The optimization of an energy network often leads to a large-scale nonconvex optimization problem that is known to be very difficult to solve. Often such problem is simplified into an easier convex or mixed-integer convex optimization problem, but the solution of the simplified problem is unlikely to be optimal or even feasible for the original problem. Recent advances in decomposition based global optimization provides a promising way to solve large-scale nonconvex optimization problems within reason time. This short course introduces the principles of these methods and explains how the principles can be applied to the optimization of energy networks. 

The course consists of two parts. In the first part, we discuss the difficulty of nonconvex optimization and the disadvantage of the classical branch-and-bound based global optimization methods, and introduce the concepts of complicating variables and linking variables, which lead to the notion of decomposition, i.e., solving a nonconvex optimization problem via solving a sequence of alternative problems that are smaller in size and easier in nature. Then we introduce the details of generalized Benders decomposition (GBD), including the reformulation into a master problem using strong Lagrangian duality, the construction of upper and lower bounding problems, and the finite convergence property. We also show the significant advantage of GBD for solving multiscenario problems. Then we introduce two variants of GBD. The first variant, called nonconvex generalized Benders decomposition (NGBD), is able to solve problems where the linking variables are integers and some non-linking variables form nonconvex functions. We introduce a convex relaxation technique that is used in NGBD to deal with the nonconvex functions. The second variant, called cross decomposition (CD) or joint decomposition (JD) depending on the nature of the problem, enhances the performance of GBD via integration of Dantzig-Wolfe decomposition or Lagrangian decomposition. 

In the second part, we demonstrate how the methods discussed in the first part can be applied to energy network optimization via an industrial natural gas production network optimization problem. We first describe the engineering details of the problem, and analyze the general structure of the optimization problem. Then we discuss three specific cases, each having distinctive assumptions for decision-making. In the first case,  we show that NGBD can solve the resulting optimization problem efficiently, and the solution time scales moderately with the size of the problem. We also show that the solution of a simplified optimization problem may be infeasible for the original problem. In the second case, the optimization problem includes more nonconvex functions and NGBD cannot solve the problem within reasonable time, so we propose a rigorous reformulation procedure after which the problem can be solved efficiently via a bi-lelvel decomposition approach. In the third case, the linking variables are continuous so NGBD cannot solve the problem rigorously. We show how this problem can be solved by JD, and demonstrate the importance of domain reduction in improving the performance of JD. 

The course audience are expected to have taken an introductory optimization course. Prior experience in applying optimization to engineering decision-making problems will help the understanding of the course materials.