The study of time-inhomogeneous Markov jump processes is a traditional topic within probability theory that has recently attracted substantial attention in various applications. However, their flexibility also incurs a substantial mathematical burden which is usually circumvented by using well-known generic distributional approximations or simulations. This article provides a novel approximation method that tailors the dynamics of a time-homogeneous Markov jump process to meet those of its time-inhomogeneous counterpart on an increasingly fine Poisson grid. Strong convergence of the processes in terms of the Skorokhod J1 metric is established, and convergence rates are provided. Under traditional regularity assumptions, distributional convergence is established for unconditional proxies, to the same limit. Special attention is devoted to the case where the target process has one absorbing state and the remaining ones transient, for which the absorption times also converge. Some applications are outlined, such as univariate hazard-rate density estimation, ruin probabilities, and multivariate phase-type density evaluation.Funding: M. Bladt and O. Peralta would like to acknowledge financial support from the Swiss National Science Foundation Project 200021_191984. O. Peralta acknowledges financial support from NSF Award #1653354 and AXA Research Fund Award on “Mitigating risk in the wake of the pandemic”.
研究时间同构马尔可夫跳跃过程是概率论中的一个传统课题，最近在各种应用中引起了广泛关注。然而，它们的灵活性也带来了巨大的数学负担，通常通过使用众所周知的通用分布近似或模拟来规避。本文提供了一种新颖的近似方法，可在越来越细的泊松网格上调整时间均质马尔可夫跳跃过程的动态，以满足其时间均质对应过程的动态。根据 Skorokhod J1 度量，建立了过程的强收敛性，并提供了收敛率。在传统的正则性假设下，无条件代理的分布收敛性也被确定为相同的极限。特别关注的是目标过程有一个吸收状态和其余瞬态的情况，对于这种情况，吸收时间也会收敛。本文概述了一些应用，如单变量危险率密度估计、毁坏概率和多变量相型密度评估：M. Bladt 和 O. Peralta 感谢瑞士国家科学基金会项目 200021_191984 的资助。O. Peralta 感谢美国国家科学基金会奖 #1653354 和 AXA 研究基金奖 "在大流行后降低风险 "的资助。
{"title":"Strongly Convergent Homogeneous Approximations to Inhomogeneous Markov Jump Processes and Applications","authors":"Martin Bladt, Oscar Peralta","doi":"10.1287/moor.2022.0153","DOIUrl":"https://doi.org/10.1287/moor.2022.0153","url":null,"abstract":"The study of time-inhomogeneous Markov jump processes is a traditional topic within probability theory that has recently attracted substantial attention in various applications. However, their flexibility also incurs a substantial mathematical burden which is usually circumvented by using well-known generic distributional approximations or simulations. This article provides a novel approximation method that tailors the dynamics of a time-homogeneous Markov jump process to meet those of its time-inhomogeneous counterpart on an increasingly fine Poisson grid. Strong convergence of the processes in terms of the Skorokhod J<jats:sub>1</jats:sub> metric is established, and convergence rates are provided. Under traditional regularity assumptions, distributional convergence is established for unconditional proxies, to the same limit. Special attention is devoted to the case where the target process has one absorbing state and the remaining ones transient, for which the absorption times also converge. Some applications are outlined, such as univariate hazard-rate density estimation, ruin probabilities, and multivariate phase-type density evaluation.Funding: M. Bladt and O. Peralta would like to acknowledge financial support from the Swiss National Science Foundation Project 200021_191984. O. Peralta acknowledges financial support from NSF Award #1653354 and AXA Research Fund Award on “Mitigating risk in the wake of the pandemic”.","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"159 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139946213","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study a class of zero-sum games between a singular controller and a stopper over a finite-time horizon. The underlying process is a multidimensional (locally nondegenerate) controlled stochastic differential equation (SDE) evolving in an unbounded domain. We prove that such games admit a value and provide an optimal strategy for the stopper. The value of the game is shown to be the maximal solution in a suitable Sobolev class of a variational inequality of min-max type with an obstacle constraint and a gradient constraint. Although the variational inequality and the game are solved on an unbounded domain, we do not require boundedness of either the coefficients of the controlled SDE or of the cost functions in the game.Funding: A. Bovo was partially supported by the Doctoral Studentship from the University of Leeds.
我们研究的是一类在有限时间范围内奇异控制器与阻止者之间的零和博弈。基本过程是在无界域中演化的多维（局部非enerate）受控随机微分方程（SDE）。我们证明了这种博弈存在一个值，并为阻止者提供了一个最优策略。博弈值被证明是具有障碍约束和梯度约束的最小-最大类型变分不等式的合适 Sobolev 类中的最大解。虽然变分不等式和博弈都是在无界域上求解的，但我们并不要求受控 SDE 的系数或博弈中的代价函数有界：A. Bovo 部分获得了利兹大学博士生奖学金的资助。
{"title":"Variational Inequalities on Unbounded Domains for Zero-Sum Singular Controller vs. Stopper Games","authors":"Andrea Bovo, Tiziano De Angelis, Elena Issoglio","doi":"10.1287/moor.2023.0029","DOIUrl":"https://doi.org/10.1287/moor.2023.0029","url":null,"abstract":"We study a class of zero-sum games between a singular controller and a stopper over a finite-time horizon. The underlying process is a multidimensional (locally nondegenerate) controlled stochastic differential equation (SDE) evolving in an unbounded domain. We prove that such games admit a value and provide an optimal strategy for the stopper. The value of the game is shown to be the maximal solution in a suitable Sobolev class of a variational inequality of min-max type with an obstacle constraint and a gradient constraint. Although the variational inequality and the game are solved on an unbounded domain, we do not require boundedness of either the coefficients of the controlled SDE or of the cost functions in the game.Funding: A. Bovo was partially supported by the Doctoral Studentship from the University of Leeds.","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"18 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139923324","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we study the risk-sharing problem among multiple agents using lambda value at risk ([Formula: see text]) as their preferences via the tool of inf-convolution, where [Formula: see text] is an extension of value at risk ([Formula: see text]). We obtain explicit formulas of the inf-convolution of multiple [Formula: see text] with monotone Λ and explicit forms of the corresponding optimal allocations, extending the results of the inf-convolution of [Formula: see text]. It turns out that the inf-convolution of several [Formula: see text] is still a [Formula: see text] under some mild condition. Moreover, we investigate the inf-convolution of one [Formula: see text] and a general monotone risk measure without cash additivity, including [Formula: see text], expected utility, and rank-dependent expected utility as special cases. The expression of the inf-convolution and the explicit forms of the optimal allocation are derived, leading to some partial solution of the risk-sharing problem with multiple [Formula: see text] for general Λ functions. Finally, we discuss the risk-sharing problem with [Formula: see text], another definition of lambda value at risk. We focus on the inf-convolution of [Formula: see text] and a risk measure that is consistent with the second-order stochastic dominance, deriving very different expression of the inf-convolution and the forms of the optimal allocations.
{"title":"Risk Sharing with Lambda Value at Risk","authors":"Peng Liu","doi":"10.1287/moor.2023.0246","DOIUrl":"https://doi.org/10.1287/moor.2023.0246","url":null,"abstract":"In this paper, we study the risk-sharing problem among multiple agents using lambda value at risk ([Formula: see text]) as their preferences via the tool of inf-convolution, where [Formula: see text] is an extension of value at risk ([Formula: see text]). We obtain explicit formulas of the inf-convolution of multiple [Formula: see text] with monotone Λ and explicit forms of the corresponding optimal allocations, extending the results of the inf-convolution of [Formula: see text]. It turns out that the inf-convolution of several [Formula: see text] is still a [Formula: see text] under some mild condition. Moreover, we investigate the inf-convolution of one [Formula: see text] and a general monotone risk measure without cash additivity, including [Formula: see text], expected utility, and rank-dependent expected utility as special cases. The expression of the inf-convolution and the explicit forms of the optimal allocation are derived, leading to some partial solution of the risk-sharing problem with multiple [Formula: see text] for general Λ functions. Finally, we discuss the risk-sharing problem with [Formula: see text], another definition of lambda value at risk. We focus on the inf-convolution of [Formula: see text] and a risk measure that is consistent with the second-order stochastic dominance, deriving very different expression of the inf-convolution and the forms of the optimal allocations.","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"234 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139923182","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Motivated by resource allocation problems (RAPs) in power management applications, we investigate the existence of solutions to optimization problems that simultaneously minimize the class of Schur-convex functions, also called least-majorized elements. For this, we introduce a generalization of majorization and least-majorized elements, called (a, b)-majorization and least (a, b)-majorized elements, and characterize the feasible sets of problems that have such elements in terms of base and (bi-)submodular polyhedra. Hereby, we also obtain new characterizations of these polyhedra that extend classical characterizations in terms of optimal greedy algorithms from the 1970s. We discuss the implications of our results for RAPs in power management applications and derive a new characterization of convex cooperative games and new properties of optimal estimators of specific regularized regression problems. In general, our results highlight the combinatorial nature of simultaneously optimizing solutions and provide a theoretical explanation for why such solutions generally do not exist.
{"title":"A Characterization of Simultaneous Optimization, Majorization, and (Bi-)Submodular Polyhedra","authors":"Martijn H. H. Schoot Uiterkamp","doi":"10.1287/moor.2023.0054","DOIUrl":"https://doi.org/10.1287/moor.2023.0054","url":null,"abstract":"Motivated by resource allocation problems (RAPs) in power management applications, we investigate the existence of solutions to optimization problems that simultaneously minimize the class of Schur-convex functions, also called least-majorized elements. For this, we introduce a generalization of majorization and least-majorized elements, called (a, b)-majorization and least (a, b)-majorized elements, and characterize the feasible sets of problems that have such elements in terms of base and (bi-)submodular polyhedra. Hereby, we also obtain new characterizations of these polyhedra that extend classical characterizations in terms of optimal greedy algorithms from the 1970s. We discuss the implications of our results for RAPs in power management applications and derive a new characterization of convex cooperative games and new properties of optimal estimators of specific regularized regression problems. In general, our results highlight the combinatorial nature of simultaneously optimizing solutions and provide a theoretical explanation for why such solutions generally do not exist.","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"26 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139946280","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the problem of training a shallow neural network with quadratic activation functions and the generalization power of such trained networks. Assuming that the samples are generated by a full rank matrix [Formula: see text] of the hidden network node weights, we obtain the following results. We establish that all full-rank approximately stationary solutions of the risk minimization problem are also approximate global optimums of the risk (in-sample and population). As a consequence, we establish that, when trained on polynomially many samples, the gradient descent algorithm converges to the global optimum of the risk minimization problem regardless of the width of the network when it is initialized at some value [Formula: see text], which we compute. Furthermore, the network produced by the gradient descent has a near zero generalization error. Next, we establish that initializing the gradient descent algorithm below [Formula: see text] is easily achieved when the weights of the ground truth matrix [Formula: see text] are randomly generated and the matrix is sufficiently overparameterized. Finally, we identify a simple necessary and sufficient geometric condition on the size of the training set under which any global minimizer of the empirical risk has necessarily zero generalization error.Funding: The research of E. C. Kizildag is supported by Columbia University, with the Distinguished Postdoctoral Fellowship in Statistics. Support from the National Science Foundation [Grant DMS-2015517] is gratefully acknowledged.
我们考虑的是用二次激活函数训练浅层神经网络的问题，以及这种训练网络的泛化能力。假设样本是由隐藏网络节点权重的全秩矩阵 [公式：见正文] 生成的，我们会得到以下结果。我们确定，风险最小化问题的所有全秩近似静态解也是风险的近似全局最优解（样本内和群体）。因此，我们确定，当在多项式数量的样本上进行训练时，梯度下降算法会收敛到风险最小化问题的全局最优，而不管网络的宽度是多少，当它初始化为某个值时[公式：见正文]，我们计算出了这个值。此外，梯度下降算法生成的网络具有接近零的泛化误差。接下来，我们确定，当地面实况矩阵[公式：见正文]的权重是随机生成的，且矩阵被充分过度参数化时，梯度下降算法的初始化值低于[公式：见正文]是很容易实现的。最后，我们在训练集的大小上确定了一个简单的必要和充分几何条件，在此条件下，经验风险的任何全局最小化都必然具有零泛化误差：E. C. Kizildag 的研究得到了哥伦比亚大学统计学杰出博士后奖学金的支持。感谢美国国家科学基金会 [Grant DMS-2015517] 的支持。
{"title":"Stationary Points of a Shallow Neural Network with Quadratic Activations and the Global Optimality of the Gradient Descent Algorithm","authors":"David Gamarnik, Eren C. Kızıldağ, Ilias Zadik","doi":"10.1287/moor.2021.0082","DOIUrl":"https://doi.org/10.1287/moor.2021.0082","url":null,"abstract":"We consider the problem of training a shallow neural network with quadratic activation functions and the generalization power of such trained networks. Assuming that the samples are generated by a full rank matrix [Formula: see text] of the hidden network node weights, we obtain the following results. We establish that all full-rank approximately stationary solutions of the risk minimization problem are also approximate global optimums of the risk (in-sample and population). As a consequence, we establish that, when trained on polynomially many samples, the gradient descent algorithm converges to the global optimum of the risk minimization problem regardless of the width of the network when it is initialized at some value [Formula: see text], which we compute. Furthermore, the network produced by the gradient descent has a near zero generalization error. Next, we establish that initializing the gradient descent algorithm below [Formula: see text] is easily achieved when the weights of the ground truth matrix [Formula: see text] are randomly generated and the matrix is sufficiently overparameterized. Finally, we identify a simple necessary and sufficient geometric condition on the size of the training set under which any global minimizer of the empirical risk has necessarily zero generalization error.Funding: The research of E. C. Kizildag is supported by Columbia University, with the Distinguished Postdoctoral Fellowship in Statistics. Support from the National Science Foundation [Grant DMS-2015517] is gratefully acknowledged.","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"32 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139946201","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Tara Abrishami, Maria Chudnovsky, Cemil Dibek, Kristina Vušković
We give a combinatorial polynomial-time algorithm to find a maximum weight independent set in perfect graphs of bounded degree that do not contain a prism or a hole of length four as an induced subgraph. An even pair in a graph is a pair of vertices all induced paths between which are even. An even set is a set of vertices every two of which are an even pair. We show that every perfect graph that does not contain a prism or a hole of length four as an induced subgraph has a balanced separator which is the union of a bounded number of even sets, where the bound depends only on the maximum degree of the graph. This allows us to solve the maximum weight independent set problem using the well-known submodular function minimization algorithm.Funding: This work was supported by the Engineering and Physical Sciences Research Council [Grant EP/V002813/1]; the National Science Foundation [Grants DMS-1763817, DMS-2120644, and DMS-2303251]; and Alexander von Humboldt-Stiftung.
我们给出了一种组合多项式时间算法，可以在不包含长度为四的棱或洞作为诱导子图的有界度完美图中找到最大权重独立集。图中的偶数对是指一对顶点之间的所有诱导路径都是偶数。偶数集是每两个顶点都是偶数对的顶点集。我们证明，每一个不包含棱柱或长度为四的洞的完美图的诱导子图都有一个平衡分隔符，它是有界数的偶数集的联合，而界数只取决于图的最大度。这样，我们就可以使用著名的子模函数最小化算法来解决最大权重独立集问题：本研究得到了工程与物理科学研究委员会 [EP/V002813/1 号基金]、美国国家科学基金会 [DMS-1763817 号基金、DMS-2120644 号基金和 DMS-2303251 号基金] 以及 Alexander von Humboldt-Stiftung 的资助。
{"title":"Submodular Functions and Perfect Graphs","authors":"Tara Abrishami, Maria Chudnovsky, Cemil Dibek, Kristina Vušković","doi":"10.1287/moor.2021.0302","DOIUrl":"https://doi.org/10.1287/moor.2021.0302","url":null,"abstract":"We give a combinatorial polynomial-time algorithm to find a maximum weight independent set in perfect graphs of bounded degree that do not contain a prism or a hole of length four as an induced subgraph. An even pair in a graph is a pair of vertices all induced paths between which are even. An even set is a set of vertices every two of which are an even pair. We show that every perfect graph that does not contain a prism or a hole of length four as an induced subgraph has a balanced separator which is the union of a bounded number of even sets, where the bound depends only on the maximum degree of the graph. This allows us to solve the maximum weight independent set problem using the well-known submodular function minimization algorithm.Funding: This work was supported by the Engineering and Physical Sciences Research Council [Grant EP/V002813/1]; the National Science Foundation [Grants DMS-1763817, DMS-2120644, and DMS-2303251]; and Alexander von Humboldt-Stiftung.","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"200 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139755776","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We develop a comprehensive methodology for the fluctuation theory of continuous-time, skip-free Markov chains, extending and improving the recent work of Choi and Patie for discrete-time, skip-free Markov chains. As a significant application, we use it to derive a full set of fluctuation identities regarding exiting a finite or infinite interval for Markov branching processes with immigration, thereby uncovering many new results for this classic family of continuous-time Markov chains. The theory also allows us to recover in a simple manner fluctuation identities for skip-free downward compound Poisson processes.
{"title":"Fluctuation Theory of Continuous-Time, Skip-Free Downward Markov Chains with Applications to Branching Processes with Immigration","authors":"Ronnie Loeffen, Pierre Patie, Jian Wang","doi":"10.1287/moor.2022.0246","DOIUrl":"https://doi.org/10.1287/moor.2022.0246","url":null,"abstract":"We develop a comprehensive methodology for the fluctuation theory of continuous-time, skip-free Markov chains, extending and improving the recent work of Choi and Patie for discrete-time, skip-free Markov chains. As a significant application, we use it to derive a full set of fluctuation identities regarding exiting a finite or infinite interval for Markov branching processes with immigration, thereby uncovering many new results for this classic family of continuous-time Markov chains. The theory also allows us to recover in a simple manner fluctuation identities for skip-free downward compound Poisson processes.","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"245 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139755768","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
There is growing interest in nonconvex minimax problems that is driven by an abundance of applications. Our focus is on nonsmooth, nonconvex-strongly concave minimax, thus departing from the more common weakly convex and smooth models assumed in the recent literature. We present proximal gradient schemes with either parallel or alternating steps. We show that both methods can be analyzed through a single scheme within a unified analysis that relies on expanding a general convergence mechanism used for analyzing nonconvex, nonsmooth optimization problems. In contrast to the current literature, which focuses on the complexity of obtaining nearly approximate stationary solutions, we prove subsequence convergence to a critical point of the primal objective and global convergence when the latter is semialgebraic. Furthermore, the complexity results we provide are with respect to approximate stationary solutions. Lastly, we expand the scope of problems that can be addressed by generalizing one of the steps with a Bregman proximal gradient update, and together with a few adjustments to the analysis, this allows us to extend the convergence and complexity results to this broader setting.Funding: The research of E. Cohen was partially supported by a doctoral fellowship from the Israel Science Foundation [Grant 2619-20] and Deutsche Forschungsgemeinschaft [Grant 800240]. The research of M. Teboulle was partially supported by the Israel Science Foundation [Grant 2619-20] and Deutsche Forschungsgemeinschaft [Grant 800240].
{"title":"Alternating and Parallel Proximal Gradient Methods for Nonsmooth, Nonconvex Minimax: A Unified Convergence Analysis","authors":"Eyal Cohen, Marc Teboulle","doi":"10.1287/moor.2022.0294","DOIUrl":"https://doi.org/10.1287/moor.2022.0294","url":null,"abstract":"There is growing interest in nonconvex minimax problems that is driven by an abundance of applications. Our focus is on nonsmooth, nonconvex-strongly concave minimax, thus departing from the more common weakly convex and smooth models assumed in the recent literature. We present proximal gradient schemes with either parallel or alternating steps. We show that both methods can be analyzed through a single scheme within a unified analysis that relies on expanding a general convergence mechanism used for analyzing nonconvex, nonsmooth optimization problems. In contrast to the current literature, which focuses on the complexity of obtaining nearly approximate stationary solutions, we prove subsequence convergence to a critical point of the primal objective and global convergence when the latter is semialgebraic. Furthermore, the complexity results we provide are with respect to approximate stationary solutions. Lastly, we expand the scope of problems that can be addressed by generalizing one of the steps with a Bregman proximal gradient update, and together with a few adjustments to the analysis, this allows us to extend the convergence and complexity results to this broader setting.Funding: The research of E. Cohen was partially supported by a doctoral fellowship from the Israel Science Foundation [Grant 2619-20] and Deutsche Forschungsgemeinschaft [Grant 800240]. The research of M. Teboulle was partially supported by the Israel Science Foundation [Grant 2619-20] and Deutsche Forschungsgemeinschaft [Grant 800240].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"2 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139755793","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose a model uncertainty approach to heavy traffic asymptotics that allows for a high level of uncertainty. That is, the uncertainty classes of underlying distributions accommodate disturbances that are of order 1 at the usual diffusion scale as opposed to asymptotically vanishing disturbances studied previously in relation to heavy traffic. A main advantage of the approach is that the invariance principle underlying diffusion limits makes it possible to define uncertainty classes in terms of the first two moments only. The model we consider is a single-server queue with multiple job types. The problem is formulated as a zero sum stochastic game played between the system controller, who determines scheduling and attempts to minimize an expected linear holding cost, and an adversary, who dynamically controls the service time distributions of arriving jobs and attempts to maximize the cost. The heavy traffic asymptotics of the game are fully solved. It is shown that an asymptotically optimal policy for the system controller is to prioritize according to an index rule, and for the adversary, it is to select distributions based on the system’s current workload. The workload-to-distribution feedback mapping is determined by a Hamilton–Jacobi–Bellman equation, which also characterizes the game’s limit value. Unlike in the vast majority of results in the heavy traffic theory and as a direct consequence of the diffusive size disturbances, the limiting dynamics under asymptotically optimal play are captured by a stochastic differential equation where both the drift and the diffusion coefficients may be discontinuous.Funding: R. Atar is supported by the Israeli Science Foundation [Grant 1035/20].
{"title":"Scheduling in the High-Uncertainty Heavy Traffic Regime","authors":"Rami Atar, Eyal Castiel, Yonatan Shadmi","doi":"10.1287/moor.2022.0100","DOIUrl":"https://doi.org/10.1287/moor.2022.0100","url":null,"abstract":"We propose a model uncertainty approach to heavy traffic asymptotics that allows for a high level of uncertainty. That is, the uncertainty classes of underlying distributions accommodate disturbances that are of order 1 at the usual diffusion scale as opposed to asymptotically vanishing disturbances studied previously in relation to heavy traffic. A main advantage of the approach is that the invariance principle underlying diffusion limits makes it possible to define uncertainty classes in terms of the first two moments only. The model we consider is a single-server queue with multiple job types. The problem is formulated as a zero sum stochastic game played between the system controller, who determines scheduling and attempts to minimize an expected linear holding cost, and an adversary, who dynamically controls the service time distributions of arriving jobs and attempts to maximize the cost. The heavy traffic asymptotics of the game are fully solved. It is shown that an asymptotically optimal policy for the system controller is to prioritize according to an index rule, and for the adversary, it is to select distributions based on the system’s current workload. The workload-to-distribution feedback mapping is determined by a Hamilton–Jacobi–Bellman equation, which also characterizes the game’s limit value. Unlike in the vast majority of results in the heavy traffic theory and as a direct consequence of the diffusive size disturbances, the limiting dynamics under asymptotically optimal play are captured by a stochastic differential equation where both the drift and the diffusion coefficients may be discontinuous.Funding: R. Atar is supported by the Israeli Science Foundation [Grant 1035/20].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"77 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139755796","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A joint mix (JM) is a random vector with a constant component-wise sum. The dependence structure of a joint mix minimizes some common objectives, such as the variance of the component-wise sum, and it is regarded as a concept of extremal negative dependence. In this paper, we explore the connection between the joint mix structure and popular notions of negative dependence in statistics, such as negative correlation dependence, negative orthant dependence, and negative association. A joint mix is not always negatively dependent in any of these senses, but some natural classes of joint mixes are. We derive various necessary and sufficient conditions for a joint mix to be negatively dependent and study the compatibility of these notions. For identical marginal distributions, we show that a negatively dependent joint mix solves a multimarginal optimal transport problem for quadratic cost under a novel setting of uncertainty. Analysis of this optimal transport problem with heterogeneous marginals reveals a trade-off between negative dependence and the joint mix structure.Funding: T. Koike was supported by the Japan Society for the Promotion of Science [Grant JSPS KAKENHI JP21K13275]. R. Wang acknowledges financial support from the Natural Sciences and Engineering Research Council of Canada [Grants RGPIN-2018-03823 and RGPAS-2018-522590].
联合混合（JM）是一种具有恒定分量和的随机向量。联合混合的依赖结构能使一些常见目标（如分量和的方差）最小化，它被视为极端负依赖的概念。在本文中，我们将探讨联合混合结构与统计学中流行的负依赖性概念（如负相关依赖性、负正相关依赖性和负关联性）之间的联系。联合混合结构并不总是这些意义上的负依赖性，但有些自然类的联合混合结构是负依赖性的。我们推导了联合混合负相关的各种必要条件和充分条件，并研究了这些概念的兼容性。在边际分布相同的情况下，我们证明了负相关联合组合可以解决一个新颖的不确定性环境下二次成本的多边际最优运输问题。对这个具有异质性边际分布的最优运输问题的分析揭示了负依赖性与联合混合结构之间的权衡：T. Koike 得到了日本学术振兴会的资助[JSPS KAKENHI JP21K13275]。R. Wang 感谢加拿大自然科学与工程研究理事会 [Grants RGPIN-2018-03823 and RGPAS-2018-522590] 的资助。
{"title":"Joint Mixability and Notions of Negative Dependence","authors":"Takaaki Koike, Liyuan Lin, Ruodu Wang","doi":"10.1287/moor.2022.0121","DOIUrl":"https://doi.org/10.1287/moor.2022.0121","url":null,"abstract":"A joint mix (JM) is a random vector with a constant component-wise sum. The dependence structure of a joint mix minimizes some common objectives, such as the variance of the component-wise sum, and it is regarded as a concept of extremal negative dependence. In this paper, we explore the connection between the joint mix structure and popular notions of negative dependence in statistics, such as negative correlation dependence, negative orthant dependence, and negative association. A joint mix is not always negatively dependent in any of these senses, but some natural classes of joint mixes are. We derive various necessary and sufficient conditions for a joint mix to be negatively dependent and study the compatibility of these notions. For identical marginal distributions, we show that a negatively dependent joint mix solves a multimarginal optimal transport problem for quadratic cost under a novel setting of uncertainty. Analysis of this optimal transport problem with heterogeneous marginals reveals a trade-off between negative dependence and the joint mix structure.Funding: T. Koike was supported by the Japan Society for the Promotion of Science [Grant JSPS KAKENHI JP21K13275]. R. Wang acknowledges financial support from the Natural Sciences and Engineering Research Council of Canada [Grants RGPIN-2018-03823 and RGPAS-2018-522590].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"163 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139584331","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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