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Scalable Co-Design via Linear Design Problems: Compositional Theory and Algorithms

Yubo Cai, Yujun Huang, Meshal Alharbi, Gioele Zardini

Abstract

Designing complex engineered systems requires managing tightly coupled trade-offs between subsystem capabilities and resource requirements. Monotone co-design provides a compositional language for such problems, but its generality does not by itself reveal which problem classes admit exact and scalable computation. This paper isolates such a class by introducing Linear Design Problems (LDPs): design problems whose feasible functionality--resource relations are polyhedra over Euclidean posets. We show that queries on LDPs reduce exactly to Multi-Objective Linear Programs (MOLPs), thereby connecting monotone co-design semantics with polyhedral multiobjective optimization. We further prove that LDPs are closed under the fundamental co-design interconnections, implying that any interconnection of linear components induces a system-level LDP. To compute the resulting feasible sets, we develop two complementary constructions: a monolithic lifted formulation that preserves block-angular sparsity, and a compositional formulation that incrementally eliminates internal variables through polyhedral projection. Beyond the exact linear setting, we show that convex co-design resource queries admit arbitrarily accurate polyhedral outer approximations, with recession-cone error identically zero for standard nonnegative resource cones. Numerical studies on synthetic series-chain benchmarks, a gripper, and a rover co-design validate the theory.

Scalable Co-Design via Linear Design Problems: Compositional Theory and Algorithms

Abstract

Designing complex engineered systems requires managing tightly coupled trade-offs between subsystem capabilities and resource requirements. Monotone co-design provides a compositional language for such problems, but its generality does not by itself reveal which problem classes admit exact and scalable computation. This paper isolates such a class by introducing Linear Design Problems (LDPs): design problems whose feasible functionality--resource relations are polyhedra over Euclidean posets. We show that queries on LDPs reduce exactly to Multi-Objective Linear Programs (MOLPs), thereby connecting monotone co-design semantics with polyhedral multiobjective optimization. We further prove that LDPs are closed under the fundamental co-design interconnections, implying that any interconnection of linear components induces a system-level LDP. To compute the resulting feasible sets, we develop two complementary constructions: a monolithic lifted formulation that preserves block-angular sparsity, and a compositional formulation that incrementally eliminates internal variables through polyhedral projection. Beyond the exact linear setting, we show that convex co-design resource queries admit arbitrarily accurate polyhedral outer approximations, with recession-cone error identically zero for standard nonnegative resource cones. Numerical studies on synthetic series-chain benchmarks, a gripper, and a rover co-design validate the theory.

Paper Structure

This paper contains 48 sections, 13 theorems, 58 equations, 7 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Statement of Contribution
  4. Linear Co-Design Theory
  5. Mathematical preliminaries
  6. Background on orders
  7. Convex geometry
  8. Background on co-design and queries
  9. Linear co-design problems and queries
  10. Feasible-Set Construction and Query Solving for LCDPs
  11. Monolithic formulation
  12. Compositional approach and variable elimination
  13. Composition as polyhedral calculus
  14. Graph-guided elimination schedule
  15. Projection primitives and practical considerations
  16. ...and 33 more sections

Key Result

Lemma 1

Consider an ${\mathrm{dp}} \in \mathsf{LDP}\{{\textcolor{dpgreen}{F}}, {\textcolor{dpred}{R}}\}$ with feasible set ${\mathrm{ldp}} = \{ \left\langle x_{\textcolor{dpgreen}{F}}, x_{\textcolor{dpred}{R}} \right\rangle \in {\textcolor{dpgreen}{F}} \times {\textcolor{dpred}{R}} : A_{{\textcolor{dpgreen}

Figures (7)

  • Figure 3: A is a monotone relation between posets of functionalities and resources (a), and can be composed via various operations (b)--(e).
  • Figure 4: Series-chain benchmark with $m$ nodes. Each $\mathrm{DP}_i$ has $k$-dimensional functionality and resource ports, and the resource of $\mathrm{DP}_i$ feeds the functionality of $\mathrm{DP}_{i+1}$.
  • Figure 5: Gripper co-design architecture. The query $\left\langle f_{\mathrm{grasp}},f_{\mathrm{ws}} \right\rangle$ is propagated to subsystem design choices; the motor and material subsystems are coupled through the shared design variables $\left\langle L_{\mathrm{Al}},L_{\mathrm{CF}} \right\rangle$; and contributions are aggregated into system objectives $\left\langle r_{\mathrm{cost}},r_{\mathrm{mass}} \right\rangle$.
  • Figure 6: Pareto-front comparison for the rigid gripper case study. (Left) Monolithic exact frontier (3 vertices) versus MCDPL optimistic approximations at resolutions $R\in\{5,10,20,40\}$. (Right) IGD convergence versus computational time: the normalised Inverted Generational Distance of MCDPL optimistic solutions decreases monotonically with increasing resolution, while the exact monolithic solver achieves IGD $=$ 0.
  • Figure 7: Rover co-design architecture. The query $\left\langle m_{\mathrm{pay}},v_{\mathrm{req}} \right\rangle$ propagates to chassis and battery choices; battery and chassis are coupled through power and mass; and the system-level objectives are $\left\langle c_{\mathrm{sys}},m_{\mathrm{sys}} \right\rangle$.
  • ...and 2 more figures

Theorems & Definitions (57)

  • Definition 1: Poset
  • Definition 2: Opposite poset
  • Definition 3: Product poset
  • Definition 4: Upper closure
  • Definition 5: Upper set
  • Definition 6: Set operations
  • Definition 7: Basic convex sets
  • Definition 8: Polyhedron
  • Remark : H/V representations and conversion
  • Definition 9
  • ...and 47 more