The theory of linear time invariant systems is well established and allows, among other things, to formulate and solve control problems in finite time. In this context the control laws are typically taken in a space of the form L^p(0,T;U). In this paper we consider the possibility of taking control laws in (H^1(0,T;U))* , which induces non-trivial issues. We overcome these difficulties by adapting the functional setting, notably by considering a generalized final state for the systems under consideration. In addition we collect time regularity properties and we pretend that in general it is not possible to consider control laws in H^{-1}(0,T;U). Then, we apply our results to propose an interpretation of the inifinite order of defect for an observability inequality, in terms of controllability properties.
{"title":"On the control of LTI systems with rough control laws","authors":"Lucas DavronCEREMADE","doi":"arxiv-2409.11766","DOIUrl":"https://doi.org/arxiv-2409.11766","url":null,"abstract":"The theory of linear time invariant systems is well established and allows,\u0000among other things, to formulate and solve control problems in finite time. In\u0000this context the control laws are typically taken in a space of the form\u0000L^p(0,T;U). In this paper we consider the possibility of taking control laws in\u0000(H^1(0,T;U))* , which induces non-trivial issues. We overcome these\u0000difficulties by adapting the functional setting, notably by considering a\u0000generalized final state for the systems under consideration. In addition we\u0000collect time regularity properties and we pretend that in general it is not\u0000possible to consider control laws in H^{-1}(0,T;U). Then, we apply our results\u0000to propose an interpretation of the inifinite order of defect for an\u0000observability inequality, in terms of controllability properties.","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254610","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 propose a proximal variable smoothing algorithm for nonsmooth optimization problem with sum of three functions involving weakly convex composite function. The proposed algorithm is designed as a time-varying forward-backward splitting algorithm with two steps: (i) a time-varying forward step with the gradient of a smoothed surrogate function, designed with the Moreau envelope, of the sum of two functions; (ii) the backward step with a proximity operator of the remaining function. For the proposed algorithm, we present a convergence analysis in terms of a stationary point by using a newly smoothed surrogate stationarity measure. As an application of the target problem, we also present a formulation of multiple-input-multiple-output (MIMO) signal detection with phase-shift keying. Numerical experiments demonstrate the efficacy of the proposed formulation and algorithm.
{"title":"A Proximal Variable Smoothing for Nonsmooth Minimization Involving Weakly Convex Composite with MIMO Application","authors":"Keita Kume, Isao Yamada","doi":"arxiv-2409.10934","DOIUrl":"https://doi.org/arxiv-2409.10934","url":null,"abstract":"We propose a proximal variable smoothing algorithm for nonsmooth optimization\u0000problem with sum of three functions involving weakly convex composite function.\u0000The proposed algorithm is designed as a time-varying forward-backward splitting\u0000algorithm with two steps: (i) a time-varying forward step with the gradient of\u0000a smoothed surrogate function, designed with the Moreau envelope, of the sum of\u0000two functions; (ii) the backward step with a proximity operator of the\u0000remaining function. For the proposed algorithm, we present a convergence\u0000analysis in terms of a stationary point by using a newly smoothed surrogate\u0000stationarity measure. As an application of the target problem, we also present\u0000a formulation of multiple-input-multiple-output (MIMO) signal detection with\u0000phase-shift keying. Numerical experiments demonstrate the efficacy of the\u0000proposed formulation and algorithm.","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254637","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}
Boris Chervonenkis, Andrei Krasnov, Alexander Gasnikov, Aleksandr Lobanov
The challenges of black box optimization arise due to imprecise responses and limited output information. This article describes new results on optimizing multivariable functions using an Order Oracle, which provides access only to the order between function values and with some small errors. We obtained convergence rate estimates for the one-dimensional search method (golden ratio method) under the condition of oracle inaccuracy, as well as convergence results for the algorithm on a "square" (also with noise), which outperforms its alternatives. The results obtained are similar to those in problems with oracles providing significantly more information about the optimized function. Additionally, the practical application of the algorithm has been demonstrated in maximizing a preference function, where the parameters are the acidity and sweetness of the drink. This function is expected to be convex or at least quasi-convex.
{"title":"Nesterov's method of dichotomy via Order Oracle: The problem of optimizing a two-variable function on a square","authors":"Boris Chervonenkis, Andrei Krasnov, Alexander Gasnikov, Aleksandr Lobanov","doi":"arxiv-2409.11077","DOIUrl":"https://doi.org/arxiv-2409.11077","url":null,"abstract":"The challenges of black box optimization arise due to imprecise responses and\u0000limited output information. This article describes new results on optimizing\u0000multivariable functions using an Order Oracle, which provides access only to\u0000the order between function values and with some small errors. We obtained\u0000convergence rate estimates for the one-dimensional search method (golden ratio\u0000method) under the condition of oracle inaccuracy, as well as convergence\u0000results for the algorithm on a \"square\" (also with noise), which outperforms\u0000its alternatives. The results obtained are similar to those in problems with\u0000oracles providing significantly more information about the optimized function.\u0000Additionally, the practical application of the algorithm has been demonstrated\u0000in maximizing a preference function, where the parameters are the acidity and\u0000sweetness of the drink. This function is expected to be convex or at least\u0000quasi-convex.","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254634","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 study the relationship between maximum principle (MP) and dynamic programming principle (DPP) for forward-backward control system under consistent convex expectation dominated by G-expectation. Under the smooth assumptions for the value function, we get the relationship between MP and DPP under a reference probability by establishing a useful estimate. If the value function is not smooth, then we obtain the first-order sub-jet and super-jet of the value function at any t. However, the processing method in this case is much more difficult than that when t equals 0.
本文研究了由 G 期望支配的一致凸期望下前后向控制系统的最大原理（MP）与动态编程原理（DPP）之间的关系。在值函数平滑的假设条件下，我们通过建立一个有用的估计值，得到了在参考概率下 MP 与 DPP 的关系。如果值函数不平滑，则我们可以得到任意 t 下值函数的一阶子喷流和超喷流，但这种情况下的处理方法要比 t 等于 0 时的处理方法困难得多。
{"title":"Relationship between stochastic maximum principle and dynamic programming principle under convex expectation","authors":"Xiaojuan Li, Mingshang Hu","doi":"arxiv-2409.10987","DOIUrl":"https://doi.org/arxiv-2409.10987","url":null,"abstract":"In this paper, we study the relationship between maximum principle (MP) and\u0000dynamic programming principle (DPP) for forward-backward control system under\u0000consistent convex expectation dominated by G-expectation. Under the smooth\u0000assumptions for the value function, we get the relationship between MP and DPP\u0000under a reference probability by establishing a useful estimate. If the value\u0000function is not smooth, then we obtain the first-order sub-jet and super-jet of\u0000the value function at any t. However, the processing method in this case is\u0000much more difficult than that when t equals 0.","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254635","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 investigate, simultaneously, the null-controllability via the feedback control method and the turnpike property of dynamic systems arising from population dynamics models where the control is localized on the non-local term. These models describe the dynamics of one or several populations with age dependence and spatial structure involving time. By considering control functions localized with respect to the spatial variable at the time (t) but active for age ( a=0 ), we prove that the entire population can be steered to zero in any positive time ( T>A ) for any data in ( L^2(Omegatimes(0,A)).) Regarding turnpike property, we use the results of null-controllability and the Phillips'theorem for stability and we design an appropriate dichotomy transformation, based on solutions of the algebraic Riccati and Lyapunov equations. We give numerical examples to support the analytic results.
{"title":"Birth control and turnpike property of Lotka-McKendrick models","authors":"Marius Bargo, Yacouba Simpore","doi":"arxiv-2409.11247","DOIUrl":"https://doi.org/arxiv-2409.11247","url":null,"abstract":"In this paper, we investigate, simultaneously, the null-controllability via\u0000the feedback control method and the turnpike property of dynamic systems\u0000arising from population dynamics models where the control is localized on the\u0000non-local term. These models describe the dynamics of one or several\u0000populations with age dependence and spatial structure involving time. By\u0000considering control functions localized with respect to the spatial variable at\u0000the time (t) but active for age ( a=0 ), we prove that the entire\u0000population can be steered to zero in any positive time ( T>A ) for any data\u0000in ( L^2(Omegatimes(0,A)).) Regarding turnpike property, we use the results\u0000of null-controllability and the Phillips'theorem for stability and we design an\u0000appropriate dichotomy transformation, based on solutions of the algebraic\u0000Riccati and Lyapunov equations. We give numerical examples to support the\u0000analytic results.","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254609","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 presents a method for calculating the Region of Attraction (ROA) of nonlinear dynamical systems, both with and without control. The ROA is determined by solving a hierarchy of semidefinite programs (SDPs) defined on a splitting of the time and state space. Previous works demonstrated that this splitting could significantly enhance approximation accuracy, although the improvement was highly dependent on the ad-hoc selection of split locations. In this work, we eliminate the need for this ad-hoc selection by introducing an optimization-based method that performs the splits through conic differentiation of the underlying semidefinite programming problem. We provide the differentiability conditions for the split ROA problem, prove the absence of a duality gap, and demonstrate the effectiveness of our method through numerical examples.
{"title":"Towards Optimal Spatio-Temporal Decomposition of Control-Related Sum-of-Squares Programs","authors":"Vít Cibulka, Milan Korda, Tomáš Haniš","doi":"arxiv-2409.11196","DOIUrl":"https://doi.org/arxiv-2409.11196","url":null,"abstract":"This paper presents a method for calculating the Region of Attraction (ROA)\u0000of nonlinear dynamical systems, both with and without control. The ROA is\u0000determined by solving a hierarchy of semidefinite programs (SDPs) defined on a\u0000splitting of the time and state space. Previous works demonstrated that this\u0000splitting could significantly enhance approximation accuracy, although the\u0000improvement was highly dependent on the ad-hoc selection of split locations. In\u0000this work, we eliminate the need for this ad-hoc selection by introducing an\u0000optimization-based method that performs the splits through conic\u0000differentiation of the underlying semidefinite programming problem. We provide\u0000the differentiability conditions for the split ROA problem, prove the absence\u0000of a duality gap, and demonstrate the effectiveness of our method through\u0000numerical examples.","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254611","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 surge in data availability has inundated decision-makers with an overwhelming array of choices. While existing approaches focus on optimizing decisions based on quantifiable metrics, practical decision-making often requires balancing measurable quantitative criteria with unmeasurable qualitative factors embedded in the broader context. In such cases, algorithms can generate high-quality recommendations, but the final decision rests with the human, who must weigh both dimensions. We define the process of selecting the optimal set of algorithmic recommendations in this context as human-centered decision making. To address this challenge, we introduce a novel framework called generative curation, which optimizes the true desirability of decision options by integrating both quantitative and qualitative aspects. Our framework uses a Gaussian process to model unknown qualitative factors and derives a diversity metric that balances quantitative optimality with qualitative diversity. This trade-off enables the generation of a manageable subset of diverse, near-optimal actions that are robust to unknown qualitative preferences. To operationalize this framework, we propose two implementation approaches: a generative neural network architecture that produces a distribution $pi$ to efficiently sample a diverse set of near-optimal actions, and a sequential optimization method to iteratively generates solutions that can be easily incorporated into complex optimization formulations. We validate our approach with extensive datasets, demonstrating its effectiveness in enhancing decision-making processes across a range of complex environments, with significant implications for policy and management.
{"title":"Balancing Optimality and Diversity: Human-Centered Decision Making through Generative Curation","authors":"Michael Lingzhi Li, Shixiang Zhu","doi":"arxiv-2409.11535","DOIUrl":"https://doi.org/arxiv-2409.11535","url":null,"abstract":"The surge in data availability has inundated decision-makers with an\u0000overwhelming array of choices. While existing approaches focus on optimizing\u0000decisions based on quantifiable metrics, practical decision-making often\u0000requires balancing measurable quantitative criteria with unmeasurable\u0000qualitative factors embedded in the broader context. In such cases, algorithms\u0000can generate high-quality recommendations, but the final decision rests with\u0000the human, who must weigh both dimensions. We define the process of selecting\u0000the optimal set of algorithmic recommendations in this context as\u0000human-centered decision making. To address this challenge, we introduce a novel\u0000framework called generative curation, which optimizes the true desirability of\u0000decision options by integrating both quantitative and qualitative aspects. Our\u0000framework uses a Gaussian process to model unknown qualitative factors and\u0000derives a diversity metric that balances quantitative optimality with\u0000qualitative diversity. This trade-off enables the generation of a manageable\u0000subset of diverse, near-optimal actions that are robust to unknown qualitative\u0000preferences. To operationalize this framework, we propose two implementation\u0000approaches: a generative neural network architecture that produces a\u0000distribution $pi$ to efficiently sample a diverse set of near-optimal actions,\u0000and a sequential optimization method to iteratively generates solutions that\u0000can be easily incorporated into complex optimization formulations. We validate\u0000our approach with extensive datasets, demonstrating its effectiveness in\u0000enhancing decision-making processes across a range of complex environments,\u0000with significant implications for policy and management.","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254607","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 demand for high-performance computing in machine learning and artificial intelligence has led to the development of specialized hardware accelerators like Tensor Processing Units (TPUs), Graphics Processing Units (GPUs), and Field-Programmable Gate Arrays (FPGAs). A key strategy to enhance these accelerators is the reduction of precision in arithmetic operations, which increases processing speed and lowers latency - crucial for real-time AI applications. Precision reduction minimizes memory bandwidth requirements and energy consumption, essential for large-scale and mobile deployments, and increases throughput by enabling more parallel operations per cycle, maximizing hardware resource utilization. This strategy is equally vital for solving NP-hard quadratic unconstrained binary optimization (QUBO) problems common in machine learning, which often require high precision for accurate representation. Special hardware solvers, such as quantum annealers, benefit significantly from precision reduction. This paper introduces a fully principled Branch-and-Bound algorithm for reducing precision needs in QUBO problems by utilizing dynamic range as a measure of complexity. Experiments validate our algorithm's effectiveness on an actual quantum annealer.
{"title":"Dynamic Range Reduction via Branch-and-Bound","authors":"Thore Gerlach, Nico Piatkowski","doi":"arxiv-2409.10863","DOIUrl":"https://doi.org/arxiv-2409.10863","url":null,"abstract":"The demand for high-performance computing in machine learning and artificial\u0000intelligence has led to the development of specialized hardware accelerators\u0000like Tensor Processing Units (TPUs), Graphics Processing Units (GPUs), and\u0000Field-Programmable Gate Arrays (FPGAs). A key strategy to enhance these\u0000accelerators is the reduction of precision in arithmetic operations, which\u0000increases processing speed and lowers latency - crucial for real-time AI\u0000applications. Precision reduction minimizes memory bandwidth requirements and\u0000energy consumption, essential for large-scale and mobile deployments, and\u0000increases throughput by enabling more parallel operations per cycle, maximizing\u0000hardware resource utilization. This strategy is equally vital for solving\u0000NP-hard quadratic unconstrained binary optimization (QUBO) problems common in\u0000machine learning, which often require high precision for accurate\u0000representation. Special hardware solvers, such as quantum annealers, benefit\u0000significantly from precision reduction. This paper introduces a fully\u0000principled Branch-and-Bound algorithm for reducing precision needs in QUBO\u0000problems by utilizing dynamic range as a measure of complexity. Experiments\u0000validate our algorithm's effectiveness on an actual quantum annealer.","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254488","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 letter presents contraction analysis of continuation method for suboptimal model predictive control. A contraction metric is proposed that reflects hierarchical dynamics inherent in the continuation method and, thus, leads us to a pair of tractable matrix inequalities characterizing contraction of closed-loop system controlled by the continuation method. A numerical example verifies our contraction analysis and demonstrates the tractability of the presented matrix inequalities.
{"title":"Contraction Analysis of Continuation Method for Suboptimal Model Predictive Control","authors":"Ryotaro Shima, Yuji Ito, Tatsuya Miyano","doi":"arxiv-2409.10970","DOIUrl":"https://doi.org/arxiv-2409.10970","url":null,"abstract":"This letter presents contraction analysis of continuation method for\u0000suboptimal model predictive control. A contraction metric is proposed that\u0000reflects hierarchical dynamics inherent in the continuation method and, thus,\u0000leads us to a pair of tractable matrix inequalities characterizing contraction\u0000of closed-loop system controlled by the continuation method. A numerical\u0000example verifies our contraction analysis and demonstrates the tractability of\u0000the presented matrix inequalities.","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254636","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}
For bilinear control systems in $mathbb{R}^d$ we prove, under an accessibility hypothesis, the existence of a nontrivial compact set $Dsubsetmathbb{R}^d$ satisfying $mathcal{O}_t(D)=e^{tR}D$ for all $t>0$, where $Rinmathbb{R}$ is a fixed constant and $mathcal{O}_t(D)$ denotes the orbit from $D$ at time $t$. This property generalizes the trajectory of an eigenvector on a linear dynamical system, and merits such a set the name "eigenset".
{"title":"Existence of eigensets on bilinear control systems","authors":"Eduardo Celso Viscovini","doi":"arxiv-2409.11194","DOIUrl":"https://doi.org/arxiv-2409.11194","url":null,"abstract":"For bilinear control systems in $mathbb{R}^d$ we prove, under an\u0000accessibility hypothesis, the existence of a nontrivial compact set\u0000$Dsubsetmathbb{R}^d$ satisfying $mathcal{O}_t(D)=e^{tR}D$ for all $t>0$,\u0000where $Rinmathbb{R}$ is a fixed constant and $mathcal{O}_t(D)$ denotes the\u0000orbit from $D$ at time $t$. This property generalizes the trajectory of an\u0000eigenvector on a linear dynamical system, and merits such a set the name\u0000\"eigenset\".","PeriodicalId":501286,"journal":{"name":"arXiv - MATH - Optimization and Control","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254613","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：