A distributed method for convex quadratic programming problems arising in optimal control of distributed systems

We propose a distributed algorithm for strictly convex quadratic programming (QP) problems with a generic coupling topology. The coupling constraints are dualized via Lagrangian relaxation. This allows for a distributed evaluation of the non-smooth dual function and its derivatives. We propose to use both the gradient and the curvature information within a non-smooth variant of Newton's method to find the optimal dual variables. Our novel approach is designed such that the large Newton system never needs to be formed. Instead, we employ an iterative method to solve the Newton system in a distributed manner. The effectiveness of the method is demonstrated on an academic optimal control problem. A comparison with state-of-the-art first order dual methods is given.

[1]  Moritz Diehl,et al.  Multiple shooting for distributed systems with applications in hydro electricity production , 2011 .

[2]  Christian Kirches,et al.  qpOASES: a parametric active-set algorithm for quadratic programming , 2014, Mathematical Programming Computation.

[3]  Robert A. van de Geijn,et al.  Distributed memory matrix-vector multiplication and conjugate gradient algorithms , 1993, Supercomputing '93. Proceedings.

[4]  Stephen J. Wright,et al.  Numerical Optimization (Springer Series in Operations Research and Financial Engineering) , 2000 .

[5]  Claude Lemaréchal,et al.  Lagrangian Relaxation , 2000, Computational Combinatorial Optimization.

[6]  Mohamed-Jalal Fadili,et al.  A quasi-Newton proximal splitting method , 2012, NIPS.

[7]  Michael Jünger,et al.  Computational Combinatorial Optimization , 2001, Lecture Notes in Computer Science.

[8]  Matthias Gerdts,et al.  Hamburger Beiträge zur Angewandten Mathematik A nonsmooth Newton ’ s method for discretized optimal control problems with state and control constraints , 2007 .

[9]  Wei Yu,et al.  Optimal multiuser spectrum balancing for digital subscriber lines , 2006, IEEE Transactions on Communications.

[10]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[11]  Emmanuel J. Candès,et al.  Adaptive Restart for Accelerated Gradient Schemes , 2012, Foundations of Computational Mathematics.

[12]  M. Diehl,et al.  A Parallel Active-Set Strategy to Solve Sparse Parametric Quadratic Programs arising in MPC , 2012 .

[13]  Fu Lin,et al.  On the dual decomposition of linear quadratic optimal control problems for vehicular formations , 2010, 49th IEEE Conference on Decision and Control (CDC).

[14]  J. Shewchuk.  An Introduction to the Conjugate Gradient Method Without the Agonizing Pain , 1994 .

[15]  Yurii Nesterov,et al.  Introductory Lectures on Convex Optimization - A Basic Course , 2014, Applied Optimization.

[16]  Liqun Qi,et al.  A nonsmooth version of Newton's method , 1993, Math. Program..

[17]  John N. Tsitsiklis,et al.  Parallel and distributed computation , 1989 .

[18]  Johan A. K. Suykens,et al.  Application of the proximal center decomposition method to distributed model predictive control , 2008, 2008 47th IEEE Conference on Decision and Control.

[19]  Manfred Morari,et al.  Towards computational complexity certification for constrained MPC based on Lagrange Relaxation and the fast gradient method , 2011, IEEE Conference on Decision and Control and European Control Conference.

[20]  A.G. Beccuti,et al.  A distributed solution approach to centralized emergency voltage control , 2006, 2006 American Control Conference.

[21]  M. Moonen,et al.  Improved Dual Decomposition Based Optimization for DSL Dynamic Spectrum Management , 2010, IEEE Transactions on Signal Processing.

[22]  Wu Li,et al.  A New Algorithm for Solving Strictly Convex Quadratic Programs , 1997, SIAM J. Optim..

[23]  H. Bock,et al.  A Multiple Shooting Algorithm for Direct Solution of Optimal Control Problems , 1984 .

[24]  MORITZ DIEHL,et al.  A Real-Time Iteration Scheme for Nonlinear Optimization in Optimal Feedback Control , 2005, SIAM J. Control. Optim..

[25]  Stephen P. Boyd,et al.  Distributed estimation via dual decomposition , 2007, 2007 European Control Conference (ECC).