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

TL;DR

This paper introduces Linear Design Problems (LDPs) for scalable co-design, linking them to multi-objective linear programming and providing methods for compositional and monolithic computation of feasible system configurations.

Contribution

It defines LDPs as a new class of design problems that are closed under interconnections and can be solved exactly or approximated efficiently, advancing scalable co-design theory.

Findings

LDPs reduce to Multi-Objective Linear Programs (MOLPs).

Interconnections of linear components induce system-level LDPs.

Numerical studies validate the theoretical results.

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.