In 1931--1932, Erwin Schr"odinger studied a hot gas Gedankenexperiment (an instance of large deviations of the empirical distribution). Schr"odinger's problem represents an early example of a fundamental inference method, the so-called maximum entropy method, having roots in Boltzmann's work and being developed in subsequent years by Jaynes, Burg, Dempster, and Csisz'ar. The problem, known as the Schr" odinger bridge problem (SBP) with ``uniform"" prior, was more recently recognized as a regularization of the Monge-Kantorovich optimal mass transport (OMT) problem, leading to effective computational schemes for the latter. Specifically, OMT with quadratic cost may be viewed as a zerotemperature limit of the problem posed by Schr"odinger in the early 1930s. The latter amounts to minimization of Helmholtz's free energy over probability distributions that are constrained to possess two given marginals. The problem features a delicate compromise, mediated by a temperature parameter, between minimizing the internal energy and maximizing the entropy. These concepts are central to a rapidly expanding area of modern science dealing with the so-called Sinkhorn algorithm, which appears as a special case of an algorithm first studied in the more challenging continuous space setting by the French analyst Robert Fortet in 1938--1940 specifically for Schr"odinger bridges. Due to the constraint on end-point distributions, dynamic programming is not a suitable tool to attack these problems. Instead, Fortet's iterative algorithm and its discrete counterpart, the Sinkhorn iteration, permit computation of the optimal solution by iteratively solving the so-called Schr" odinger system. Convergence of the iteration is guaranteed by contraction along the steps in suitable metrics, such as Hilbert's projective metric. In both the continuous as well as the discrete time and space settings, stochastic control provides a reformulation of and a context for the dynamic versions of general Schr" odinger bridge problems and of their zero-temperature limit, the OMT problem. These problems, in turn, naturally lead to steering problems for flows of one-time marginals which represent a new paradigm for controlling uncertainty. The zero-temperature problem in the continuous-time and space setting turns out to be the celebrated Benamou--Brenier characterization of theMcCann displacement interpolation flow in OMT. The formalism and techniques behind these control problems on flows of probability distributions have attracted significant attention in recent years as they lead to a variety of new applications in spacecraft guidance, control of robot or biological swarms, sensing, active cooling, and network routing as well as in computer and data science. This multifaceted and versatile framework, intertwining SBP and OMT, provides the substrate for the historical and technical overview ast Received by the editors May 22, 2020; accepted for publication (in revised form) October 29, 2020;
1931- 1932年，埃尔温·薛定谔研究了热气体格丹肯实验(经验分布大偏差的一个实例)。薛定谔的问题代表了一种基本推理方法的早期例子，即所谓的最大熵法，它植根于玻尔兹曼的工作，并在随后的几年里由杰恩斯、伯格、登普斯特和cissar发展。该问题被称为具有“均匀”先验的Schr odinger桥问题(SBP)，最近被认为是Monge-Kantorovich最优质量传递(OMT)问题的正则化，导致后者的有效计算方案。具体来说，具有二次代价的OMT可以看作是20世纪30年代初Schr odinger提出的问题的零温度极限。后者相当于亥姆霍兹自由能在概率分布上的最小化，这些概率分布被限制为具有两个给定的边际。这个问题的特点是一个微妙的折衷，由温度参数调解，在最小化内能和最大化熵之间。这些概念是处理所谓的Sinkhorn算法的现代科学快速扩展领域的核心，该算法是1938年至1940年法国分析师Robert Fortet在更具挑战性的连续空间设置中专门研究Schr odinger桥的算法的一个特例。由于端点分布的限制，动态规划不是解决这些问题的合适工具。相反，Fortet的迭代算法和它的离散对应，Sinkhorn迭代，允许通过迭代求解所谓的Schr odinger系统来计算最优解。迭代的收敛性是通过在适当的度量(如希尔伯特射影度量)中沿着步长收缩来保证的。在连续和离散的时间和空间设置中，随机控制为一般Schr odinger桥问题及其零温度极限OMT问题的动态版本提供了一个重新表述和上下文。这些问题，反过来，自然会导致一次性边际流动的控制问题，这代表了控制不确定性的新范式。连续时空条件下的零温度问题是OMT中themcann位移插值流的著名的Benamou—Brenier表征。近年来，这些概率分布流控制问题背后的形式主义和技术引起了人们的极大关注，因为它们导致了航天器制导、机器人或生物群控制、传感、主动冷却和网络路由以及计算机和数据科学等领域的各种新应用。这个多方面和多功能的框架，将SBP和OMT交织在一起，为历史和技术概述提供了基础。接受发表(修订版)2020年10月29日;于2021年5月6日以电子方式发布。https://doi.org/10.1137/20M1339982资助:本研究由美国国家科学基金会(NSF)资助1807664,1839441,1901599和1942523,AFOSR资助FA9550-17-1-0435，以及帕多瓦大学研究项目CPDA 140897部分支持。美国佐治亚理工学院航空航天工程学院，佐治亚州亚特兰大30332 (yongchen@gatech.edu)。美国加州大学尔湾分校机械与航天工程系，CA 92697 USA (tryphon@uci.edu)。“图里奥·列维-奇维塔数学学系”，“帕多瓦大学”，意大利帕多瓦35121 (pavon@math.unipd.it)。249 D噢问oa de D 11/0 9/21 62 47。1。2 13 1。11 R ed tr ib ut io n苏bj ec t t o如果M李ce ns e或c op年ig ht;请参阅TTP: //请参阅TTP: //请参阅TTP: //请参阅TTP: //请参阅TTP: //请参阅TTP: //请参阅TTP
{"title":"Stochastic Control Liaisons: Richard Sinkhorn Meets Gaspard Monge on a Schrödinger Bridge","authors":"Yongxin Chen, T. Georgiou, M. Pavon","doi":"10.1137/20M1339982","DOIUrl":"https://doi.org/10.1137/20M1339982","url":null,"abstract":"In 1931--1932, Erwin Schr\"odinger studied a hot gas Gedankenexperiment (an instance of large deviations of the empirical distribution). Schr\"odinger's problem represents an early example of a fundamental inference method, the so-called maximum entropy method, having roots in Boltzmann's work and being developed in subsequent years by Jaynes, Burg, Dempster, and Csisz'ar. The problem, known as the Schr\" odinger bridge problem (SBP) with ``uniform\"\" prior, was more recently recognized as a regularization of the Monge-Kantorovich optimal mass transport (OMT) problem, leading to effective computational schemes for the latter. Specifically, OMT with quadratic cost may be viewed as a zerotemperature limit of the problem posed by Schr\"odinger in the early 1930s. The latter amounts to minimization of Helmholtz's free energy over probability distributions that are constrained to possess two given marginals. The problem features a delicate compromise, mediated by a temperature parameter, between minimizing the internal energy and maximizing the entropy. These concepts are central to a rapidly expanding area of modern science dealing with the so-called Sinkhorn algorithm, which appears as a special case of an algorithm first studied in the more challenging continuous space setting by the French analyst Robert Fortet in 1938--1940 specifically for Schr\"odinger bridges. Due to the constraint on end-point distributions, dynamic programming is not a suitable tool to attack these problems. Instead, Fortet's iterative algorithm and its discrete counterpart, the Sinkhorn iteration, permit computation of the optimal solution by iteratively solving the so-called Schr\" odinger system. Convergence of the iteration is guaranteed by contraction along the steps in suitable metrics, such as Hilbert's projective metric. In both the continuous as well as the discrete time and space settings, stochastic control provides a reformulation of and a context for the dynamic versions of general Schr\" odinger bridge problems and of their zero-temperature limit, the OMT problem. These problems, in turn, naturally lead to steering problems for flows of one-time marginals which represent a new paradigm for controlling uncertainty. The zero-temperature problem in the continuous-time and space setting turns out to be the celebrated Benamou--Brenier characterization of theMcCann displacement interpolation flow in OMT. The formalism and techniques behind these control problems on flows of probability distributions have attracted significant attention in recent years as they lead to a variety of new applications in spacecraft guidance, control of robot or biological swarms, sensing, active cooling, and network routing as well as in computer and data science. This multifaceted and versatile framework, intertwining SBP and OMT, provides the substrate for the historical and technical overview ast Received by the editors May 22, 2020; accepted for publication (in revised form) October 29, 2020;","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":null,"pages":null},"PeriodicalIF":10.2,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86866105","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"How Many Steps Still Left to x*?","authors":"E. Catinas","doi":"10.1137/19M1244858","DOIUrl":"https://doi.org/10.1137/19M1244858","url":null,"abstract":"","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":null,"pages":null},"PeriodicalIF":10.2,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81935311","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Xavier Allamigeon, P. Benchimol, S. Gaubert, M. Joswig
. Tropical geometry has been recently used to obtain new complexity results in convex optimization and game theory. In this paper, we present an application of this approach to a famous class of algorithms for linear programming, i.e., log-barrier interior point methods. We show that these methods are not strongly polynomial by constructing a family of linear programs with 3 r + 1 inequalities in dimension 2 r for which the number of iterations performed is in Omega (2 r ). The total curvature of the central path of these linear programs is also exponential in r , disproving a continuous analogue of the Hirsch conjecture proposed by Deza, Terlaky, and Zinchenko. These results are obtained by analyzing the tropical central path, which is the piecewise linear limit of the central paths of parameterized families of classical linear programs viewed through ``logarithmic glasses."" This allows us to provide combinatorial lower bounds for the number of iterations and the total curvature in a general setting.
{"title":"What Tropical Geometry Tells Us about the Complexity of Linear Programming","authors":"Xavier Allamigeon, P. Benchimol, S. Gaubert, M. Joswig","doi":"10.1137/20M1380211","DOIUrl":"https://doi.org/10.1137/20M1380211","url":null,"abstract":". Tropical geometry has been recently used to obtain new complexity results in convex optimization and game theory. In this paper, we present an application of this approach to a famous class of algorithms for linear programming, i.e., log-barrier interior point methods. We show that these methods are not strongly polynomial by constructing a family of linear programs with 3 r + 1 inequalities in dimension 2 r for which the number of iterations performed is in Omega (2 r ). The total curvature of the central path of these linear programs is also exponential in r , disproving a continuous analogue of the Hirsch conjecture proposed by Deza, Terlaky, and Zinchenko. These results are obtained by analyzing the tropical central path, which is the piecewise linear limit of the central paths of parameterized families of classical linear programs viewed through ``logarithmic glasses.\"\" This allows us to provide combinatorial lower bounds for the number of iterations and the total curvature in a general setting.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":null,"pages":null},"PeriodicalIF":10.2,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84651133","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a class of in nite-dimensional optimization problems in which a distributed vector-valued variable should pointwise almost everywhere take values from a given nite set M ⊂ R . Such hybrid discrete–continuous problems occur in, e.g., topology optimization or medical imaging and are challenging due to their lack of weak lower semicontinuity. To circumvent this di culty, we introduce as a regularization term a convex integral functional with an integrand that has a polyhedral epigraph with vertices corresponding to the values ofM; similar to the L1 norm in sparse regularization, this “vector multibang penalty” promotes solutions with the desired structure while allowing the use of tools from convex optimization for the analysis as well as the numerical solution of the resulting problem. We show well-posedness of the regularized problem and analyze stability properties of its solution in a general setting. We then illustrate the approach for three speci c model optimization problems of broader interest: optimal control of the Bloch equation, optimal control of an elastic deformation, and a multimaterial branched transport problem. In the rst two cases, we derive explicit characterizations of the penalty and its generalized derivatives for a concrete class of sets M. For the third case, we discuss the algorithmic computation of these derivatives for general sets. These derivatives are then used in a superlinearly convergent semismooth Newton method applied to a sequence of regularized optimization problems. We illustrate the behavior of this approach for the threemodel problemswith numerical examples.
{"title":"Convex Relaxation of Discrete Vector-Valued Optimization Problems","authors":"Christian Clason, Carla Tameling, B. Wirth","doi":"10.1137/21M1426237","DOIUrl":"https://doi.org/10.1137/21M1426237","url":null,"abstract":"We consider a class of in nite-dimensional optimization problems in which a distributed vector-valued variable should pointwise almost everywhere take values from a given nite set M ⊂ R . Such hybrid discrete–continuous problems occur in, e.g., topology optimization or medical imaging and are challenging due to their lack of weak lower semicontinuity. To circumvent this di culty, we introduce as a regularization term a convex integral functional with an integrand that has a polyhedral epigraph with vertices corresponding to the values ofM; similar to the L1 norm in sparse regularization, this “vector multibang penalty” promotes solutions with the desired structure while allowing the use of tools from convex optimization for the analysis as well as the numerical solution of the resulting problem. We show well-posedness of the regularized problem and analyze stability properties of its solution in a general setting. We then illustrate the approach for three speci c model optimization problems of broader interest: optimal control of the Bloch equation, optimal control of an elastic deformation, and a multimaterial branched transport problem. In the rst two cases, we derive explicit characterizations of the penalty and its generalized derivatives for a concrete class of sets M. For the third case, we discuss the algorithmic computation of these derivatives for general sets. These derivatives are then used in a superlinearly convergent semismooth Newton method applied to a sequence of regularized optimization problems. We illustrate the behavior of this approach for the threemodel problemswith numerical examples.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":null,"pages":null},"PeriodicalIF":10.2,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75823974","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Rotations in Three Dimensions","authors":"M. Maritz","doi":"10.1201/9781420041767-8","DOIUrl":"https://doi.org/10.1201/9781420041767-8","url":null,"abstract":"","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":null,"pages":null},"PeriodicalIF":10.2,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83026145","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
C. Angstmann, A. Erickson, B. Henry, A. V. McGann, J. M. Murray, J. A. Nichols
{"title":"A General Framework for Fractional Order Compartment Models","authors":"C. Angstmann, A. Erickson, B. Henry, A. V. McGann, J. M. Murray, J. A. Nichols","doi":"10.1137/21M1398549","DOIUrl":"https://doi.org/10.1137/21M1398549","url":null,"abstract":"","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":null,"pages":null},"PeriodicalIF":10.2,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85639858","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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