Periodic Reporting for period 1 - SCARCE (Scalable Control Approximations for Resource Constrained Environments)
Periodo di rendicontazione: 2023-07-01 al 2025-12-31
This project aims at making a breakthrough contribution in optimal control and decision making for nonlinear processes that take place on hierarchical network structures and are dynamic in time and/or space. While setting has a wide range of potential domains of applicability with high societal relevance: thermal, electric, or fluid dynamics in energy networks, traffic and logistics, disease spreading dynamics, or cell signalling in biomedicine. This project focuses on energy scarcity control tasks and will pursue the following objectives: To contribute new theory, to develop numerical approximation methods, to implement algorithmic methods in software, and to conduct proof-of-concept studies. Research in the young field of mixed-integer optimal control (MIOC) has recently seen increased momentum together with numerical approximation algorithms and control theory. Despite initial successes, key questions remain unsolved because of a lack of analytical understanding, a lack of tractable formulations, the unavailability of efficient large-scale solvers, or the insufficiency of existing implementations. This project focuses on pivotal but poorly understood topics: decomposition, relaxation, and combinatorial integral approximation; domains admitting homo- genization and limiting processes using weak topologies; tractable approximations of direct costs of decisions; efficient distributed and parallel nonlinear solvers; and robustness of approximate nonlinear decision policies under uncertainty. Due to non-trivial interactions emerging in theory and the unavailability of comprehensive algorithms, these topics cannot be suitably handled by merely combining the respective states of the art. A focused effort to decisively extend MIOC to optimal decision policies for dynamics on hierarchical networks is therefore a timely endeavour that will help to address the challenging demands of practitioners. Proof-of-concept studies for energy scarcity control scenarios in power systems, in heat/cooling, and in mobility will assess the applicability of the solutions proposed.
Work during the initial two years has focused on the theoretical and algorithmic work packages 1 to 4.
In WP1 (Dynamics on Hierarchical Networks), we embrace the notion of graphons, which express a continuous generalization of networks. A graphon is able to encode a full hierarchy of synthetic networks together into a unified object, offering a minimal stage for theoretical exploration.
In this setting, we prove that a smaller approximate graph closely reproduces the nonlinear dynamics on the original network, which in turn yields an approximate guide for the control variables needed to reach a desired state on the original network.
In WP2 (Decomposition and Topologies), we focus on the reconstruction of integer control approximation for a relaxed formulation of a partial differential equation (PDE) constrained optimisation problem. The main achievement generalises the previous approach of reconstructing the integer controls along a structured space-filling curve throughout the domain the PDE is defined in. Now, this generalisation allows any unstructured and randomised space-filling curve to direct the sum-up rounding algorithm throughout the space and leveraging a stochastic argument, we prove a convergence with high probability.
In WP3 (Homogenization and Reconstruction) the focus so far has been on the implications of integer reconstruction on the closed-loop optimal control behaviour of dynamical systems governed by nonlinear differential equations.
A large-scale graph dynamical systems can be represented as a semi-linear system of nonlinear differential equations. We apply partial outer convexification and relaxation to handle mixed-integer and finite control set optimal control problems, and design asymptotically stabilising model predictive control in the relaxed domain.
In WP4 (Decisions and Cost) we have addressed the problem of reducing the costs of the remaining rounding decisions in the context of (open-loop) mixed-integer optimal control by repeatedly reoptimising the relaxed and shortened OCP after each rounding step.
In WP1 (Dynamics on Hierarchical Networks), we embrace the notion of graphons, which express a continuous generalization of networks. A graphon is able to encode a full hierarchy of synthetic networks together into a unified object, offering a minimal stage for theoretical exploration.
In this setting, we prove that a smaller approximate graph closely reproduces the nonlinear dynamics on the original network, which in turn yields an approximate guide for the control variables needed to reach a desired state on the original network.
In WP2 (Decomposition and Topologies), we focus on the reconstruction of integer control approximation for a relaxed formulation of a partial differential equation (PDE) constrained optimisation problem. The main achievement generalises the previous approach of reconstructing the integer controls along a structured space-filling curve throughout the domain the PDE is defined in. Now, this generalisation allows any unstructured and randomised space-filling curve to direct the sum-up rounding algorithm throughout the space and leveraging a stochastic argument, we prove a convergence with high probability.
In WP3 (Homogenization and Reconstruction) the focus so far has been on the implications of integer reconstruction on the closed-loop optimal control behaviour of dynamical systems governed by nonlinear differential equations.
A large-scale graph dynamical systems can be represented as a semi-linear system of nonlinear differential equations. We apply partial outer convexification and relaxation to handle mixed-integer and finite control set optimal control problems, and design asymptotically stabilising model predictive control in the relaxed domain.
In WP4 (Decisions and Cost) we have addressed the problem of reducing the costs of the remaining rounding decisions in the context of (open-loop) mixed-integer optimal control by repeatedly reoptimising the relaxed and shortened OCP after each rounding step.