Boris Houska

Alexander Engelmann, Yuning Jiang, Tillmann Mühlpfordt et al.

The present paper discusses the application of the recently proposed Augmented Lagrangian Alternating Direction Inexact Newton (ALADIN) method to non-convex AC Optimal Power Flow Problems (OPF) in a distributed fashion. In contrast to the often used Alternating Direction of Multipliers Method (ADMM), ALADIN guarantees locally quadratic convergence for AC OPF. Numerical results for 5 to 300 bus test cases indicate that ALADIN is able to outperform ADMM and to reduce the number of iterations by about one order of magnitude. We compare ALADIN to numerical results for ADMM documented in the literature. The improved convergence speed comes at the cost of increasing the communication effort per iteration. Therefore, we propose a variant of ALADIN that uses inexact Hessians to reduce communication. Additionally, we provide a detailed comparison of these ALADIN variants to ADMM from an algorithmic and communication perspective. Moreover, we prove that ALADIN converges locally at quadratic rate even for the relevant case of suboptimally solved local NLPs.

Alexander Engelmann, Yuning Jiang, Boris Houska et al.

Decentralized optimization algorithms are important in different contexts, such as distributed optimal power flow or distributed model predictive control, as they avoid central coordination and enable decomposition of large-scale problems. In case of constrained non-convex optimization only a few algorithms are currently are available; often their performance is limited, or they lack convergence guarantees. This paper proposes a framework for decentralized non-convex optimization via bi-level distribution of the Augmented Lagrangian Alternating Direction Inexact Newton (ALADIN) algorithm. Bi-level distribution means that the outer ALADIN structure is combined with an inner distribution/decentralization level solving a condensed variant of ALADIN's convex coordination QP by decentralized algorithms. We prove sufficient conditions ensuring local convergence while allowing for inexact decentralized/distributed solutions of the coordination QP. Moreover, we show how a decentralized variant of conjugate gradient or decentralized ADMM schemes can be employed at the inner level. We draw upon case studies from power systems and robotics to illustrate the performance of the proposed framework.

Xu Du, Alexander Engelmann, Yuning Jiang et al.

This paper proposes a structure exploiting algorithm for solving non-convex power system state estimation problems in distributed fashion. Because the power flow equations in large electrical grid networks are non-convex equality constraints, we develop a tailored state estimator based on Augmented Lagrangian Alternating Direction Inexact Newton (ALADIN) method, which can handle the nonlinearities efficiently. Here, our focus is on using Gauss-Newton Hessian approximations within ALADIN in order to arrive at at an efficient (computationally and communicationally) variant of ALADIN for network maximum likelihood estimation problems. Analyzing the IEEE 30-Bus system we illustrate how the proposed algorithm can be used to solve highly non-trivial network state estimation problems. We also compare the method with existing distributed parameter estimation codes in order to illustrate its performance.

Yanlin Zha, Mario E. Villanueva, Boris Houska

This paper presents a novel set-based computing method, called interval superposition arithmetic, for enclosing the image set of multivariate factorable functions on a given domain. In order to construct such enclosures, the proposed arithmetic operates over interval superposition models which are parameterized by a matrix with interval components. Every point in the domain of a factorable function is then associated with a sequence of components of this matrix and the superposition, i.e. Minkowski sum, of these elements encloses the image of the function at this point. Interval superposition arithmetic has a linear runtime complexity with respect to the number of variables. Besides presenting a detailed theoretical analysis of the accuracy and convergence properties of interval superposition arithmetic, the paper illustrates its advantages compared to existing set arithmetics via numerical examples.

Junyan Su, Yanlin Zha, Kai Wang et al.

The problem of guaranteed parameter estimation (GPE) consists in enclosing the set of all possible parameter values, such that the model predictions match the corresponding measurements within prescribed error bounds. One of the bottlenecks in GPE algorithms is the construction of enclosures for the image-set of factorable functions. In this paper, we introduce a novel set-based computing method called interval superposition arithmetics (ISA) for the construction of enclosures of such image sets and its use in GPE algorithms. The main benefits of using ISA in the context of GPE lie in the improvement of enclosure accuracy and in the implied reduction of number set-membership tests of the set-inversion algorithm.

Nicolas Hoischen, Petar Bevanda, Stefan Sosnowski et al.

This paper proposes a fully data-driven approach for optimal control of nonlinear control-affine systems represented by a stochastic diffusion. The focus is on the scenario where both the nonlinear dynamics and stage cost functions are unknown, while only a control penalty function and constraints are provided. To this end, we embed state probability densities into a reproducing kernel Hilbert space (RKHS) to leverage recent advances in operator regression, thereby identifying Markov transition operators associated with controlled diffusion processes. This operator learning approach integrates naturally with convex operator-theoretic Hamilton-Jacobi-Bellman recursions that scale linearly with state dimensionality, effectively solving a wide range of nonlinear optimal control problems. Numerical results demonstrate its ability to address diverse nonlinear control tasks, including the depth regulation of an autonomous underwater vehicle.

Nicolas Hoischen, Petar Bevanda, Max Beier et al.

Continuous-time stochastic processes underlie many natural and engineered systems. In healthcare, autonomous driving, and industrial control, direct interaction with the environment is often unsafe or impractical, motivating offline reinforcement learning from historical data. However, there is limited statistical understanding of the approximation errors inherent in learning policies from offline datasets. We address this by linking reinforcement learning to the Hamilton-Jacobi-Bellman equation and proposing an operator-theoretic algorithm based on a simple dynamic programming recursion. Specifically, we represent our world model in terms of the infinitesimal generator of controlled diffusion processes learned in a reproducing kernel Hilbert space. By integrating statistical learning methods and operator theory, we establish global convergence of the value function and derive finite-sample guarantees with bounds tied to system properties such as smoothness and stability. Our theoretical and numerical results indicate that operator-based approaches may hold promise in solving offline reinforcement learning using continuous-time optimal control.

Sampath Kumar Mulagaleti, Boris Houska, Mario Zanon et al.

This paper presents a novel approach to synthesize stabilizing termi- nal ingredients for linear model predictive control (MPC) schemes, with the aim of increasing the region of attraction while reducing suboptimal- ity with respect to the solution of the infinite-horizon optimal control problem. It is based on the construction of a novel terminal region using methods from the field of configuration-constrained polytopic computing, along with a terminal cost that is exactly equal to the infinite-horizon linear-quadratic regulator cost in a nontrivial neighborhood of the steady- state. The practical performance of the controller is illustrated through various case studies, and comparisons with state-of-the-art approaches are presented.

Yanlin Zha, Mario Eduardo Villanueva, Boris Houska et al.

This article presents an arithmetic, called superposition relaxation, for bracketing the graph of a multivariate factorable function on a compact domain between a pair of underestimating and overestimating functions that are both separable. Propagation rules are established for affine and nonlinear composition operations, with a focus on exploiting global monotonicity and convexity properties in the composition. The local convergence properties of this arithmetic are also analyzed in both the pointwise and Hausdorff sense, including conditions under which quadratic pointwise convergence propagates through composition. Parameterizations of the univariate summands in a superposition relaxation either as piecewise-constant or continuous piecewise-linear functions are discussed for a practical implementation. It is shown through numerical case studies that superposition relaxations can be consistently tighter than McCormick relaxations, including for the relaxation of artificial neural networks. But superposition relaxations also incur a higher computational cost than McCormick relaxations. Further investigations are thus warranted as applications in global optimization seek to balance a relaxation's tightness with its computational cost.

Petar Bevanda, Nicolas Hoischen, Tobias Wittmann et al.

This paper presents a novel operator-theoretic approach for optimal control of nonlinear stochastic systems within reproducing kernel Hilbert spaces. Our learning framework leverages data samples of system dynamics and stage cost functions, with only control penalties and constraints provided. The proposed method directly learns the infinitesimal generator of a controlled stochastic diffusion in an infinite-dimensional hypothesis space. We demonstrate that our approach seamlessly integrates with modern convex operator-theoretic Hamilton-Jacobi-Bellman recursions, enabling a data-driven solution to the optimal control problems. Furthermore, our learning framework includes nonparametric estimators for uncontrolled infinitesimal generators as a special case. Numerical experiments, ranging from synthetic differential equations to simulated robotic systems, showcase the advantages of our approach compared to both modern data-driven and classical nonlinear programming methods for optimal control.