A parallel quadratic programming method for dynamic optimization problems

Abstract

Quadratic programming problems (QPs) that arise from dynamic optimization problems typically exhibit a very particular structure. We address the ubiquitous case where these QPs are strictly convex and propose a dual Newton strategy that exploits the block-bandedness similarly to an interior-point method. Still, the proposed method features warmstarting capabilities of active-set methods. We give details for an efficient implementation, including tailored numerical linear algebra, step size computation, parallelization, and infeasibility handling. We prove convergence of the algorithm for the considered problem class. A numerical study based on the open-source implementation qpDUNES shows that the algorithm outperforms both well-established general purpose QP solvers as well as state-of-the-art tailored control QP solvers significantly on the considered benchmark problems.

A Feedback Optimal Control Algorithm with Optimal Measurement Time Points

“…Let an optimal solution (x * , G * , w * , µ * ) of the optimization problem (12) with W = [w min , w max ] be given. Here µ ω, * is the Lagrange multiplier of the constraint (9). Let F * −1 (t f ) exist.…”

“…As a fast feedback of the controller is important in many applications, clever approaches doing most of the necessary calculations before a new measurement arrives have been proposed in the literature. The most important numerical concepts comprise real-time iterations [4,5], multi-level iterations [6], parallel multi-level iterations [7], an exploitation of the KKT structures [8,9], adaptive control [10], automatic code export [11,12], and usage of parametric QPs [13,14]. For a benchmark problem, the continuously stirred tank reactor of [15], a speedup of approximately 150,000 has been achieved comparing the 60 seconds per iteration reported in 1997 [16] and the 400 microseconds per iteration reported in 2011 by [17].…”

Primal decomposition of the optimal coordination of vehicles at traffic intersections

“…It follows that the proposed SQP strategy ought to be deployed using tools from non-smooth Newton schemes, where globalization and possible regularizations are used to guarantee convergence, e.g. as in [7].…”

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

“…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.…”

“…For specific applications, the dual Hessian can be structured and the cost for its factorization can therefore be lowered; see e.g. [9]. However, most problems yield a dense dual Hessian, making its factorization computationally expensive, or even a bottleneck when solving problems that have a large number of complicating constraints between the subproblems.…”