Zonotope-Valued Utilities & Incomplete Preferences

• Zonotope-valued utilities are a set-valued model using Minkowski sums of intervals to encode multidimensional preferences with inherent uncertainty.
• They transform conflicting criteria into a geometric object whose structure captures trade-offs and interval incomparability in decision-making.
• Algorithmic methods like vertex enumeration and support evaluation render these geometric computations tractable despite complex set inclusion challenges.

Searching arXiv for recent and foundational papers on zonotope-valued utilities and related zonotope computation. Zonotope-valued utility is a set-valued representation of preference in which each alternative is assigned not a scalar or a single vector, but a zonotope in $\mathbb{R}^m$. In the formulation proposed for multidimensional incomplete preferences, the utility map has the form $U:X\to \mathcal K(\mathbb R^m)$, with $U(x)$ a compact convex zonotope generated as a Minkowski sum of interval contributions, and preference is represented by requiring the extended set difference $U(x)\ominus_e U(y)$ to lie in the non-negative orthant (Ramezanzadeh, 8 Jul 2025). This construction places utility theory in the broader geometry of zonotopes, which are Minkowski sums of finitely many line segments and admit generator, vertex, and support-function descriptions with unusually explicit algorithmic structure (Deza et al., 2019).

1. Conceptual basis and motivation

The framework is motivated by multidimensional incomplete preferences, where scalar-valued utility $u:X\to\mathbb R$ is too restrictive because it collapses conflicting criteria, partial trade-offs, and genuine incomparability into a single number (Ramezanzadeh, 8 Jul 2025). Standard vector utility $u:X\to\mathbb R^m$ preserves multiple criteria, but still assigns each alternative a single point. The zonotope-valued proposal targets a stronger form of incompleteness: evaluations may be interval-like within each dimension, trade-off rates need not be fully fixed, and alternatives may remain incomparable because different dimensions pull in different directions.

The set-valuedness is attributed to several sources. One is structural incompleteness: criteria conflict, interval-order comparisons are only partial, and aggregation across dimensions remains underdetermined. Another is non-probabilistic uncertainty, such as imprecise utility evaluation due to incomplete information about criteria weights. The paper also mentions probabilistic uncertainty arising from stochastic decision environments, although the formal development is centered more clearly on structural incompleteness and epistemic imprecision than on an explicit stochastic model (Ramezanzadeh, 8 Jul 2025).

A zonotope is used because it is a compact convex set generated by additive combination of intervals. This makes it suitable for combining several interval-order components without forcing pointwise collapse. The result is not merely a family of admissible scores, but a geometric object whose shape encodes trade-offs, uncertainty, and incomparability. In this sense, zonotope-valued utility extends interval-valued utility from one dimension to a Minkowski-sum geometry in several dimensions (Ramezanzadeh, 8 Jul 2025).

2. Mathematical construction

The formal representation begins with a decomposition of the preorder $\succeq$ into finitely many interval orders,

$\succeq=\bigcap_{k=1}^m \succeq_k.$

For each component $\succeq_k$, there are endpoint functions

$\underline u_k,\overline u_k:X\to \mathbb R$

such that

$U:X\to \mathcal K(\mathbb R^m)$0

This yields an interval

$U:X\to \mathcal K(\mathbb R^m)$1

for each alternative and each component (Ramezanzadeh, 8 Jul 2025).

The zonotope-valued utility is then defined by

$U:X\to \mathcal K(\mathbb R^m)$2

where $U:X\to \mathcal K(\mathbb R^m)$3 are basis vectors with strictly nonnegative components (Ramezanzadeh, 8 Jul 2025). Each interval-order component contributes one segment direction, and the whole utility object is their Minkowski sum.

This fits the standard zonotope description. A zonotope is, in general, a Minkowski sum of finitely many line segments,

$U:X\to \mathcal K(\mathbb R^m)$4

or equivalently a box image under a linear map (Deza et al., 2019). If $U:X\to \mathcal K(\mathbb R^m)$5, then

$U:X\to \mathcal K(\mathbb R^m)$6

so the utility zonotope can be rewritten as a translated generator form (Ramezanzadeh, 8 Jul 2025). This is the point at which the decision-theoretic construction meets the standard computational geometry of zonotopes.

The minimal ambient dimension is not arbitrary. The representation theorem identifies it with the interval dimension: $U:X\to \mathcal K(\mathbb R^m)$7 the smallest number of interval orders whose intersection yields $U:X\to \mathcal K(\mathbb R^m)$8 (Ramezanzadeh, 8 Jul 2025). The framework is therefore not just geometric, but order-theoretically calibrated.

3. Order representation, extended set difference, and axiomatization

Preference is represented by an orthant condition on the extended set difference: $U:X\to \mathcal K(\mathbb R^m)$9 The operator $U(x)$0 is used because ordinary Minkowski difference is not generally well behaved on compact convex sets. For $U(x)$1,

$U(x)$2

where

$U(x)$3

and $U(x)$4 is the Hausdorff metric (Ramezanzadeh, 8 Jul 2025). The appendix result establishes that $U(x)$5 for all compact convex $U(x)$6, so the extended difference is always defined.

The induced order on compact convex sets is

$U(x)$7

and the preference relation on alternatives is the pullback of this set order (Ramezanzadeh, 8 Jul 2025). Strict preference is given in the usual way by

$U(x)$8

Geometrically, the paper characterizes $U(x)$9 by either of two equivalent conditions: $U(x)\ominus_e U(y)$0 lies strictly to the northeast of $U(x)\ominus_e U(y)$1, or there exists a separating hyperplane with normal $U(x)\ominus_e U(y)$2, $U(x)\ominus_e U(y)$3, such that

$U(x)\ominus_e U(y)$4

with at least one strict inequality (Ramezanzadeh, 8 Jul 2025).

The representation theorem is axiomatic. For countable $U(x)\ominus_e U(y)$5, the assumptions are interval-order decomposition, non-degeneracy, strict preference consistency, and separability. For uncountable $U(x)\ominus_e U(y)$6, separability is replaced by numerical representability of each interval order. Under these conditions there exists a zonotope-valued utility map $U(x)\ominus_e U(y)$7 such that

$U(x)\ominus_e U(y)$8

with

$U(x)\ominus_e U(y)$9

and $u:X\to\mathbb R$0 equal to the interval dimension (Ramezanzadeh, 8 Jul 2025).

The construction also has a continuity statement: if $u:X\to\mathbb R$1 is Hausdorff and each endpoint map $u:X\to\mathbb R$2 is continuous, then $u:X\to\mathbb R$3 is continuous in the Hausdorff metric (Ramezanzadeh, 8 Jul 2025).

4. Geometric interpretation

The geometry of a zonotope-valued utility is explicit. For a fixed alternative $u:X\to\mathbb R$4, the set

$u:X\to\mathbb R$5

contains all aggregate evaluations obtained by selecting one admissible value from each interval-order component and summing along the generator directions (Ramezanzadeh, 8 Jul 2025). Each segment $u:X\to\mathbb R$6 is a one-dimensional range of admissible contribution; the zonotope is the additive envelope of those contributions.

The choice of generators $u:X\to\mathbb R$7 controls how criteria are geometrically combined. With the standard basis in two dimensions, $u:X\to\mathbb R$8 is a rectangle. With a different positive basis such as $u:X\to\mathbb R$9, $u:X\to\mathbb R^m$0, the same interval data produces a parallelogram. In three dimensions, the standard basis gives a rectangular prism, while nontrivial positive generators such as

$u:X\to\mathbb R^m$1

produce a more complex polyhedron (Ramezanzadeh, 8 Jul 2025). The paper interprets these changes as encoding different weighting and interaction structures among dimensions.

Incomparability is represented geometrically rather than residually. If neither

$u:X\to\mathbb R^m$2

holds, the alternatives are incomparable (Ramezanzadeh, 8 Jul 2025). The paper’s overlapping-zonotope example shows that incomparability arises when the utility regions overlap or interpenetrate in a way that precludes northeast dominance and prevents separation by a hyperplane with strictly positive normal.

This geometric reading aligns zonotope-valued utility with a general fact about zonotopes: faces are exposed by common maximizing directions, and linear functionals interact with generators in a separable way (Deza et al., 2019). In the utility setting, this means that changing a scalarization direction can move the maximizer from one face of a utility zonotope to another, while the face structure records which component contributions are active or indifferent.

5. Computational structure and algorithmics

Once utility sets are represented by generators, several computations become unusually explicit. For a zonotope

$u:X\to\mathbb R^m$3

maximizing a linear functional decomposes generatorwise: $u:X\to\mathbb R^m$4 and one optimizer is

$u:X\to\mathbb R^m$5

The centered form yields the sign rule

$u:X\to\mathbb R^m$6

when no degeneracy occurs. The paper states directly that “linear optimization on a zonotope is linear time solvable in the number of its generators” (Deza et al., 2019). For zonotope-valued utility interpretations, this means any linear preference or scoring functional can be maximized by generatorwise sign tests.

The same paper gives an exact vertex criterion. For $u:X\to\mathbb R^m$7 and

$u:X\to\mathbb R^m$8

one has

$u:X\to\mathbb R^m$9

equivalently via a linear feasibility system in a supporting direction $\succeq$0 (Deza et al., 2019). This yields an output-sensitive vertex-enumeration algorithm with complexity

$\succeq$1

refined in the proof as

$\succeq$2

for a $\succeq$3-dimensional zonotope with $\succeq$4 vertices and $\succeq$5 generators (Deza et al., 2019). The same toolkit recovers generators from vertices, decides whether a vertex-described polytope is a zonotope, and computes the greatest zonotopal summand $\succeq$6 of a general polytope (Deza et al., 2019).

A complementary randomized approach samples vertices through sign patterns. For a generator matrix $\succeq$7, the zonotope is

$\succeq$8

and if $\succeq$9 has no zero component, then

$\succeq=\bigcap_{k=1}^m \succeq_k.$0

is a vertex. The inverse image of a vertex is the interior of its normal cone, so Gaussian sampling recovers vertices with probability equal to the Gaussian measure of their normal cones (Stinson et al., 2016). This favors “sharp” vertices with large normal cones and supports approximation by the convex hull of recovered vertices when complete enumeration is impractical.

Comparison of utility sets by inclusion is substantially harder. Zonotope Non-Containment is $\succeq=\bigcap_{k=1}^m \succeq_k.$1-hard with respect to the ambient dimension $\succeq=\bigcap_{k=1}^m \succeq_k.$2, and under ETH it is not solvable in

$\succeq=\bigcap_{k=1}^m \succeq_k.$3

time for any computable $\succeq=\bigcap_{k=1}^m \succeq_k.$4 (Froese et al., 26 Sep 2025). The paper presenting this result does not discuss utilities directly, but a plausible implication for zonotope-valued utility is that exact comparison by set inclusion,

$\succeq=\bigcap_{k=1}^m \succeq_k.$5

is unlikely to be fixed-parameter tractable in dimension. The same paper gives the baseline enumeration algorithm

$\succeq=\bigcap_{k=1}^m \succeq_k.$6

for zonotope containment and $\succeq=\bigcap_{k=1}^m \succeq_k.$7-Max on Zonotopes, and a randomized order-reduction theorem producing $\succeq=\bigcap_{k=1}^m \succeq_k.$8 with

$\succeq=\bigcap_{k=1}^m \succeq_k.$9

such that, with high probability,

$\succeq_k$0

(Froese et al., 26 Sep 2025). Approximation, rather than exact global comparison, is therefore central once dimension grows.

6. Related zonotopal frameworks

Zonotope-valued utility belongs to a wider family of zonotopal set representations, but it is not identical to them. A nearby line of work studies scalar valuations of random zonotopes. If

$\succeq_k$1

is a random zonotope generated by i.i.d. segments and $\succeq_k$2, then

$\succeq_k$3

and normalized valuations satisfy a central limit theorem (Schneider, 2022). This is directly relevant to scalar utility functionals on zonotopes, but it is not itself a theory of zonotope-valued utility.

Another adjacent framework is the hybrid zonotope, defined as a set with continuous generators, binary generators, and equality constraints on latent factors. It is used to over-approximate graphs of nonlinear mappings through automated functional decomposition, with applications to LSTM networks and discrete hybrid automata (Glunt et al., 19 Mar 2025). This is a set-valued computational representation rather than a preference representation, but it shows how zonotopal objects can encode mixed discrete-continuous structure beyond ordinary convex zonotopes.

Applied work in localization and estimation uses related ideas. Mosaic Zonotope Shadow Matching begins with zonotopes and constrained zonotopes for GNSS shadow geometry, then passes to a weighted mosaic of feasible polytopes carrying an exact shadow matching distribution and confidence collections (Neamati et al., 2022). Interval estimation for bounded Jacobian nonlinear systems combines a Luenberger-like observer with zonotope propagation for the estimation error, producing a point-valued center together with a feasible zonotope set and an interval fusion rule (Xu et al., 19 Sep 2025). These constructions are not utility theory, but they illustrate how zonotopal sets function as primary value objects in risk-aware inference and control.

The distinction is important. Zonotope-valued utility, in the strict sense of incomplete preference representation, assigns zonotopes to alternatives and reconstructs $\succeq_k$4 through extended set difference and orthant dominance (Ramezanzadeh, 8 Jul 2025). Scalar valuations on zonotopes, hybrid zonotopes for graph approximation, and zonotope-derived belief mosaics are mathematically adjacent but conceptually different.

7. Scope, limitations, and open directions

The framework has clear scope conditions. It relies on a finite interval-order decomposition, and its representation theorem identifies the minimal dimension with interval dimension (Ramezanzadeh, 8 Jul 2025). This gives the model structural discipline, but also limits it to preorders that admit such a decomposition. The paper also notes that the crucial comparison criterion depends on the extended set difference $\succeq_k$5, which is more sophisticated than ordinary vector comparison and raises an immediate computational question: how to compute $\succeq_k$6 efficiently in applied settings (Ramezanzadeh, 8 Jul 2025).

A second limitation concerns uncertainty. The framework explicitly accommodates non-probabilistic uncertainty and structural incompleteness, while probabilistic uncertainty is mentioned as a source of set-valued payoffs but is not developed into a full stochastic decision model (Ramezanzadeh, 8 Jul 2025). This suggests a distinction between the current theory, which is primarily geometric and order-theoretic, and possible future stochastic or dynamic extensions.

On the computational side, exact comparison of zonotope-valued utilities by containment inherits strong worst-case barriers from zonotope containment complexity (Froese et al., 26 Sep 2025). This shifts attention toward tractable subproblems: support evaluation in fixed directions, output-sensitive vertex enumeration, randomized recovery of influential vertices, and low-order approximation. The direct paper on zonotope-valued utility also points toward further questions about estimating interval-order components from data, developing dynamic or social-choice versions of the model, and interpreting the generator directions $\succeq_k$7 normatively (Ramezanzadeh, 8 Jul 2025).

Taken together, these results place zonotope-valued utility at the intersection of decision theory, order theory, and polyhedral computation. Its distinctive claim is not merely that preferences may be multidimensional, but that incompleteness itself can be represented as zonotopal geometry: $\succeq_k$8 Within that geometry, trade-offs, interval uncertainty, boundary structure, and incomparability become properties of convex sets rather than anomalies of scalarization (Ramezanzadeh, 8 Jul 2025).