Piecewise Affine Models

• Piecewise affine models are mappings defined on finite-dimensional spaces using a finite union of convex polyhedral regions, each governed by an affine function.
• They provide a hybrid framework that combines the simplicity of linear models with the power to capture nonlinearity and discontinuities in real-world systems.
• These models underpin effective numerical optimization and control strategies by leveraging sup/inf representations and vector lattice properties.

Piecewise affine (PWA) models constitute a mathematically rigorous framework for representing mappings between finite-dimensional normed spaces by finite unions of convex polyhedral domains, each supporting an affine mapping. They are foundational in nonsmooth analysis, hybrid system theory, numerical optimization, and constructive approximation, bridging the tractability of linear models with the expressive power needed to capture nonlinearity and discontinuity in real-world systems.

1. Formal Definition and Characterizations

Let $X$ and $Y$ be finite-dimensional normed spaces. A mapping $P \colon X \to Y$ is called piecewise affine if there exists a finite family of convex polyhedral sets $\{M_1, \ldots, M_k\}$ covering $X$ and affine mappings $A_1, \ldots, A_k$ such that: $P(x) = A_i(x)$ for $x \in M_i$, and $X = \bigcup_{i=1}^k M_i$. Each $A_i$ is affine: $A_i(x) = L_i(x) + b_i$ for a linear map $L_i$ and vector $b_i$.

Two central characterizations appear:

• Geometric: The graph of a PWA mapping is the union of finitely many convex polyhedral subsets of $X \times Y$. If $P$ is PWA, then

$$\mathrm{graph}(P) = \bigcup_{i=1}^k \{(x, y) \mid x \in M_i,\, y = A_i(x)\}.$$

• Analytic: For any partial order on $Y$ defined by a polyhedral convex cone, both the $\preceq$-epigraph and hypograph of $P$ can be written as the union of finitely many convex polyhedral sets in $X \times Y$:

$$\mathrm{epi}_\preceq P = \bigcup_{i=1}^k \mathrm{epi}_\preceq A_i, \qquad \mathrm{hyp}_\preceq P = \bigcup_{i=1}^k \mathrm{hyp}_\preceq A_i.$$

For scalar-valued $p \colon X \to \mathbb{R}$, this reduces to $\mathrm{epi}\, p = \{(x, a) \mid p(x) \leq a\}$ and $\mathrm{hyp}\, p = \{(x, a) \mid p(x) \geq a\}$, both polyhedral.

The paper demonstrates the mutual equivalence between the existence of a covering by convex polyhedral domains and the existence of a solid polyhedral partition with disjoint relative interiors, on each of which $P$ is affine.

2. Geometric and Analytical Properties

Geometrically, PWA mappings are constructed by “gluing together” a finite number of affine mappings across polyhedral regions. Although the global graph may be non-convex, its piecewise structure ensures polyhedrality.

Analytically:

• Both the epigraphs and hypographs with respect to a polyhedral order admit a polyhedral description.
• The PWA structure extends directly: convex polyhedral sets supporting affine restrictions allow direct computation of optimization quantities, closure operations, and support for variational analysis.

An illustrative formula for a PWA mapping is:

$P(x) = A_i(x) \quad \text{if} \quad x \in M_i.$

3. Vector Lattice Structure and Algebraic Properties

When $Y$ is a vector lattice (i.e., ordered by a minihedral cone) or equivalently a vector lattice space, the set $F(X, Y)$ of all mappings under pointwise operations is itself a vector lattice.

Key structural result: the set of all PWA mappings $PA(X, Y)$ forms the smallest vector sublattice of $F(X, Y)$ containing all affine mappings. That is, every PWA mapping can be written via finite combinations of pointwise $\sup$ and $\inf$ of affine mappings.

For scalar-valued mappings, this translates to:

• The pointwise $\max$ and $\min$ (supremum and infimum) of affine functions (e.g., $\max\{x_1, x_2\}$, $\min\{x_1, x_2\}$) are PWA and generate the full lattice via finite operations.

This vector lattice structure underpins many approximation, optimization, and control-theoretic algorithms leveraging sup/inf combinations and provides lattice-theoretic convergence and order-theoretic compactness properties.

4. Convexity, Ordering, and Functional Representation

Convexity interacts deeply with the order structure. When $Y$ is ordered by a polyhedral convex (minihedral) cone, a mapping $P \colon X \to Y$ is order-convex if for all $x, y \in X$ and $\lambda \in [0, 1]$,

$$P(\lambda x + (1-\lambda)y) \leq \lambda P(x) + (1-\lambda)P(y).$$

Key facts:

• Any convex PWA mapping (with respect to the ordering) is representable as the least upper bound (pointwise supremum) of finitely many affine mappings:

$P(x) = \sup\{A_j(x) : j \in J\}.$

• For general (not necessarily convex) mappings, representations involve combinations of inf–sup or sup–inf expressions.

This provides explicit, constructive descriptions of PWA mappings in terms of a finite number of affine “generators,” critical for theoretical analysis and computation.

Example: For $F(x_1, x_2) = \max\{x_1, x_2\}$, the function is PWA and convex, being the supremum of two linear mappings.

5. Approximation and Numerical Implications

A major result (Theorem 5.2) is that every continuous mapping on a compact set can be uniformly approximated by PWA maps. This denseness is crucial in nonlinear analysis, finite element methods, optimization, and computational mathematics. The finite representation by polyhedral pieces enables direct discretization, tractable numerical processing, and straightforward exploitation of sparsity.

Analytical representations via finite sup/inf build direct bridges to algorithmic optimization (LP, QP) and hybrid systems analysis, where such “mode-switching” affine structures naturally arise.

PWA models are thus commonly employed to approximate cost functions in optimization, Lyapunov functions in control, activation functions in deep learning, and more generally, nonsmooth or hybrid system behaviors.

6. Applications, Theoretical Importance, and Outlook

PWA mappings feature across differential games, nonsmooth optimization, control design, numerical approximation, and machine learning. Notably:

• In differential game theory and nonsmooth systems, the PWA structure enables algorithmic synthesis and guarantees of tractability.
• The lattice-theoretic and geometric properties suggest robust computational methods for large-scale solvers, leveraging polyhedral and piecewise-structural sparsity.
• Denseness in continuous mappings [Corollary 5.1] undergirds foundational results in approximation theory with direct practical implications for model reduction, control synthesis, and realization theory.

By uniting geometrical, analytical, and order-theoretic perspectives, PWA models admit rigorous treatment and provide a precise language for reasoning about complex, multi-modal, and hybrid systems. Their substantial role in both theory and practice makes them a central object in applied mathematics, engineering, and computational analysis.