An Alternating Trust Region Algorithm for Distributed Linearly Constrained Nonlinear Programs, Application to the AC Optimal Power Flow

A novel trust region method for solving linearly constrained nonlinear programs is presented. The proposed technique is amenable to a distributed implementation, as its salient ingredient is an alternating projected gradient sweep in place of the Cauchy point computation. It is proven that the algorithm yields a sequence that globally converges to a critical point. As a result of some changes to the standard trust region method, namely a proximal regularisation of the trust region subproblem, it is shown that the local convergence rate is linear with an arbitrarily small ratio. Thus, convergence is locally almost superlinear, under standard regularity assumptions. The proposed method is successfully applied to compute local solutions to alternating current optimal power flow problems in transmission and distribution networks. Moreover, the new mechanism for computing a Cauchy point compares favourably against the standard projected search as for its activity detection properties

OSQP: An Operator Splitting Solver for Quadratic Programs

We present a general-purpose solver for convex quadratic programs based on the alternating direction method of multipliers, employing a novel operator splitting technique that requires the solution of a quasi-definite linear system with the same coefficient matrix at almost every iteration. Our algorithm is very robust, placing no requirements on the problem data such as positive definiteness of the objective function or linear independence of the constraint functions. It can be configured to be division-free once an initial matrix factorization is carried out, making it suitable for real-time applications in embedded systems. In addition, our technique is the first operator splitting method for quadratic programs able to reliably detect primal and dual infeasible problems from the algorithm iterates. The method also supports factorization caching and warm starting, making it particularly efficient when solving parametrized problems arising in finance, control, and machine learning. Our open-source C implementation OSQP has a small footprint, is library-free, and has been extensively tested on many problem instances from a wide variety of application areas. It is typically ten times faster than competing interior-point methods, and sometimes much more when factorization caching or warm start is used. OSQP has already shown a large impact with tens of thousands of users both in academia and in large corporations

High performance implementation of MPC schemes for fast systems

In recent years, the number of applications of model predictive control (MPC) is rapidly increasing due to the better control performance that it provides in comparison to traditional control methods. However, the main limitation of MPC is the computational e ort required for the online solution of an optimization problem. This shortcoming restricts the use of MPC for real-time control of dynamic systems with high sampling rates. This thesis aims to overcome this limitation by implementing high-performance MPC solvers for real-time control of fast systems. Hence, one of the objectives of this work is to take the advantage of the particular mathematical structures that MPC schemes exhibit and use parallel computing to improve the computational e ciency. Firstly, this thesis focuses on implementing e cient parallel solvers for linear MPC (LMPC) problems, which are described by block-structured quadratic programming (QP) problems. Speci cally, three parallel solvers are implemented: a primal-dual interior-point method with Schur-complement decomposition, a quasi-Newton method for solving the dual problem, and the operator splitting method based on the alternating direction method of multipliers (ADMM). The implementation of all these solvers is based on C++. The software package Eigen is used to implement the linear algebra operations. The Open Message Passing Interface (Open MPI) library is used for the communication between processors. Four case-studies are presented to demonstrate the potential of the implementation. Hence, the implemented solvers have shown high performance for tackling large-scale LMPC problems by providing the solutions in computation times below milliseconds. Secondly, the thesis addresses the solution of nonlinear MPC (NMPC) problems, which are described by general optimal control problems (OCPs). More precisely, implementations are done for the combined multiple-shooting and collocation (CMSC) method using a parallelization scheme. The CMSC method transforms the OCP into a nonlinear optimization problem (NLP) and de nes a set of underlying sub-problems for computing the sensitivities and discretized state values within the NLP solver. These underlying sub-problems are decoupled on the variables and thus, are solved in parallel. For the implementation, the software package IPOPT is used to solve the resulting NLP problems. The parallel solution of the sub-problems is performed based on MPI and Eigen. The computational performance of the parallel CMSC solver is tested using case studies for both OCPs and NMPC showing very promising results. Finally, applications to autonomous navigation for the SUMMIT robot are presented. Specially, reference tracking and obstacle avoidance problems are addressed using an NMPC approach. Both simulation and experimental results are presented and compared to a previous work on the SUMMIT, showing a much better computational e ciency and control performance.Tesi

Distributed algorithms for nonlinear tree-sparse problems

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Computational Methods for Nonlinear Systems Analysis With Applications in Mathematics and Engineering

An investigation into current methods and new approaches for solving systems of nonlinear equations was performed. Nontraditional methods for implementing arc-length type solvers were developed in search of a more robust capability for solving general systems of nonlinear algebraic equations. Processes for construction of parameterized curves representing the many possible solutions to systems of equations versus finding single or point solutions were established. A procedure based on these methods was then developed to identify static equilibrium states for solutions to multi-body-dynamic systems. This methodology provided for a pictorial of the overall solution to a given system, which demonstrated the possibility of multiple candidate equilibrium states for which a procedure for selection of the proper state was proposed. Arc-length solvers were found to identify and more readily trace solution curves as compared to other solvers making such an approach practical. Comparison of proposed methods was made to existing methods found in the literature and commercial software with favorable results. Finally, means for parallel processing of the Jacobian matrix inherent to the arc-length and other nonlinear solvers were investigated, and an efficient approach for implementation was identified. Several case studies were performed to substantiate results. Commercial software was also used in some instances for additional results verification

Interior-point methods for PDE-constrained optimization

In applied sciences PDEs model an extensive variety of phenomena. Typically the final goal of simulations is a system which is optimal in a certain sense. For instance optimal control problems identify a control to steer a system towards a desired state. Inverse problems seek PDE parameters which are most consistent with measurements. In these optimization problems PDEs appear as equality constraints. PDE-constrained optimization problems are large-scale and often nonconvex. Their numerical solution leads to large ill-conditioned linear systems. In many practical problems inequality constraints implement technical limitations or prior knowledge. In this thesis interior-point (IP) methods are considered to solve nonconvex large-scale PDE-constrained optimization problems with inequality constraints. To cope with enormous fill-in of direct linear solvers, inexact search directions are allowed in an inexact interior-point (IIP) method. This thesis builds upon the IIP method proposed in [Curtis, Schenk, Wächter, SIAM Journal on Scientific Computing, 2010]. SMART tests cope with the lack of inertia information to control Hessian modification and also specify termination tests for the iterative linear solver. The original IIP method needs to solve two sparse large-scale linear systems in each optimization step. This is improved to only a single linear system solution in most optimization steps. Within this improved IIP framework, two iterative linear solvers are evaluated: A general purpose algebraic multilevel incomplete L D L^T preconditioned SQMR method is applied to PDE-constrained optimization problems for optimal server room cooling in three space dimensions and to compute an ambient temperature for optimal cooling. The results show robustness and efficiency of the IIP method when compared with the exact IP method. These advantages are even more evident for a reduced-space preconditioned (RSP) GMRES solver which takes advantage of the linear system's structure. This RSP-IIP method is studied on the basis of distributed and boundary control problems originating from superconductivity and from two-dimensional and three-dimensional parameter estimation problems in groundwater modeling. The numerical results exhibit the improved efficiency especially for multiple PDE constraints. An inverse medium problem for the Helmholtz equation with pointwise box constraints is solved by IP methods. The ill-posedness of the problem is explored numerically and different regularization strategies are compared. The impact of box constraints and the importance of Hessian modification on the optimization algorithm is demonstrated. A real world seismic imaging problem is solved successfully by the RSP-IIP method