---
title: 'Hybrid Zonotopes: Compact Nonconvex Sets'
url: https://www.emergentmind.com/topics/hybrid-zonotopes
type: topic
---

# Hybrid Zonotopes: Compact Nonconvex Sets

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 [2106.14831] [2304.02755] [2210.09902].

## 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 [2106.14831].

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 [2106.14831].

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 \([0,1]\) and binary factors in \(\{0,1\}\), while preserving the same center-generator-constraint architecture [2602.00325] [2512.11493].

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 [2106.14831] [2210.09902].

For a linear map \(R\),
\[
R\mathcal Z_h=\langle RG^c,RG^b,Rc,A^c,A^b,b\rangle.
\]
For two hybrid zonotopes \(\mathcal Z_{h,1}\) and \(\mathcal Z_{h,2}\), the Minkowski sum is obtained by concatenating generators and block-diagonalizing the constraint matrices:
\[
\mathcal Z_{h,1}\oplus\mathcal Z_{h,2}
=
\Bigl\langle
[G_1^c\;G_2^c],
[G_1^b\;G_2^b],
c_1+c_2,
\operatorname{blockdiag}(A_1^c,A_2^c),
\operatorname{blockdiag}(A_1^b,A_2^b),
[b_1;b_2]
\Bigr\rangle
\]
[2106.14831].

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 \(G^c\), \(G^b\), and \(c\) corresponding to discarded state components [2602.00325].

| Operation | Closed-form update | Representative cost |
|---|---|---|
| Linear image | Map \(c,G^c,G^b\) by \(A\) | \(O(n(n_g+n_b))\) |
| Minkowski sum | Concatenate generators, stack constraints | \(O(n_g^2+n_b^2+n_c^2)\) |
| Half-space intersection | Add one continuous generator and one constraint per inequality | \(O(n_i\,n\,(n_g+n_b))\) |
| Projection | Remove rows | \(O(n_g+n_b+n_c)\) |

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 [2602.00325]. By contrast, explicit union constructions do add binary generators and constraints, as in mode-union constructions for piecewise-affine reachability [2504.04362].

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 [2106.14831].

## 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:
\[
n_g(k)=n_g(0)+k(n_{g,u}+n_{g,w}+n_e),\quad
n_b(k)=n_b(0)+k(n_{b,u}+n_{b,w}),\quad
n_c(k)=n_c(0)+k(n_e)
\]
[2106.14831]. 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 [2210.09902] [2304.06827].

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 [2504.04362]. 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 [2504.04362].

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 \(0.029\) s per step, followed by Generalized Intersection at \(0.034\) s and Implicit Intersection at \(0.043\) s; all three were equivalent in the error bounds established in the paper [2504.04362].

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 [2210.09902] [2304.06827].

## 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 \(v_i\in[-a,a]\), the graph of a single ReLU unit \((v_i,x_i)\) with \(x_i=\max\{0,v_i\}\) can be represented exactly as a hybrid zonotope with
\[
n_g=4,\quad n_b=1,\quad n_c=2.
\]
Stacking across all neurons yields an exact graph representation of the network with
\[
n_g = 4\sum_{\ell=1}^L n_\ell,\quad
n_b = \sum_{\ell=1}^L n_\ell,\quad
n_c = 3\sum_{\ell=1}^L n_\ell
\]
[2304.02755].

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 [2303.10513]. Safety verification reduces to emptiness of an HZ intersection and can be posed as a mixed-integer linear program [2303.10513] [2210.03244].

For feedforward neural controllers, the reachability literature also stresses convex-relaxation quality. Relaxing binary factors from \(\{-1,1\}\) to \([-1,1]\) 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 [2210.03244].

The framework has since been extended to recurrent networks. For closed-loop ReLU-RNNs, state-pair sets \(\mathcal S_x(X,t)\subseteq\mathbb R^n\times\mathbb R^n\) are computed as hybrid zonotopes without unrolling. To trade exactness for scalability, unstable ReLUs are ranked by a triangle-area score
\[
\mathrm{score}(\alpha,\beta)=-\frac{\alpha\beta}{2},
\]
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 [2603.11547].

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 [2310.15426].

## 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 [2602.00325] 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, \(n_g=12\) continuous generators, \(n_b=11\) binary generators, and \(n_c=7\) linear constraints. Over a \(25\)-step horizon, the final mixed-integer quadratic program has \(192\) continuous variables and \(234\) binary variables and solves in about \(0.75\) s [2602.00325].

The same paper’s door-key example illustrates how temporal operators are tied directly to binary region-membership coordinates. The environment map at \(k=0\) is represented with \(n_g=24\), \(n_b=10\), and \(n_c=13\), 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 \(N=25\), the resulting MIQP has approximately \(150\) binary variables and solves in under \(1\) s, versus hundreds of binaries and seconds–minutes for a big-\(M\) encoding [2602.00325].

Linear Temporal Logic verification has likewise been formulated through backward reachability with hybrid zonotopes. In an autonomous parking example with state \(x=[x,y,v_x,v_y]\) and input \(u=[a_x,a_y]\), a hybrid-zonotope-based temporal logic tree is constructed in \(8.56\) s for a \(4\text{D}+2\text{D}\) problem and in \(3.94\) s for a \(2\)D-only version, while representing nonconvex, disjoint lane geometries natively [2404.03308].

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 [2602.17574].

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 [2304.07924]. Multi-sensor fusion for connected and automated vehicles goes further by adding a confidence coordinate: the fused set
\[
H
=
\bigcup_{\emptyset\neq S\subseteq\{1,\dots,n\}}
\Bigl(
\bigcap_{i\in S}\bar{\mathcal X}^i_t
\times
\Bigl\{\tfrac1n\sum_{i\in S}c^i_t\Bigr\}
\Bigr)
\]
is encoded as a single hybrid zonotope. In the reported implementations, average computation per time step was \(0.28\) s in MATLAB and \(94\) ms/step in a C++ real-vehicle experiment [2512.11493].

## 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 [2106.14831]. 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 [2602.00325].

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 [2504.04362]. The motion-planning paper explicitly lists “generator growth,” “constraint growth,” and “no direct support for general nonlinear maps” among the limitations of the representation [2602.00325]. 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:
\[
\mathrm{relax}(Z_h)=\mathrm{conv}(Z_h).
\]
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 [2503.17483].

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 \(E=R=I\), the representation recovers a hybrid zonotope [2506.13567]. 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-\(M\) 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 [2602.00325] [2503.17483] [2506.13567].

Source: https://www.emergentmind.com/topics/hybrid-zonotopes