A primal-dual Newton method for distributed Quadratic Programming

Abstract

This paper considers the problem of solving Quadratic Programs (QP) arising in the context of distributed optimization and optimal control. A dual decomposition approach is used, where the problem is decomposed and solved in parallel, while the coupling constraints are enforced via manipulating the dual variables. In this paper, the local problems are solved using a primal-dual interior point method and the dual variables are updated using a Newton iteration, providing a fast convergence rate. Linear predictors for the local primaldual variables and the dual variables are introduced to help the convergence of the algorithm. We observe a fast and consistent practical convergence for the proposed algorithm.

Numerical Structure of the Hessian of the Lagrange Dual Function for a Class of Convex Problems

“…Newton steps can then be used to efficiently trace the central path. Similar methods were proposed in, e.g., [25,14,12]. In this section, we extend the results in section 3 to the method proposed in [21].…”

“…Using structure-exploiting factorization techniques for banded matrices, the quasi-Newton method can be used to solve problems at a lower computational cost compared to an exact Newton method. The results were presented for problems without inequality constraints, and extended to the methods presented in [21,25,14]. The results were illustrated via a numerical example.…”

“…Additionally, a strategy for solving a sequence of relaxed dual problems in order to find an arbitrary good approximation of the solution to (4.1) is provided in Algorithm 4.2; for a detailed description see, e.g., [21,25,14].…”

“…This allows for a straightforward and compact analysis of the convergence of the quasiNewton strategy in the neighborhood of the solution. The results are then extended to inequality constrained problems, using the log-barrier method that was proposed in [21], and further developed in [25,14,12]. However, it should be emphasized that the numerical structure is a result of matrix properties and extends to the nonsmooth methods proposed in, e.g., [16,9].…”

“…All these methods make use of only the first-order derivatives of the dual function to obtain a search direction and their theoretical and practical convergence can therefore not be faster than sublinear. The authors of, e.g., [21,16,25,17,14,12,9] overcome this limitation by using Newton strategies in the dual space.…”