%0 Generic %1 censi2015mathematical %A Censi, Andrea %D 2015 %K 2015 arxiv design mathematics mit optimization paper robotics %T A Mathematical Theory of Co-Design %U http://arxiv.org/abs/1512.08055 %X 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.

@misc{censi2015mathematical, 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.}, added-at = {2018-06-05T19:49:12.000+0200}, author = {Censi, Andrea}, biburl = {https://www.bibsonomy.org/bibtex/20cf78c230c2faa67e47e5c6ca4d608e8/achakraborty}, description = {[1512.08055] A Mathematical Theory of Co-Design}, interhash = {b1e4bc62f7c6c2ca972e8965989eb669}, intrahash = {0cf78c230c2faa67e47e5c6ca4d608e8}, keywords = {2015 arxiv design mathematics mit optimization paper robotics}, note = {cite arxiv:1512.08055}, timestamp = {2018-06-05T19:49:28.000+0200}, title = {A Mathematical Theory of Co-Design}, url = {http://arxiv.org/abs/1512.08055}, year = 2015 }