A method for computationally solving design and manufacturing problems is presented, which combines two mathematical programming research areas: numerical optimization and graph rewriting. While optimization can determine values of parameters within a design, engineering designers must also determine what components to use within a design and how those components are to be connected. To date, existing numerical methods have had limited success in helping engineers with such topological decisions. This research sets out to develop new algorithmic approaches to simultaneously define a best topology, which is represented as a graph, and the best parameters within the graph. This is accomplished by combining artificial intelligence tree-searching methods with gradient and stochastic optimization. To date, the graphs synthesis methods have been used to represent sheet metal parts, gear trains, neural networks, planar mechanisms, truss structures, fluidic channels and the connection of components in large assemblies (i.e. liaison graph). This seminar will present an overview of the methods along with recent results from projects sponsored by NSF and DARPA.