026) uses reachability analysis to implicitly express the set of states satisfying an MTL specification and then optimizes over that representation. The hybrid zonotope map encoding in the time-varying traveling-salesperson scenario uses, at each time step, $G^c$9 continuous generators, $G^b$0 binary generators, and $G^b$1 linear constraints. Over a $G^b$2-step horizon, the final mixed-integer quadratic program has $G^b$3 continuous variables and $G^b$4 binary variables and solves in about $G^b$5 s (Thompson et al., 30 Jan 2026).

The same paper’s door-key example illustrates how temporal operators are tied directly to binary region-membership coordinates. The environment map at $G^b$6 is represented with $G^b$7, $G^b$8, and $G^b$9, and the requirement “do not enter door 1 until key 1 visited” is expressed by coupling the corresponding binary coordinates through MTL “until” constraints. For horizon $A^c,A^b,b$0, the resulting MIQP has approximately $A^c,A^b,b$1 binary variables and solves in under $A^c,A^b,b$2 s, versus hundreds of binaries and seconds–minutes for a big-$A^c,A^b,b$3 encoding (Thompson et al., 30 Jan 2026).

Linear Temporal Logic verification has likewise been formulated through backward reachability with hybrid zonotopes. In an autonomous parking example with state $A^c,A^b,b$4 and input $A^c,A^b,b$5, a hybrid-zonotope-based temporal logic tree is constructed in $A^c,A^b,b$6 s for a $A^c,A^b,b$7 problem and in $A^c,A^b,b$8 s for a $A^c,A^b,b$9D-only version, while representing nonconvex, disjoint lane geometries natively (Hadjiloizou et al., 2024).

Planning for piecewise-affine hybrid systems has been pushed further by combining hybrid zonotopes with a mixed-integer ADMM heuristic. In that setting, the representation is used not only for reachability but also for optimization over reachable trajectories, with sharp and condensed union identities affecting convex-relaxation quality and memory usage. The reported application is a combined behavior and motion planning scenario for autonomous driving on embedded hardware (Robbins et al., 19 Feb 2026).

Estimation problems use hybrid zonotopes in a parallel manner. Set-valued state estimation for nonlinear systems constructs HZ over-approximations of dynamics and measurement graphs and alternates prediction and correction through generalized intersections and projections, with memory complexity growing linearly in time (Siefert et al., 2023). Multi-sensor fusion for connected and automated vehicles goes further by adding a confidence coordinate: the fused set

$2^{n_b}$0

is encoded as a single hybrid zonotope. In the reported implementations, average computation per time step was $2^{n_b}$1 s in MATLAB and $2^{n_b}$2 ms/step in a C++ real-vehicle experiment (Narri et al., 12 Dec 2025).

6. Complexity growth, sharpness, limitations, and extensions

The compactness of hybrid zonotopes is relative rather than absolute. The foundational reachability result emphasizes linear growth of HZ description size with time, even when the underlying nonconvex reachable set may correspond to exponentially many convex pieces (Bird et al., 2021). However, several later works document the remaining bottlenecks. Generator growth occurs because Minkowski sums and Cartesian products concatenate generators; constraint growth occurs because each half-space or graph-intersection step adds affine constraints and often new continuous generators; and long horizons can still produce large mixed-integer objects (Thompson et al., 30 Jan 2026).

The nonlinear and data-driven literatures therefore present a more cautious picture than early exact MLD reachability might suggest. The piecewise-affine data-driven paper reports exponential growth in computation time with the number of steps in practice, despite the improved treatment of mode transitions (Xie et al., 6 Apr 2025). The motion-planning paper explicitly lists “generator growth,” “constraint growth,” and “no direct support for general nonlinear maps” among the limitations of the representation (Thompson et al., 30 Jan 2026). A common misconception is therefore that hybrid zonotopes remove combinatorial complexity; the literature supports a narrower claim that they compress and regularize it.

A distinct line of work formalizes convex-relaxation quality through the notion of sharpness. A hybrid zonotope is sharp when its convex relaxation equals its convex hull: $2^{n_b}$3 Sharpness is important because branch-and-bound algorithms rely on convex relaxations, and tighter relaxations improve convergence. Affine maps, Minkowski sums, Cartesian products, and certain union constructions preserve sharpness, while the reformulation-linearization technique can be used to produce a sharp realization of a non-sharp hybrid zonotope (Glunt et al., 21 Mar 2025).

The ecosystem has also expanded beyond classical HZs. Hybrid Polynomial Zonotopes attach polynomial exponents to hybrid generators and strictly generalize constrained polynomial zonotopes, constrained zonotopes, and hybrid zonotopes; when $2^{n_b}$4, the representation recovers a hybrid zonotope (Xie et al., 16 Jun 2025). This suggests a current research trajectory: preserving the mode-logic expressiveness of HZs while improving tightness under higher-order nonaffine maps.

Taken together, the literature presents hybrid zonotopes as a specific compromise. They are more expressive than zonotopes and constrained zonotopes for nonconvex and logic-rich problems, more algebraically tractable than explicit unions of polytopes, and frequently tighter than big-$2^{n_b}$5 or naive convex encodings. Their limitations are equally clear: binary and constraint growth remain substantive, exactness is application-dependent, and practical deployments rely on reduction, relaxation, or solver-aware reformulations rather than on the bare representation alone (Thompson et al., 30 Jan 2026, Glunt et al., 21 Mar 2025, Xie et al., 16 Jun 2025).