Giulio Fattore, Marco Peruzzo, Giacomo Sartori, Mattia Zorzi
This paper addresses the problem of learning the impulse responses characterizing forward models by means of a regularized kernel-based Prediction Error Method (PEM). The common approach to accomplish that is to approximate the system with a high-order stable ARX model. However, such choice induces a certain undesired prior information in the system that we want to estimate. To overcome this issue, we propose a new kernel-based paradigm which is formulated directly in terms of the impulse responses of the forward model and leading to the identification of a high-order MAX model. The most challenging step is the estimation of the kernel hyperparameters optimizing the marginal likelihood. The latter, indeed, does not admit a closed form expression. We propose a method for evaluating the marginal likelihood which makes possible the hyperparameters estimation. Finally, some numerical results showing the effectiveness of the method are presented.
本文通过基于正则化核的预测误差法（PEM）来解决学习前向模型脉冲响应特征的问题。常用的方法是用高阶稳定 ARX 模型来逼近系统。然而，这种选择会在我们想要估计的系统中引起某些不想要的先验信息。为了克服这个问题，我们提出了一种基于核的新范式，它直接根据前向模型的脉冲响应进行表述，从而识别出高阶 MAX 模型。最具挑战性的步骤是优化边际似然的核超参数估计。我们提出了一种评估边际似然的方法，这使得超参数估计成为可能。最后，一些数值结果显示了该方法的有效性。
{"title":"A kernel-based PEM estimator for forward model","authors":"Giulio Fattore, Marco Peruzzo, Giacomo Sartori, Mattia Zorzi","doi":"arxiv-2409.09679","DOIUrl":"https://doi.org/arxiv-2409.09679","url":null,"abstract":"This paper addresses the problem of learning the impulse responses\u0000characterizing forward models by means of a regularized kernel-based Prediction\u0000Error Method (PEM). The common approach to accomplish that is to approximate\u0000the system with a high-order stable ARX model. However, such choice induces a\u0000certain undesired prior information in the system that we want to estimate. To\u0000overcome this issue, we propose a new kernel-based paradigm which is formulated\u0000directly in terms of the impulse responses of the forward model and leading to\u0000the identification of a high-order MAX model. The most challenging step is the\u0000estimation of the kernel hyperparameters optimizing the marginal likelihood.\u0000The latter, indeed, does not admit a closed form expression. We propose a\u0000method for evaluating the marginal likelihood which makes possible the\u0000hyperparameters estimation. Finally, some numerical results showing the\u0000effectiveness of the method are presented.","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254545","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Lossy image compression algorithms play a crucial role in various domains, including graphics, and image processing. As image information density increases, so do the resources required for processing and transmission. One of the most prominent approaches to address this challenge is color quantization, proposed by Orchard et al. (1991). This technique optimally maps each pixel of an image to a color from a limited palette, maintaining image resolution while significantly reducing information content. Color quantization can be interpreted as a clustering problem (Krishna et al. (1997), Wan (2019)), where image pixels are represented in a three-dimensional space, with each axis corresponding to the intensity of an RGB channel. However, scaling of traditional algorithms like K-Means can be challenging for large data, such as modern images with millions of colors. This paper reframes color quantization as a three-dimensional stochastic transportation problem between the set of image pixels and an optimal color palette, where the number of colors is a predefined hyperparameter. We employ Stochastic Quantization (SQ) with a seeding technique proposed by Arthur et al. (2007) to enhance the scalability of color quantization. This method introduces a probabilistic element to the quantization process, potentially improving efficiency and adaptability to diverse image characteristics. To demonstrate the efficiency of our approach, we present experimental results using images from the ImageNet dataset. These experiments illustrate the performance of our Stochastic Quantization method in terms of compression quality, computational efficiency, and scalability compared to traditional color quantization techniques.
{"title":"Lossy Image Compression with Stochastic Quantization","authors":"Anton Kozyriev, Vladimir Norkin","doi":"arxiv-2409.09488","DOIUrl":"https://doi.org/arxiv-2409.09488","url":null,"abstract":"Lossy image compression algorithms play a crucial role in various domains,\u0000including graphics, and image processing. As image information density\u0000increases, so do the resources required for processing and transmission. One of\u0000the most prominent approaches to address this challenge is color quantization,\u0000proposed by Orchard et al. (1991). This technique optimally maps each pixel of\u0000an image to a color from a limited palette, maintaining image resolution while\u0000significantly reducing information content. Color quantization can be\u0000interpreted as a clustering problem (Krishna et al. (1997), Wan (2019)), where\u0000image pixels are represented in a three-dimensional space, with each axis\u0000corresponding to the intensity of an RGB channel. However, scaling of\u0000traditional algorithms like K-Means can be challenging for large data, such as\u0000modern images with millions of colors. This paper reframes color quantization\u0000as a three-dimensional stochastic transportation problem between the set of\u0000image pixels and an optimal color palette, where the number of colors is a\u0000predefined hyperparameter. We employ Stochastic Quantization (SQ) with a\u0000seeding technique proposed by Arthur et al. (2007) to enhance the scalability\u0000of color quantization. This method introduces a probabilistic element to the\u0000quantization process, potentially improving efficiency and adaptability to\u0000diverse image characteristics. To demonstrate the efficiency of our approach,\u0000we present experimental results using images from the ImageNet dataset. These\u0000experiments illustrate the performance of our Stochastic Quantization method in\u0000terms of compression quality, computational efficiency, and scalability\u0000compared to traditional color quantization techniques.","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254546","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, the initial error affection and error correction in linear quadratic mean field games (MPLQMFGs) under erroneous initial distribution information are investigated. First, a LQMFG model is developed where agents are coupled by dynamics and cost functions. Next, by studying the evolutionary of LQMFGs under erroneous initial distributions information, the affection of initial error on the game and agents' strategies are given. Furthermore, under deterministic situation, we provide a sufficient condition for agents to correct initial error and give their optimal strategies when agents are allowed to change their strategies at a intermediate time. Besides, the situation where agents are allowed to predict MF and adjust their strategies in real-time is considered. Finally, simulations are performed to verify above conclusions.
{"title":"Initial Error Affection and Error Correction in Linear Quadratic Mean Field Games under Erroneous Initial Information","authors":"Yuxin Jin, Lu Ren, Wang Yao, Xiao Zhang","doi":"arxiv-2409.09375","DOIUrl":"https://doi.org/arxiv-2409.09375","url":null,"abstract":"In this paper, the initial error affection and error correction in linear\u0000quadratic mean field games (MPLQMFGs) under erroneous initial distribution\u0000information are investigated. First, a LQMFG model is developed where agents\u0000are coupled by dynamics and cost functions. Next, by studying the evolutionary\u0000of LQMFGs under erroneous initial distributions information, the affection of\u0000initial error on the game and agents' strategies are given. Furthermore, under\u0000deterministic situation, we provide a sufficient condition for agents to\u0000correct initial error and give their optimal strategies when agents are allowed\u0000to change their strategies at a intermediate time. Besides, the situation where\u0000agents are allowed to predict MF and adjust their strategies in real-time is\u0000considered. Finally, simulations are performed to verify above conclusions.","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254549","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is mainly concerned with the observability inequalities for heat equations with time-dependent Lipschtiz potentials. The observability inequality for heat equations asserts that the total energy of a solution is bounded above by the energy localized in a subdomain with an observability constant. For a bounded measurable potential $V = V(x,t)$, the factor in the observability constant arising from the Carleman estimate is best known to be $exp(C|V|_{infty}^{2/3})$ (even for time-independent potentials). In this paper, we show that, for Lipschtiz potentials, this factor can be replaced by $exp(C(|nabla V|_{infty}^{1/2} +|partial_tV|_{infty}^{1/3} ))$, which improves the previous bound $exp(C|V|_{infty}^{2/3})$ in some typical scenarios. As a consequence, with such a Lipschitz potential, we obtain a quantitative regular control in a null controllability problem. In addition, for the one-dimensional heat equation with some time-independent bounded measurable potential $V = V(x)$, we obtain the optimal observability constant.
{"title":"Observability inequalities for heat equations with potentials","authors":"Jiuyi Zhu, Jinping Zhuge","doi":"arxiv-2409.09476","DOIUrl":"https://doi.org/arxiv-2409.09476","url":null,"abstract":"This paper is mainly concerned with the observability inequalities for heat\u0000equations with time-dependent Lipschtiz potentials. The observability\u0000inequality for heat equations asserts that the total energy of a solution is\u0000bounded above by the energy localized in a subdomain with an observability\u0000constant. For a bounded measurable potential $V = V(x,t)$, the factor in the\u0000observability constant arising from the Carleman estimate is best known to be\u0000$exp(C|V|_{infty}^{2/3})$ (even for time-independent potentials). In this\u0000paper, we show that, for Lipschtiz potentials, this factor can be replaced by\u0000$exp(C(|nabla V|_{infty}^{1/2} +|partial_tV|_{infty}^{1/3} ))$, which\u0000improves the previous bound $exp(C|V|_{infty}^{2/3})$ in some typical\u0000scenarios. As a consequence, with such a Lipschitz potential, we obtain a\u0000quantitative regular control in a null controllability problem. In addition,\u0000for the one-dimensional heat equation with some time-independent bounded\u0000measurable potential $V = V(x)$, we obtain the optimal observability constant.","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254548","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Dynamic optimization problems involving discrete decisions have several applications, yet lead to challenging optimization problems that must be addressed efficiently. Combining discrete variables with potentially nonlinear constraints stemming from dynamics within an optimization model results in mathematical programs for which off-the-shelf techniques might be insufficient. This work uses a novel approach, the Logic-based Discrete-Steepest Descent Algorithm (LD-SDA), to solve Discrete Dynamic Optimization problems. The problems are formulated using Boolean variables that enforce differential systems of constraints and encode logic constraints that the optimization problem needs to satisfy. By posing the problem as a generalized disjunctive program with dynamic equations within the disjunctions, the LD-SDA takes advantage of the problem's inherent structure to efficiently explore the combinatorial space of the Boolean variables and selectively include relevant differential equations to mitigate the computational complexity inherent in dynamic optimization scenarios. We rigorously evaluate the LD-SDA with benchmark problems from the literature that include dynamic transitioning modes and find it to outperform traditional methods, i.e., mixed-integer nonlinear and generalized disjunctive programming solvers, in terms of efficiency and capability to handle dynamic scenarios. This work presents a systematic method and provides an open-source software implementation to address these discrete dynamic optimization problems by harnessing the information within its logical-differential structure.
{"title":"Addressing Discrete Dynamic Optimization via a Logic-Based Discrete-Steepest Descent Algorithm","authors":"Zedong Peng, Albert Lee, David E. Bernal Neira","doi":"arxiv-2409.09237","DOIUrl":"https://doi.org/arxiv-2409.09237","url":null,"abstract":"Dynamic optimization problems involving discrete decisions have several\u0000applications, yet lead to challenging optimization problems that must be\u0000addressed efficiently. Combining discrete variables with potentially nonlinear\u0000constraints stemming from dynamics within an optimization model results in\u0000mathematical programs for which off-the-shelf techniques might be insufficient.\u0000This work uses a novel approach, the Logic-based Discrete-Steepest Descent\u0000Algorithm (LD-SDA), to solve Discrete Dynamic Optimization problems. The\u0000problems are formulated using Boolean variables that enforce differential\u0000systems of constraints and encode logic constraints that the optimization\u0000problem needs to satisfy. By posing the problem as a generalized disjunctive\u0000program with dynamic equations within the disjunctions, the LD-SDA takes\u0000advantage of the problem's inherent structure to efficiently explore the\u0000combinatorial space of the Boolean variables and selectively include relevant\u0000differential equations to mitigate the computational complexity inherent in\u0000dynamic optimization scenarios. We rigorously evaluate the LD-SDA with\u0000benchmark problems from the literature that include dynamic transitioning modes\u0000and find it to outperform traditional methods, i.e., mixed-integer nonlinear\u0000and generalized disjunctive programming solvers, in terms of efficiency and\u0000capability to handle dynamic scenarios. This work presents a systematic method\u0000and provides an open-source software implementation to address these discrete\u0000dynamic optimization problems by harnessing the information within its\u0000logical-differential structure.","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254552","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we consider the adaptive linear quadratic Gaussian control problem, where both the linear transformation matrix of the state $A$ and the control gain matrix $B$ are unknown. The proposed adaptive optimal control only assumes that $(A, B)$ is stabilizable and $(A, Q^{1/2})$ is detectable, where $Q$ is the weighting matrix of the state in the quadratic cost function. This condition significantly weakens the classic assumptions used in the literature. To tackle this problem, a weighted least squares algorithm is modified by using random regularization method, which can ensure uniform stabilizability and uniform detectability of the family of estimated models. At the same time, a diminishing excitation is incorporated into the design of the proposed adaptive control to guarantee strong consistency of the desired components of the estimates. Finally, by utilizing this family of estimates, even if not all components of them converge to the true values, it is demonstrated that a certainty equivalence control with such a diminishing excitation is optimal for an ergodic quadratic cost function.
{"title":"Optimal Adaptive Control of Linear Stochastic Systems with Quadratic Cost Function","authors":"Nian Liu, Cheng Zhao, Shaolin Tan, Jinhu Lü","doi":"arxiv-2409.09250","DOIUrl":"https://doi.org/arxiv-2409.09250","url":null,"abstract":"In this paper, we consider the adaptive linear quadratic Gaussian control\u0000problem, where both the linear transformation matrix of the state $A$ and the\u0000control gain matrix $B$ are unknown. The proposed adaptive optimal control only\u0000assumes that $(A, B)$ is stabilizable and $(A, Q^{1/2})$ is detectable, where\u0000$Q$ is the weighting matrix of the state in the quadratic cost function. This\u0000condition significantly weakens the classic assumptions used in the literature.\u0000To tackle this problem, a weighted least squares algorithm is modified by using\u0000random regularization method, which can ensure uniform stabilizability and\u0000uniform detectability of the family of estimated models. At the same time, a\u0000diminishing excitation is incorporated into the design of the proposed adaptive\u0000control to guarantee strong consistency of the desired components of the\u0000estimates. Finally, by utilizing this family of estimates, even if not all\u0000components of them converge to the true values, it is demonstrated that a\u0000certainty equivalence control with such a diminishing excitation is optimal for\u0000an ergodic quadratic cost function.","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254550","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Duo Xu, Rody Aerts, Petros Karamanakos, Mircea Lazar
This paper considers learning online (implicit) nonlinear model predictive control (MPC) laws using neural networks and Laguerre functions. Firstly, we parameterize the control sequence of nonlinear MPC using Laguerre functions, which typically yields a smoother control law compared to the original nonlinear MPC law. Secondly, we employ neural networks to learn the coefficients of the Laguerre nonlinear MPC solution, which comes with several benefits, namely the dimension of the learning space is dictated by the number of Laguerre functions and the complete predicted input sequence can be used to learn the coefficients. To mitigate constraints violation for neural approximations of nonlinear MPC, we develop a constraints-informed loss function that penalizes the violation of polytopic state constraints during learning. Box input constraints are handled by using a clamp function in the output layer of the neural network. We demonstrate the effectiveness of the developed framework on a nonlinear buck-boost converter model with sampling rates in the sub-millisecond range, where online nonlinear MPC would not be able to run in real time. The developed constraints-informed neural-Laguerre approximation yields similar performance with long-horizon online nonlinear MPC, but with execution times of a few microseconds, as validated on a field-programmable gate array (FPGA) platform.
{"title":"Constraints-Informed Neural-Laguerre Approximation of Nonlinear MPC with Application in Power Electronics","authors":"Duo Xu, Rody Aerts, Petros Karamanakos, Mircea Lazar","doi":"arxiv-2409.09436","DOIUrl":"https://doi.org/arxiv-2409.09436","url":null,"abstract":"This paper considers learning online (implicit) nonlinear model predictive\u0000control (MPC) laws using neural networks and Laguerre functions. Firstly, we\u0000parameterize the control sequence of nonlinear MPC using Laguerre functions,\u0000which typically yields a smoother control law compared to the original\u0000nonlinear MPC law. Secondly, we employ neural networks to learn the\u0000coefficients of the Laguerre nonlinear MPC solution, which comes with several\u0000benefits, namely the dimension of the learning space is dictated by the number\u0000of Laguerre functions and the complete predicted input sequence can be used to\u0000learn the coefficients. To mitigate constraints violation for neural\u0000approximations of nonlinear MPC, we develop a constraints-informed loss\u0000function that penalizes the violation of polytopic state constraints during\u0000learning. Box input constraints are handled by using a clamp function in the\u0000output layer of the neural network. We demonstrate the effectiveness of the\u0000developed framework on a nonlinear buck-boost converter model with sampling\u0000rates in the sub-millisecond range, where online nonlinear MPC would not be\u0000able to run in real time. The developed constraints-informed neural-Laguerre\u0000approximation yields similar performance with long-horizon online nonlinear\u0000MPC, but with execution times of a few microseconds, as validated on a\u0000field-programmable gate array (FPGA) platform.","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254596","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Vincent Guigues, Anton J. Kleywegt, Victor Hugo Nascimento
The operation of an ambulance fleet involves ambulance selection decisions about which ambulance to dispatch to each emergency, and ambulance reassignment decisions about what each ambulance should do after it has finished the service associated with an emergency. For ambulance selection decisions, we propose four new heuristics: the Best Myopic (BM) heuristic, a NonMyopic (NM) heuristic, and two greedy heuristics (GHP1 and GHP2). Two variants of the greedy heuristics are also considered. We also propose an optimization problem for an extension of the BM heuristic, useful when a call for several patients arrives. For ambulance reassignment decisions, we propose several strategies to choose which emergency in queue to send an ambulance to or which ambulance station to send an ambulance to when it finishes service. These heuristics are also used in a rollout approach: each time a new decision has to be made (when a call arrives or when an ambulance finishes service), a two-stage stochastic program is solved. The proposed heuristics are used to efficiently compute the second stage cost of these problems. We apply the rollout approach with our heuristics to data of the Emergency Medical Service (EMS) of a large city, and show that these methods outperform other heuristics that have been proposed for ambulance dispatch decisions. We also show that better response times can be obtained using the rollout approach instead of using the heuristics without rollout. Moreover, each decision is computed in a few seconds, which allows these methods to be used for the real-time management of a fleet of ambulances.
{"title":"New Heuristics for the Operation of an Ambulance Fleet under Uncertainty","authors":"Vincent Guigues, Anton J. Kleywegt, Victor Hugo Nascimento","doi":"arxiv-2409.09158","DOIUrl":"https://doi.org/arxiv-2409.09158","url":null,"abstract":"The operation of an ambulance fleet involves ambulance selection decisions\u0000about which ambulance to dispatch to each emergency, and ambulance reassignment\u0000decisions about what each ambulance should do after it has finished the service\u0000associated with an emergency. For ambulance selection decisions, we propose\u0000four new heuristics: the Best Myopic (BM) heuristic, a NonMyopic (NM)\u0000heuristic, and two greedy heuristics (GHP1 and GHP2). Two variants of the\u0000greedy heuristics are also considered. We also propose an optimization problem\u0000for an extension of the BM heuristic, useful when a call for several patients\u0000arrives. For ambulance reassignment decisions, we propose several strategies to\u0000choose which emergency in queue to send an ambulance to or which ambulance\u0000station to send an ambulance to when it finishes service. These heuristics are\u0000also used in a rollout approach: each time a new decision has to be made (when\u0000a call arrives or when an ambulance finishes service), a two-stage stochastic\u0000program is solved. The proposed heuristics are used to efficiently compute the\u0000second stage cost of these problems. We apply the rollout approach with our\u0000heuristics to data of the Emergency Medical Service (EMS) of a large city, and\u0000show that these methods outperform other heuristics that have been proposed for\u0000ambulance dispatch decisions. We also show that better response times can be\u0000obtained using the rollout approach instead of using the heuristics without\u0000rollout. Moreover, each decision is computed in a few seconds, which allows\u0000these methods to be used for the real-time management of a fleet of ambulances.","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254553","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider nonlinearly constrained optimization problems and discuss a generic double-loop framework consisting of four algorithmic ingredients that unifies a broad range of nonlinear optimization solvers. This framework has been implemented in the open-source solver Uno, a Swiss Army knife-like C++ optimization framework that unifies many nonlinearly constrained nonconvex optimization solvers. We illustrate the framework with a sequential quadratic programming (SQP) algorithm that maintains an acceptable upper bound on the constraint violation, called a funnel, that is monotonically decreased to control the feasibility of the iterates. Infeasible quadratic subproblems are handled by a feasibility restoration strategy. Globalization is controlled by a line search or a trust-region method. We prove global convergence of the trust-region funnel SQP method, building on known results from filter methods. We implement the algorithm in Uno, and we provide extensive test results for the trust-region line-search funnel SQP on small CUTEst instances.
我们考虑了非线性约束优化问题，并讨论了由四种算法成分组成的通用双环框架，该框架统一了广泛的非线性优化求解器。该框架已在开源求解器 Uno 中实现，Uno 是一个类似瑞士军刀的 C++ 优化框架，它统一了许多非线性约束非凸优化求解器。我们用一种顺序二次编程（SQP）算法来说明该框架，该算法对违反约束的情况保持一个可接受的上限，称为漏斗，该漏斗单调递减，以控制迭代的可行性。不可行的二次子问题由可行性恢复策略处理。全局化由直线搜索或信任区域方法控制。我们以滤波方法的已知结果为基础，证明了信任区域漏斗 SQP 方法的全局收敛性。我们在 Uno 中实现了该算法，并在小型 CUTEst 实例上提供了信任区域线性搜索漏斗 SQP 的大量测试结果。
{"title":"A Unified Funnel Restoration SQP Algorithm","authors":"David Kiessling, Sven Leyffer, Charlie Vanaret","doi":"arxiv-2409.09208","DOIUrl":"https://doi.org/arxiv-2409.09208","url":null,"abstract":"We consider nonlinearly constrained optimization problems and discuss a\u0000generic double-loop framework consisting of four algorithmic ingredients that\u0000unifies a broad range of nonlinear optimization solvers. This framework has\u0000been implemented in the open-source solver Uno, a Swiss Army knife-like C++\u0000optimization framework that unifies many nonlinearly constrained nonconvex\u0000optimization solvers. We illustrate the framework with a sequential quadratic\u0000programming (SQP) algorithm that maintains an acceptable upper bound on the\u0000constraint violation, called a funnel, that is monotonically decreased to\u0000control the feasibility of the iterates. Infeasible quadratic subproblems are\u0000handled by a feasibility restoration strategy. Globalization is controlled by a\u0000line search or a trust-region method. We prove global convergence of the\u0000trust-region funnel SQP method, building on known results from filter methods.\u0000We implement the algorithm in Uno, and we provide extensive test results for\u0000the trust-region line-search funnel SQP on small CUTEst instances.","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254551","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The convergence rate is a crucial issue in opinion dynamics, which characterizes how quickly opinions reach a consensus and tells when the collective behavior can be formed. However, the key factors that determine the convergence rate of opinions are elusive, especially when individuals interact with complex interaction types such as friend/foe, ally/adversary, or trust/mistrust. In this paper, using random matrix theory and low-rank perturbation theory, we present a new body of theory to comprehensively study the convergence rate of opinion dynamics. First, we divide the complex interaction types into five typical scenarios: mutual trust $(+/+)$, mutual mistrust $(-/-)$, trust$/$mistrust $(+/-)$, unilateral trust $(+/0)$, and unilateral mistrust $(-/0)$. For diverse interaction types, we derive the mathematical expression of the convergence rate, and further establish the direct connection between the convergence rate and population size, the density of interactions (network connectivity), and individuals' self-confidence level. Second, taking advantage of these connections, we prove that for the $(+/+)$, $(+/-)$, $(+/0)$, and random mixture of different interaction types, the convergence rate is proportional to the population size and network connectivity, while it is inversely proportional to the individuals' self-confidence level. However, for the $(-/-)$ and $(-/0)$ scenarios, we draw the exact opposite conclusions. Third, for the $(+/+,-/-)$ and $(-/-,-/0)$ scenarios, we derive the optimal proportion of different interaction types to ensure the fast convergence of opinions. Finally, simulation examples are provided to illustrate the effectiveness and robustness of our theoretical findings.
{"title":"Convergence rate of opinion dynamics with complex interaction types","authors":"Lingling Yao, Aming Li","doi":"arxiv-2409.09100","DOIUrl":"https://doi.org/arxiv-2409.09100","url":null,"abstract":"The convergence rate is a crucial issue in opinion dynamics, which\u0000characterizes how quickly opinions reach a consensus and tells when the\u0000collective behavior can be formed. However, the key factors that determine the\u0000convergence rate of opinions are elusive, especially when individuals interact\u0000with complex interaction types such as friend/foe, ally/adversary, or\u0000trust/mistrust. In this paper, using random matrix theory and low-rank\u0000perturbation theory, we present a new body of theory to comprehensively study\u0000the convergence rate of opinion dynamics. First, we divide the complex\u0000interaction types into five typical scenarios: mutual trust $(+/+)$, mutual\u0000mistrust $(-/-)$, trust$/$mistrust $(+/-)$, unilateral trust $(+/0)$, and\u0000unilateral mistrust $(-/0)$. For diverse interaction types, we derive the\u0000mathematical expression of the convergence rate, and further establish the\u0000direct connection between the convergence rate and population size, the density\u0000of interactions (network connectivity), and individuals' self-confidence level.\u0000Second, taking advantage of these connections, we prove that for the $(+/+)$,\u0000$(+/-)$, $(+/0)$, and random mixture of different interaction types, the\u0000convergence rate is proportional to the population size and network\u0000connectivity, while it is inversely proportional to the individuals'\u0000self-confidence level. However, for the $(-/-)$ and $(-/0)$ scenarios, we draw\u0000the exact opposite conclusions. Third, for the $(+/+,-/-)$ and $(-/-,-/0)$\u0000scenarios, we derive the optimal proportion of different interaction types to\u0000ensure the fast convergence of opinions. Finally, simulation examples are\u0000provided to illustrate the effectiveness and robustness of our theoretical\u0000findings.","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254592","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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