This grant provides funding for the development a mathematical algorithm based on Lagrangian duality theory to solve complex and large engineering design problems that have been decomposed, without imposing difficult-to-satisfy conditions of convexity and differentiability generally required for convergence. This will be done by combining classical Lagrangian duality (LD) and augmented Lagrangian duality (ALD) into a dual algorithm. One of the traditional drawbacks of LD has been that it is only applicable to convex problems. ALD extends the theory to non-convex problems but at the expense of separability. Combining classical LD with ALD will provide a simple method for coordination without imposing restrictive conditions. The developed algorithm will use sub-gradient and cutting plane methods to circumvent numerical issues associated with decomposed engineering design problems. A proof of convergence to the solution of the original non-decomposed problem will be provided. The work will then be extended to multiple-level problems with multiple subsystems on each level. If successful, the results of this research will lead to a new coordination strategy applicable to a wide class of complex decomposed engineering design problems. More specifically, the algorithm will completely determine the weights applied on coupling terms generally required for convergence. This will relieve designers from the delicate task of determining appropriate weights, and allow them to focus on the design aspect instead. In addition, the algorithm will render each subproblem temporarily independent so that they can be solved concurrently, which is a significant advantage in most complex engineering applications. This research involves the application of mathematically grounded theories to the engineering design process. As a result of this project, students, educators and researchers will be more knowledgeable on the potential of Lagrangian duality to solve real-life engineering design problems.
