Splitting Methods for Distributed Optimization and Control

This thesis contributes towards the design and analysis of fast and distributed optimization algorithms based on splitting techniques, such as proximal gradient methods or alternation minimization algorithms, with the application of solving model predictive control (MPC) problems. The first part of the thesis focuses on developing an efficient algorithm based on the fast alternating minimization algorithm to solve MPC problems with polytopic and second-order cone constraints. Due to the requirement of bounding the online computation time in the context of real-time MPC, complexity bounds on the number of iterations to achieve a certain accuracy are derived. In addition, a discussion of the computation of the complexity bounds is provided. To further improve the convergence speed of the proposed algorithm, an o-line pre-conditioning method is presented for MPC problems with polyhedral and ellipsoidal constraints. The inexact alternating minimization algorithm, as well as its accelerated variant, is proposed in the second part of the thesis. Different from standard algorithms, inexact methods allow for errors in the update at each iteration. Complexity upper-bounds on the number of iterations in the presence of errors are derived. By employing the complexity bounds, sufficient conditions on the errors, which guarantee the convergence of the algorithms, are presented. The proposed algorithms are applied for solving distributed optimization problems in the presence of local computation and communication errors, with an emphasis on distributed MPC applications. The convergence properties of the algorithms for this special case are analysed. Motivated by the complexity upper-bounds of the inexact proximal gradient method, two distributed optimization algorithms with an iteratively refining quantization design are proposed for solving distributed optimization problems with a limited communication data-rate. We show that if the parameters of the quantizers satisfy certain conditions, then the quantization error decreases linearly, while at each iteration only a fixed number of bits is transmitted, and the convergence of the distributed algorithms is guaranteed. The proposed methods are further extended to distributed optimization problems with time-varying parameters

Complexity Certification of the Fast Alternating Minimization Algorithm for Linear Model Predictive Control

In this paper, the fast alternating minimization algorithm (FAMA) is proposed to solve model predictive control (MPC) problems with polytopic and second-order cone constraints. We extend previous theoretical results of FAMA to a more general case, where convex constraints are allowed to be imposed on the strongly convex objective and all convergence properties of FAMA are still preserved. Two splitting strategies for MPC problems are presented. Both of them satisfy the assumptions of FAMA and result in efficient implementations by reducing each iteration of FAMA to simple operations. We derive computational complexity certificates for both splitting strategies, by providing bounds on the number of iterations for both primal and dual variables, which are of particular relevance in the context of real-time MPC to bound the required online computation time. For MPC problems with polyhedral and ellipsoidal constraints, an off-line preconditioning method is presented to further improve the convergence speed of FAMA by reducing the complexity bound and enlarging the step-size of the algorithm. Finally, we demonstrate the performance of FAMA compared to other splitting methods using a quadrotor example

Complexity Certification of the Fast Alternating Minimization Algorithm for Linear MPC

In this technical note, the fast alternating minimization algorithm (FAMA) is proposed to solve model predictive control (MPC) problems with polytopic and second-order cone constraints. Two splitting strategies with efficient implementations for MPC problems are presented. We derive computational complexity certificates for both splitting strategies, by providing complexity upper-bounds on the number of iterations required to provide a certain accuracy of the dual function value and, most importantly, of the primal solution. This is of particular relevance in the context of real-time MPC in order to bound the required on-line computation time. We further address the computation of the complexity bounds, requiring the solution of a non-convex minimization problem. Finally, we demonstrate the performance of FAMA compared to other splitting methods using a quadrotor example

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

Cooperative control for multi-vehicle swarms

The cooperative control of large-scale multi-agent systems has gained a significant interest in recent years from the robotics and control communities for multi-vehicle control. One motivator for the growing interest is the application of spatially and temporally distributed multiple unmanned aerial vehicle (UAV) systems for distributed sensing and collaborative operations. In this research, the multi-vehicle control problem is addressed using a decentralised control system. The work aims to provide a decentralised control framework that synthesises the self-organised and coordinated behaviour of natural swarming systems into cooperative UAV systems. The control system design framework is generalised for application into various other multi-agent systems including cellular robotics, ad-hoc communication networks, and modular smart-structures. The approach involves identifying su itable relationships that describe the behaviour of the UAVs within the swarm and the interactions of these behaviours to produce purposeful high-level actions for system operators. A major focus concerning the research involves the development of suitable analytical tools that decomposes the general swarm behaviours to the local vehicle level. The control problem is approached using two-levels of abstraction; the supervisory level, and the local vehicle level. Geometric control techniques based on differential geometry are used at the supervisory level to reduce the control problem to a small set of permutation and size invariant abstract descriptors. The abstract descriptors provide an open-loop optimal state and control trajectory for the collective swarm and are used to describe the intentions of the vehicles. Decentralised optimal control is implemented at the local vehicle level to synthesise self-organised and cooperative behaviour. A deliberative control scheme is implemented at the local vehicle le vel that demonstrates autonomous, cooperative and optimal behaviour whilst the preserving precision and reliability at the local vehicle level