Augmented Lagrangian Coordination for Decomposed Design Problems

Designing large-scale systems frequently involves solving a complex mathematical program that requires, for various reasons, decomposition into a number of smaller systems. Practical studies have proved the effectiveness of multilevel hierarchical methods at early stages of design; these methods divide a large program into multiple levels and multiple systems at each level and the result is known as a multilevel or hierarchical mathematical program. Using insight provided by Lagrangian duality theory, the paper presents a new algorithm for bi-level programs that does not impose difficult-to-satisfy conditions of convexity and differentiability generally required for convergence. In addition, the algorithm renders each sub-problem independent so that they can be solved concurrently, which is a significant advantage in certain applications. This is done by combining classical Lagrangian duality (LD) and the augmented Lagrangian duality (ALD) in an algorithm to solve a bi-level problem. 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 provides a simple method for decomposition without imposing restrictive conditions. The method is applied to two mathematical examples to illustrate its potential as well as associated numerical issues.