Tree-Sparse Modeling and Solution of Multistage Stochastic Programs

Multistage stochastic programs are prototypical for nonlinear programs with an inherent tree structure inducing characteristic sparsity patterns in the KKT systems of interior methods. We present an integrated modeling and solution approach for such tree-sparse programs. Three closely related natural formulations having desirable control-theoretic properties lead to KKT system solution algorithms with linear complexity. Application examples from computational finance and process engineering demonstrate the efficiency of the approach

An elastic primal active-set method for structured QPs

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On the relationship between bilevel decomposition algorithms and direct interior-point methods

Engineers have been using bilevel decomposition algorithms to solve certain nonconvex large-scale optimization problems arising in engineering design projects. These algorithms transform the large-scale problem into a bilevel program with one upperlevel problem (the master problem) and several lower-level problems (the subproblems). Unfortunately, there is analytical and numerical evidence that some of these commonly used bilevel decomposition algorithms may fail to converge even when the starting point is very close to the minimizer. In this paper, we establish a relationship between a particular bilevel decomposition algorithm, which only performs one iteration of an interior-point method when solving the subproblems, and a direct interior-point method, which solves the problem in its original (integrated) form. Using this relationship, we formally prove that the bilevel decomposition algorithm converges locally at a superlinear rate. The relevance of our analysis is that it bridges the gap between the incipient local convergence theory of bilevel decomposition algorithms and the mature theory of direct interior-point methods

Exploiting Chordality in Optimization Algorithms for Model Predictive Control

In this chapter we show that chordal structure can be used to devise efficient optimization methods for many common model predictive control problems. The chordal structure is used both for computing search directions efficiently as well as for distributing all the other computations in an interior-point method for solving the problem. The chordal structure can stem both from the sequential nature of the problem as well as from distributed formulations of the problem related to scenario trees or other formulations. The framework enables efficient parallel computations.Comment: arXiv admin note: text overlap with arXiv:1502.0638

Numerical Methods for Scenario Tree Nonlinear Model Predictive Control

In this thesis we propose new methods in the field of numerical mathematics and stochastics for a model-based optimization method to control dynamical systems under uncertainty. In model-based control the model-plant mismatch is often large and unforeseen external influences on the dynamics must be taken into account. Therefore we extend the dynamical system by a stochastic component and approximate it by scenario trees. The combination of Nonlinear Model Predictive Control (NMPC) and the scenario tree approach to robustify with respect to the uncertainty is of growing interest. In engineering practice scenario tree NMPC yields a beneficial balance of the conservatism introduced by the robustification with respect to the uncertainty and the controller performance. However, there is a high numerical effort to solve the occuring optimization problems, which heavily depends on the design of the scenario tree used to approximate the uncertainty. A big challenge is then to control the system in real-time. The contribution of this work to the field of numerical optimization is a structure exploiting method for the large-scale optimization problems based on dual decomposition that yields smaller subproblems. They can be solved in a massively parallel fashion and additionally their discretization structure can be exploited numerically. Furthermore, this thesis presents novel methods to generate suitable scenario trees to approximate the uncertainty. Our scenario tree generation based on quadrature rules for sparse grids allows for scenario tree NMPC in high-dimensional uncertainty spaces with approximation properties of the quadrature rules. A further novel approach of this thesis to generate scenario trees is based on the interpretation of the underlying stochastic process as a Markov chain. Under the Markovian assumption we provide guarantees for the scenario tree approximation of the uncertainty. Finally, we present numerical results for scenario tree NMPC. We consider dynamical systems from the chemical industry and demonstrate that the methods developed in this thesis solve optimization problems with large scenario trees in real-time

Distributed algorithms for nonlinear tree-sparse problems

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Distributed Optimization with Application to Power Systems and Control

In many engineering domains, systems are composed of partially independent subsystems—power systems are composed of distribution and transmission systems, teams of robots are composed of individual robots, and chemical process systems are composed of vessels, heat exchangers and reactors. Often, these subsystems should reach a common goal such as satisfying a power demand with minimum cost, flying in a formation, or reaching an optimal set-point. At the same time, limited information exchange is desirable—for confidentiality reasons but also due to communication constraints. Moreover, a fast and reliable decision process is key as applications might be safety-critical. Mathematical optimization techniques are among the most successful tools for controlling systems optimally with feasibility guarantees. Yet, they are often centralized—all data has to be collected in one central and computationally powerful entity. Methods from distributed optimization control the subsystems in a distributed or decentralized fashion, reducing or avoiding central coordination. These methods have a long and successful history. Classical distributed optimization algorithms, however, are typically designed for convex problems. Hence, they are only partially applicable in the above domains since many of them lead to optimization problems with non-convex constraints. This thesis develops one of the first frameworks for distributed and decentralized optimization with non-convex constraints. Based on the Augmented Lagrangian Alternating Direction Inexact Newton (ALADIN) algorithm, a bi-level distributed ALADIN framework is presented, solving the coordination step of ALADIN in a decentralized fashion. This framework can handle various decentralized inner algorithms, two of which we develop here: a decentralized variant of the Alternating Direction Method of Multipliers (ADMM) and a novel decentralized Conjugate Gradient algorithm. Decentralized conjugate gradient is to the best of our knowledge the first decentralized algorithm with a guarantee of convergence to the exact solution in a finite number of iterates. Sufficient conditions for fast local convergence of bi-level ALADIN are derived. Bi-level ALADIN strongly reduces the communication and coordination effort of ALADIN and preserves its fast convergence guarantees. We illustrate these properties on challenging problems from power systems and control, and compare performance to the widely used ADMM. The developed methods are implemented in the open-source MATLAB toolbox ALADIN-—one of the first toolboxes for decentralized non-convex optimization. ALADIN- comes with a rich set of application examples from different domains showing its broad applicability. As an additional contribution, this thesis provides new insights why state-of-the-art distributed algorithms might encounter issues for constrained problems