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Optimization and Control (math.OC)

Tue, 18 Jul 2023

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1.Solution of the Optimal Control Problem for the Cahn-Hilliard Equation Using Finite Difference Approximation

Authors:Gobinda Garai, Bankim C. Mandal

Abstract: This paper is concerned with the designing, analyzing and implementing linear and nonlinear discretization scheme for the distributed optimal control problem (OCP) with the Cahn-Hilliard (CH) equation as constrained. We propose three difference schemes to approximate and investigate the solution behaviour of the OCP for the CH equation. We present the convergence analysis of the proposed discretization. We verify our findings by presenting numerical experiments.

2.Harnessing the mathematics of matrix decomposition to solve planted and maximum clique problem

Authors:Salma Omer, Montaz Ali

Abstract: We consider the problem of identifying a maximum clique in a given graph. We have proposed a mathematical model for this problem. The model resembles the matrix decomposition of the adjacency matrix of a given graph. The objective function of the mathematical model includes a weighted $\ell_{1}$-norm of the sparse matrix of the decomposition, which has an advantage over the known $\ell_{1}-$norm in reducing the error. The use of dynamically changing the weights for the $\ell_{1}$-norm has been motivated. We have used proximal operators within the iterates of the ADMM (alternating direction method of multipliers) algorithm to solve the optimization problem. Convergence of the proposed ADMM algorithm has been provided. The theoretical guarantee of the maximum clique in the form of the low-rank matrix has also been established using the golfing scheme to construct approximate dual certificates. We have constructed conditions that guarantee the recovery and uniqueness of the solution, as well as a tight bound on the dual matrix that validates optimality conditions. Numerical results for planted cliques are presented showing clear advantages of our model when compared with two recent mathematical models. Results are also presented for randomly generated graphs with minimal errors. These errors are found using a formula we have proposed based on the size of the clique. Moreover, we have applied our algorithm to real-world graphs for which cliques have been recovered successfully. The validity of these clique sizes comes from the decomposition of input graph into a rank-one matrix (corresponds to the clique) and a sparse matrix.

3.Globally solving the Gromov-Wasserstein problem for point clouds in low dimensional Euclidean spaces

Authors:Martin Ryner, Jan Kronqvist, Johan Karlsson

Abstract: This paper presents a framework for computing the Gromov-Wasserstein problem between two sets of points in low dimensional spaces, where the discrepancy is the squared Euclidean norm. The Gromov-Wasserstein problem is a generalization of the optimal transport problem that finds the assignment between two sets preserving pairwise distances as much as possible. This can be used to quantify the similarity between two formations or shapes, a common problem in AI and machine learning. The problem can be formulated as a Quadratic Assignment Problem (QAP), which is in general computationally intractable even for small problems. Our framework addresses this challenge by reformulating the QAP as an optimization problem with a low-dimensional domain, leveraging the fact that the problem can be expressed as a concave quadratic optimization problem with low rank. The method scales well with the number of points, and it can be used to find the global solution for large-scale problems with thousands of points. We compare the computational complexity of our approach with state-of-the-art methods on synthetic problems and apply it to a near-symmetrical problem which is of particular interest in computational biology.

4.Decentralized Stochastic Linear-Quadratic Optimal Control with Risk Constraint and Partial Observation

Authors:Jia Hui, Yuan-Hua Ni

Abstract: This paper addresses a risk-constrained decentralized stochastic linear-quadratic optimal control problem with one remote controller and one local controller, where the risk constraint is posed on the cumulative state weighted variance in order to reduce the oscillation of system trajectory. In this model, local controller can only partially observe the system state, and sends the estimate of state to remote controller through an unreliable channel, whereas the channel from remote controller to local controllers is perfect. For the considered constrained optimization problem, we first punish the risk constraint into cost function through Lagrange multiplier method, and the resulting augmented cost function will include a quadratic mean-field term of state. In the sequel, for any but fixed multiplier, explicit solutions to finite-horizon and infinite-horizon mean-field decentralized linear-quadratic problems are derived together with necessary and sufficient condition on the mean-square stability of optimal system. Then, approach to find the optimal Lagrange multiplier is presented based on bisection method. Finally, two numerical examples are given to show the efficiency of the obtained results.

5.A Sweeping Process Control Problem Subject To Mixed Constraints

Authors:Karla L. Cortez, Nathalie T. Khalil, Julio E. Solis

Abstract: In this study, we investigate optimal control problems that involve sweeping processes with a drift term and mixed inequality constraints. Our goal is to establish necessary optimality conditions for these problems. We address the challenges that arise due to the combination of sweeping processes and inequality mixed constraints in two contexts: regular and non-regular. This requires working with different types of multipliers, such as finite positive Radon measures for the sweeping term and integrable functions for regular mixed constraints. For non-regular mixed constraints, the multipliers correspond to purely finitely additive set functions.

6.BOP-Elites, a Bayesian Optimisation Approach to Quality Diversity Search with Black-Box descriptor functions

Authors:Paul Kent, Adam Gaier, Jean-Baptiste Mouret, Juergen Branke

Abstract: Quality Diversity (QD) algorithms such as MAP-Elites are a class of optimisation techniques that attempt to find many high performing points that all behave differently according to a user-defined behavioural metric. In this paper we propose the Bayesian Optimisation of Elites (BOP-Elites) algorithm. Designed for problems with expensive black-box fitness and behaviour functions, it is able to return a QD solution-set with excellent final performance already after a relatively small number of samples. BOP-Elites models both fitness and behavioural descriptors with Gaussian Process (GP) surrogate models and uses Bayesian Optimisation (BO) strategies for choosing points to evaluate in order to solve the quality-diversity problem. In addition, BOP-Elites produces high quality surrogate models which can be used after convergence to predict solutions with any behaviour in a continuous range. An empirical comparison shows that BOP-Elites significantly outperforms other state-of-the-art algorithms without the need for problem-specific parameter tuning.

7.Disturbance decoupled functional observers for fault estimation in nonlinear systems

Authors:Sunjeev Venkateswaran, Costas Kravaris

Abstract: This work deals with the problem of designing disturbance decupled observers for the estimation of a function of the states in nonlinear systems. Necessary and sufficient conditions for the existence of lower order disturbance decoupled functional observers with linear dynamics and linear output map are derived. Based on this methodology, a fault-estimation scheme based on disturbance decoupled observers will be presented. Throughout the paper, the application of the results will be illustrated through a chemical reactor case study

8.Grid-Forming Hybrid Angle Control: Behavior, Stability, Variants and Verification

Authors:Ali Tayyebi, Denis Vettoretti, Adolfo Anta, Florian Dörfler

Abstract: This work explores the stability, behavior, variants, and a controller-hardware-in-the-loop (C-HiL) verification of the recently proposed grid-forming (GFM) hybrid angle control (HAC). We revisit the foundation of GFM HAC, and highlight its behavioral properties in relation to the conventional synchronous machine (SM). Next, we introduce the required complementary controls to be combined with the HAC to realize a GFM behavior. The characterization of the analytical operating point and nonlinear energy-based stability analysis of a grid-connected converter under the HAC is presented. Further, we consider various output filter configurations and derive an approximation for the original control proposal. Moreover, we provide details on the integration of GFM HAC into a complex converter control architecture and introduce several variants of the standard HAC. Finally, the performance of GFM HAC is verified by several test scenarios in a C-HiL setup to test its behavior against real-world effect such as noise and delays.

9.Jointly Improving the Sample and Communication Complexities in Decentralized Stochastic Minimax Optimization

Authors:Xuan Zhang, Gabriel Mancino-Ball, Necdet Serhat Aybat, Yangyang Xu

Abstract: We propose a novel single-loop decentralized algorithm called DGDA-VR for solving the stochastic nonconvex strongly-concave minimax problem over a connected network of $M$ agents. By using stochastic first-order oracles to estimate the local gradients, we prove that our algorithm finds an $\epsilon$-accurate solution with $\mathcal{O}(\epsilon^{-3})$ sample complexity and $\mathcal{O}(\epsilon^{-2})$ communication complexity, both of which are optimal and match the lower bounds for this class of problems. Unlike competitors, our algorithm does not require multiple communications for the convergence results to hold, making it applicable to a broader computational environment setting. To the best of our knowledge, this is the first such algorithm to jointly optimize the sample and communication complexities for the problem considered here.