Hierarchical distributed optimization of constraint-coupled convex and mixed-integer programs using approximations of the dual function

Abstract

In this paper, two new algorithms for dual decomposition-based distributed optimization are presented. Both algorithms rely on the quadratic approximation of the dual function of the primal optimization problem. The dual variables are updated in each iteration through a maximization of the approximated dual function. The first algorithm approximates the dual function by solving a regression problem, based on the values of the dual function collected from previous iterations. The second algorithm updates the parameters of the quadratic approximation via a quasi-Newton scheme. Both algorithms employ step size constraints for the update of the dual variables. Furthermore, the subgradients from previous iterations are stored in order to construct cutting planes, similar to bundle methods for nonsmooth optimization. However, instead of using the cutting planes to formulate a piece-wise linear over-approximation of the dual function, they are used to construct valid inequalities for the update step. In order to demonstrate the efficiency of the algorithms, they are evaluated on a large set of constrained quadratic, convex and mixed-integer benchmark problems and compared to the subgradient method, the bundle trust method, the alternating direction method of multipliers and the quadratic approximation coordination algorithm. The results show that the proposed algorithms perform better than the compared algorithms both in terms of the required number of iterations and in the number of solved benchmark problems in most cases.

Keywords

• Distributed optimization
• Dual decomposition
• Nonsmooth optimization
• Subgradient method
• Bundle method
• ADMM