[Paper Review] A Mathematical Theory of Co-Design

Andrea Censi|arXiv (Cornell University)|Dec 25, 2015

Manufacturing Process and Optimization22 references20 citations

TL;DR

This paper proposes a mathematical theory of co-design for complex systems using Monotone Co-Design Problems (MCDPs), where functionality and resources are modeled as complete partial orders with Scott-continuous feasibility relations. It introduces a systematic solution via least fixed-point iteration over antichains, enabling rigorous, optimal design of interdependent subsystems even in non-convex, non-differentiable settings.

ABSTRACT

One of the challenges of modern engineering, and robotics in particular, is designing complex systems, composed of many subsystems, rigorously and with optimality guarantees. This paper introduces a theory of co-design that describes "design problems", defined as tuples of "functionality space", "implementation space", and "resources space", together with a feasibility relation that relates the three spaces. Design problems can be interconnected together to create "co-design problems", which describe possibly recursive co-design constraints among subsystems. A co-design problem induces a family of optimization problems of the type "find the minimal resources needed to implement a given functionality"; the solution is an antichain (Pareto front) of resources. A special class of co-design problems are Monotone Co-Design Problems (MCDPs), for which functionality and resources are complete partial orders and the feasibility relation is monotone and Scott continuous. The induced optimization problems are multi-objective, nonconvex, nondifferentiable, noncontinuous, and not even defined on continuous spaces; yet, there exists a complete solution. The antichain of minimal resources can be characterized as a least fixed point, and it can be computed using Kleene's algorithm. The computation needed to solve a co-design problem can be bounded by a function of a graph property that quantifies the interdependence of the subproblems. These results make us much more optimistic about the problem of designing complex systems in a rigorous way.

Motivation & Objective

• Address the challenge of designing complex, interdependent systems—especially in robotics—where subsystems impose recursive co-design constraints.
• Formalize co-design problems as tuples of functionality, implementation, and resources spaces with feasibility relations.
• Develop a principled framework for solving arbitrarily interconnected co-design problems with optimality guarantees.
• Establish conditions under which minimal resource solutions exist and can be systematically computed.
• Generalize prior work on single-cycle co-design to arbitrary graph-structured interconnections, including feedback loops.

Proposed method

• Model each design problem as a feasibility relation between functionality space, implementation space, and resources space.
• Define Monotone Co-Design Problems (MCDPs) where functionality and resources are complete partial orders and the feasibility relation is monotone and Scott continuous.
• Represent the solution to an MCDP as the least fixed point of a system of equations over antichains (Pareto fronts) of minimal resources.
• Apply Kleene’s fixed-point iteration to compute the solution, starting from the bottom element of the antichain lattice.
• Bound the computational complexity using a graph property quantifying interdependence between subproblems.
• Leverage techniques from abstract interpretation, such as chaotic and asynchronous iterations, to potentially accelerate convergence.

Experimental results

Research questions

• RQ1 How can co-design constraints between interdependent subsystems in complex systems be formally modeled and solved with optimality guarantees?
• RQ2 What mathematical structure ensures the existence and computability of minimal resource solutions in multi-objective, non-convex co-design problems?
• RQ3 Can a systematic solution procedure be defined for MCDPs that are closed under arbitrary interconnection, including recursive dependencies?
• RQ4 How does the structure of interdependence in a co-design diagram affect the computational complexity of solving the problem?
• RQ5 To what extent can existing methods from abstract interpretation be adapted to solve MCDPs with multi-valued, non-unique solutions?

Key findings

• Monotone Co-Design Problems (MCDPs) are closed under arbitrary interconnection, including feedback, ensuring the framework is robust for complex system design.
• The solution to an MCDP is an antichain (Pareto front) of minimal resource allocations, characterized as the least fixed point of a Scott-continuous operator on antichains.
• The fixed-point iteration converges in a finite number of steps, with complexity bounded by a graph property measuring subsystem interdependence.
• MCDPs generalize continuous monotone partial order programs (CMPOPs), which have unique solutions, while MCDPs allow multiple minimal solutions due to multi-valued feasibility relations.
• The framework enables systematic, optimal design of heterogeneous, multi-domain systems such as robotic platforms with recursive co-design constraints.
• Although the optimization problems are non-convex, non-differentiable, and not defined on continuous spaces, they admit a complete and constructive solution procedure.