A Duality Theory with Zero Duality Gap for Nonlinear Programming

Duality is an important notion for constrained optimization which provides a theoretical foundation for a number of constraint decomposition schemes such as separable programming and for deriving lower bounds in space decomposition algorithms such as branch and bound. However, the conventional duality theory has the fundamental limit that it leads to duality gaps for nonconvex optimization problems, especially discrete and mixed-integer problems where the feasible sets are nonconvex. In this paper, we propose a novel extended duality theory for nonlinear optimization that overcomes some limitations of previous dual methods. Based on a new dual function, the extended duality theory leads to zero duality gap for general nonconvex problems defined in discrete, continuous, and mixed-integer spaces under mild conditions

Network Target Coordination for Design Optimization of Decomposed Systems

A complex engineered system is often decomposed into a number of different subsystems that interact on one another and together produce results not obtainable by the subsystems alone. Effective coordination of the interdependencies shared among these subsystems is critical to fulfill the stakeholder expectations and technical requirements of the original system. The past research has shown that various coordination methods obtain different solution accuracies and exhibit different computational efficiencies when solving a decomposed system. Addressing these coordination decisions may lead to improved complex system design. This dissertation studies coordination methods through two types of decomposition structures, hierarchical, and nonhierarchical. For coordinating hierarchically decomposed systems, linear and proximal cutting plane methods are applied based on augmented Lagrangian relaxation and analytical target cascading (ATC). Three nonconvex, nonlinear design problems are used to verify the numerical performance of the proposed coordination method and the obtained results are compared to traditional update schemes of subgradient-based algorithm. The results suggest that the cutting plane methods can significantly improve the solution accuracy and computational efficiency of the hierarchically decomposed systems. In addition, a biobjective optimization method is also used to capture optimality and feasibility. The numerical performance of the biobjective algorithm is verified by solving an analytical mass allocation problem. For coordinating nonhierarchically decomposed complex systems, network target coordination (NTC) is developed by modeling the distributed subsystems as different agents in a network. To realize parallel computing of the subsystems, NTC via a consensus alternating direction method of multipliers is applied to eliminate the use of the master problem, which is required by most distributed coordination methods. In NTC, the consensus is computed using a locally update scheme, providing the potential to realize an asynchronous solution process. The numerical performance of NTC is verified using a geometrical programming problem and two engineering problems

Lagrangian Coordination for Enhancing the Convergence of Analytical Target Cascading

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/76913/1/AIAA-15326-842.pd

Distributed power control for wireless networks via the alternating direction method of multipliers

Utility-based power control in wireless networks has been widely recognized as an effective mechanism to managing co-channel interferences. It is based on the maximization of system utility subject to power constraints, which is referred to as power control optimization problem. Global coupling between the mutual interference of wireless channels increases the difficulty of searching global optimum significantly. In this paper, we decouple the optimization problems with concave and non-concave utility functions; and transform them into a global consensus problem by introducing locally slack variables. We then propose two distributed iterative optimization algorithms for the global consensus problems with concave and non-concave objective functions, respectively, based on an alternating direction method of multipliers. Furthermore, we prove that both algorithms converge to the global optimum of the total network utility. Simulation results show the effectiveness of the algorithms. Comparison experiments show that the developed algorithms compare favourably against some other well-known algorithms