Configuration-Constrained PWA Approximation

Updated 16 December 2025

The paper demonstrates that configuration-constrained PWA approximation yields error-certifiable surrogates by enforcing fixed partition templates and convex optimization.
It constrains region shapes, adjacency structures, and affine parameters to manage complexity and ensure computational tractability.
It enables robust applications in control, optimization, and learning by balancing approximation fidelity with explicit performance and safety guarantees.

A configuration-constrained piecewise-affine (PWA) approximation is an approach to approximating nonlinear multivariate functions by PWA functions, wherein the region partitioning, adjacency structure, or implementation complexity (e.g., fixed templates, breakpoints, or architecture) is a priori constrained. This paradigm enables error-certifiable approximations under explicit complexity limits, tractable solution of the approximation problem via convex or integer programming, and certified guarantees on the error or performance of the resulting PWA surrogate. It is increasingly central in control, optimization, learning, and embedded implementation contexts.

1. Foundational Principles and Problem Formulation

A configuration-constrained PWA approximation seeks to approximate a nonlinear multivariate function $F:\mathbb{R}^n\to\mathbb{R}$ by a PWA function subject to explicit polyhedral or segment-count constraints on the partition of the input space. A typical objective is:

Given a class of permitted partitions (e.g., a template of polyhedral regions, fixed number of breakpoints, or a prescribed combinatorial incidence graph), select the best (in terms of accuracy, storage, or computational tractability) PWA function within this configuration that matches $F$ to within a user-specified error $\delta$ , or achieves prescribed storage/CPU complexity.

In general, configurations may constrain:

The number and shape of the subregions into which the domain is partitioned (e.g., axis-aligned boxes, subdivisions by arbitrary hyperplanes).
The adjacency structure of regions (fixed combinatorial graphs).
The parametrization of the PWA function, e.g., as the output of a ReLU neural network of fixed architecture.
Storage complexity or number of basis functions (maximum number of affine maps, total number of breakpoints).

Formally, the design problem involves three steps (Glunt et al., 23 Feb 2024, Gharavi et al., 22 Oct 2024, Berger et al., 2023, Houska et al., 20 Mar 2025):

Function decomposition into univariate/bivariate or otherwise simple components.
Region partitioning/synthesis under configuration constraints, possibly via templates, hyperplane cuts, or combinatorially fixed polyhedral networks.
Allocation of approximation error vs. complexity, with global error or complexity target enforced.

2. Region Partitioning and Template Specification

Templates (also known as region templates, configuration templates, or combinatorial schemes) are central to the configuration-constrained paradigm. A template defines, a priori:

The number of regions (pieces) and their possible shapes.
The face-vertex and adjacency structure of the partition (c.f. combinatorial polyhedral theory).
Which facets and vertices are present in every allowable partition in the family, via fixed matrices (facet matrices $F$ , edge matrices $E$ , vertex reconstruction matrices $V$ ) (Houska et al., 20 Mar 2025, Villanueva et al., 12 Dec 2025).

Partitioning methods include:

Axis-aligned box templates, defined as $p(x)=[x; -x]$ ; each region is an axis-aligned box parameterized by its upper/lower bounds (Berger et al., 2023).
Hinging-hyperplane or general hyperplane cut-based partitioning, where the domain $D$ is recursively subdivided by $n_c$ hyperplanes $H_i = \{x\,|\, h_i^T x = 1\}$ , yielding up to $2^{n_c}$ polytopic chambers, each described by a binary signature (Gharavi et al., 22 Oct 2024).
Polyhedral epigraph templates for convex function approximation, in which the epigraph of a convex function is approximated by a polyhedron whose structure (number of faces, edge-vertex graph) is fixed by $(F,E,V)$ , and only the facet uncertainties $z$ are varied (Houska et al., 20 Mar 2025, Villanueva et al., 12 Dec 2025).
Template-based region parametrization via finite sets of facet orientation and offsets, enabling tractable embedding of control-theoretic safety or invariance properties (Houska et al., 20 Mar 2025).

These templates allow explicit control of the complexity and storage characteristics of the PWA approximation, necessary in embedded or certification-driven applications. When templates fix the region shapes, the remaining fitting problem generally reduces to continuous optimization over affine parameters (with possible integer variables for region assignments).

Template Type	Region Description	Example References
Axis-aligned box	$p(x)=[x; -x]$	(Berger et al., 2023)
Hyperplane-cut	$n_c$ -hyperplane intersections	(Gharavi et al., 22 Oct 2024)
Polyhedral epigraph	Faces/facets + vertices fixed	(Houska et al., 20 Mar 2025, Villanueva et al., 12 Dec 2025)
ReLU DNN architecture	Polyhedral induced by activations	(Fahandezh-Saadi et al., 2020)

3. Algorithmic Methodologies

Algorithmic frameworks for configuration-constrained PWA approximation vary with use case and complexity objective. Prototypical strategies:

3.1 Functional Decomposition and Block-wise Approximation

For high-dimensional mappings, a functional decomposition is carried out—via computational graphs or Kolmogorov superposition—to produce a chain/cascade of univariate or low-arity bivariate blocks $f_i$ (Glunt et al., 23 Feb 2024). Each is then approximated independently, subject to segment/breakpoint/complexity constraints.

3.2 Breakpoint Placement and Error Allocation

For univariate blocks, efficient offline methods enable the placement of breakpoints without resorting to global nonlinear optimization:

Bisection algorithm: Recursively bisect domains so maximal secant-line error per segment matches prescribed local tolerance $\tau$ (Glunt et al., 23 Feb 2024).
Derivative-based formula: Given a bound $d_3$ on the third derivative, solve for the largest $\Delta$ matching a desired error via closed-form cubic error bounds.

Error allocation across all blocks/regions is then formulated as a small mixed-integer program (MIP), either minimizing segment count subject to error, or minimizing global error under segment count constraint.

3.3 Template-Based and Hyperplane Partition Algorithms

Template-based top-down search: Systematically explores subsets of a data set, validates region-compatibility via LP feasibility, and splits in response to infeasibility certificates generated by dual analysis (Berger et al., 2023).
Hinging-hyperplane iteration: Iteratively increases cut hyperplane count, at each stage solving a bi-level problem (hyperplane geometry + best local affine fits), until error meets target. Adjacency graphs and region counts are computed using combinatorial enumeration and LP feasibility (Gharavi et al., 22 Oct 2024).

3.4 Convex Program for Polyhedral Epigraph PWA

Fix $(F,E,V)$ for the polyhedral template, and pose a convex optimization whose variables are facet offsets $z$ , and possibly control actions at each vertex (Houska et al., 20 Mar 2025, Villanueva et al., 12 Dec 2025).
Enforce convexity, region adjacency, CLF-descent, and next-step contractivity entirely via vertex/facet-wise constraints and polyhedral incidence relations.
For robust (min-max) settings, maximize worst-case decrease subject to uncertainty polytopes in the system data.

4. Error and Performance Guarantees

A core feature is certified (rather than statistical) error or performance guarantees:

Propagation via compositions: The error in each component or segment is carried through compositions via explicit, analytic bounds. For a three-stage cascade $h\to g\to f$ , total error is $E_{f\circ g\circ h} = \tau_f + d_{f,g}\tau_g + d_{f,g}d_{g,h}\tau_h$ , with $d_{f,g}$ Lipschitz constants (Glunt et al., 23 Feb 2024).
MIP-based error-complexity trade-off: Allocation of local error tolerances $\{\tau_i\}$ across blocks or regions to minimize global error or region count, solved via MIP with piecewise-constant lookup for segment- or region-count per tolerance (Glunt et al., 23 Feb 2024).
Optimization solution sensitivity: In constrained optimization, explicit upper bounds on optimizer shift under PWA approximation are given: $\|x^*_\mathrm{NLP} - x^*_\mathrm{PWA}\| \leq 2\Delta / c_1 + r_0$ where $\Delta$ is max approximation error, $c_1$ the minimal modulus of convexity, and $r_0$ the coarsest cell diameter (Gharavi et al., 30 May 2024).
Performance and safety certificates: For explicit feedback controller synthesis, convex program solutions yield Lyapunov-type or Hamilton–Jacobi–Bellman performance bounds, including ergodic cost upper bounds, min-max dissipation, and robust invariance (Villanueva et al., 12 Dec 2025, Houska et al., 20 Mar 2025).

5. Application Domains and Illustrative Examples

Configuration-constrained PWA approximations are deployed in:

Explicit MPC/controller synthesis: When storing full explicit solutions (e.g., mp-QP) is infeasible due to exponential region growth, configuration constraints control complexity (Fahandezh-Saadi et al., 2020).
Template-based regression/fitting: Data-driven approximation of nonlinear modules (e.g., insulin–glucose ODE terms), where region shapes reflect mechanistic insight or computational desiderata; template-based fit yields sharp region counts for target error (Berger et al., 2023).
Nonlinear control and Lyapunov synthesis: Concise polyhedral epigraph templates enable convex PWA CLFs for robust or nominal control design with exact certificate conditions (Houska et al., 20 Mar 2025, Villanueva et al., 12 Dec 2025).
Symbolic abstraction and formal synthesis: Sequence of symbolic PWA systems with progressively finer quantization, enforcing configuration constraints in region sequences, with explicit simulation-metric convergence guarantees (Pola et al., 2012).

A nonlinear function $y = \sum_{\ell=1}^{4} 1/(||x-s_\ell||^2+1)$ on $x\in [-5,5]^2$ is decomposed into seven unary blocks; a segment-count allocation MIP yields a PWA fit (163 breakpoints) with certified error $\approx 0.445$ , substantially outperforming uniform-grid approaches in both accuracy and segment usage.

With 100 samples and rectangular box templates in $(x_1, x_2)$ , the template-based algorithm produces 1, 2, or 3-piece PWA fits for tolerances $0.20, 0.10, 0.05$ respectively, with corresponding computation times and performance in simulating meal response ODEs.

6. Theoretical Guarantees, Complexity, and Expressiveness

For fixed templates with $P$ regions, expressiveness directly increases with $P$ or complexity of the allowed adjacency graph. The complexity of bi-level or set-cover algorithms is generally polynomial in $N$ (number of data points) for fixed region template, but the number of regions grows combinatorially with number of cuts or hyperplanes (Zaslavsky's theorem, $P \leq \sum_{k=0}^d \binom{n_c}{k}$ for $d$ -dimensional space and $n_c$ cuts) (Gharavi et al., 22 Oct 2024).

Optimal control synthesis via polyhedral epigraph templates achieves region and storage-count by design, and all offline computations reduce to high-dimensional convex programs, with total variables and constraints scaling linearly in region and template size (Houska et al., 20 Mar 2025, Villanueva et al., 12 Dec 2025).

When ReLU neural network architectures are used as the configuration constraint, the number of affine regions is governed by known lower and upper bounds as a function of depth and width, and guarantees of representation or approximation of any convex PWA map (via max-affine realization) hold (Fahandezh-Saadi et al., 2020).

7. Significance and Outlook

Configuration-constrained PWA approximation unifies a spectrum of modern approaches in learning, control, and embedded optimization:

It enables rigorous, certifiable function/model surrogation under explicit cost, complexity, or region-count budgets.
Critical for embedded systems and real-time control, where both storage and compute time must be predictable.
Provides a framework for systematic trade-off between complexity and approximation fidelity, under formal performance, stability, or safety guarantees.

Recent advances extend these tools to robust and min-max settings, including uncertain systems, and embody them in scalable convex optimization solvers amenable to practical implementation. This suggests configuration-constrained techniques will remain central as the demand for certifiable, resource-aware model approximation intensifies in safety-critical and embedded contexts (Glunt et al., 23 Feb 2024, Gharavi et al., 22 Oct 2024, Berger et al., 2023, Houska et al., 20 Mar 2025, Villanueva et al., 12 Dec 2025).