Mathematics of Operations Research最新文献_第8页

Risk Sharing with Lambda Value at Risk 用 Lambda 风险值分担风险

IF 1.7 3区数学

Mathematics of Operations Research Pub Date : 2024-02-21 DOI: 10.1287/moor.2023.0246

Peng Liu

{"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":null,"pages":null},"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}

引用次数: 0

A Characterization of Simultaneous Optimization, Majorization, and (Bi-)Submodular Polyhedra 同时优化、大数化和（双）次模多面体的表征

IF 1.7 3区数学

Mathematics of Operations Research Pub Date : 2024-02-20 DOI: 10.1287/moor.2023.0054

Martijn H. H. Schoot Uiterkamp

引用次数: 0

Stationary Points of a Shallow Neural Network with Quadratic Activations and the Global Optimality of the Gradient Descent Algorithm 具有二次激活的浅层神经网络的驻点和梯度下降算法的全局最优性

IF 1.7 3区数学

Mathematics of Operations Research Pub Date : 2024-02-20 DOI: 10.1287/moor.2021.0082

David Gamarnik, Eren C. Kızıldağ, Ilias Zadik

{"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":null,"pages":null},"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}

引用次数: 0

Submodular Functions and Perfect Graphs 亚模态函数与完美图形

IF 1.7 3区数学

Mathematics of Operations Research Pub Date : 2024-02-15 DOI: 10.1287/moor.2021.0302

Tara Abrishami, Maria Chudnovsky, Cemil Dibek, Kristina Vušković

引用次数: 0

Fluctuation Theory of Continuous-Time, Skip-Free Downward Markov Chains with Applications to Branching Processes with Immigration 连续时间、无跳跃向下马尔可夫链的波动理论及其在有移民的分支过程中的应用

IF 1.7 3区数学

Mathematics of Operations Research Pub Date : 2024-02-12 DOI: 10.1287/moor.2022.0246

Ronnie Loeffen, Pierre Patie, Jian Wang

引用次数: 0

Alternating and Parallel Proximal Gradient Methods for Nonsmooth, Nonconvex Minimax: A Unified Convergence Analysis 非平滑、非凸最小值的交替和并行近端梯度法：统一收敛分析

IF 1.7 3区数学

Mathematics of Operations Research Pub Date : 2024-02-08 DOI: 10.1287/moor.2022.0294

Eyal Cohen, Marc Teboulle

{"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":null,"pages":null},"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}

引用次数: 0

Scheduling in the High-Uncertainty Heavy Traffic Regime 高不确定性大流量情况下的调度

IF 1.7 3区数学

Mathematics of Operations Research Pub Date : 2024-02-07 DOI: 10.1287/moor.2022.0100

Rami Atar, Eyal Castiel, Yonatan Shadmi

{"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":null,"pages":null},"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}

引用次数: 0

Joint Mixability and Notions of Negative Dependence 联合混合性和负相关概念

IF 1.7 3区数学

Mathematics of Operations Research Pub Date : 2024-01-29 DOI: 10.1287/moor.2022.0121

Takaaki Koike, Liyuan Lin, Ruodu Wang

{"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":null,"pages":null},"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}

引用次数: 0

Generalization Bounds in the Predict-Then-Optimize Framework 预测-优化框架中的泛化界限

IF 1.7 3区数学

Mathematics of Operations Research Pub Date : 2024-01-24 DOI: 10.1287/moor.2022.1330

Othman El Balghiti, Adam N. Elmachtoub, Paul Grigas, Ambuj Tewari

{"title":"Generalization Bounds in the Predict-Then-Optimize Framework","authors":"Othman El Balghiti, Adam N. Elmachtoub, Paul Grigas, Ambuj Tewari","doi":"10.1287/moor.2022.1330","DOIUrl":"https://doi.org/10.1287/moor.2022.1330","url":null,"abstract":"The predict-then-optimize framework is fundamental in many practical settings: predict the unknown parameters of an optimization problem and then solve the problem using the predicted values of the parameters. A natural loss function in this environment is to consider the cost of the decisions induced by the predicted parameters in contrast to the prediction error of the parameters. This loss function is referred to as the smart predict-then-optimize (SPO) loss. In this work, we seek to provide bounds on how well the performance of a prediction model fit on training data generalizes out of sample in the context of the SPO loss. Because the SPO loss is nonconvex and non-Lipschitz, standard results for deriving generalization bounds do not apply. We first derive bounds based on the Natarajan dimension that, in the case of a polyhedral feasible region, scale at most logarithmically in the number of extreme points but, in the case of a general convex feasible region, have linear dependence on the decision dimension. By exploiting the structure of the SPO loss function and a key property of the feasible region, which we denote as the strength property, we can dramatically improve the dependence on the decision and feature dimensions. Our approach and analysis rely on placing a margin around problematic predictions that do not yield unique optimal solutions and then providing generalization bounds in the context of a modified margin SPO loss function that is Lipschitz continuous. Finally, we characterize the strength property and show that the modified SPO loss can be computed efficiently for both strongly convex bodies and polytopes with an explicit extreme point representation.Funding: O. El Balghiti thanks Rayens Capital for their support. A. N. Elmachtoub acknowledges the support of the National Science Foundation (NSF) [Grant CMMI-1763000]. P. Grigas acknowledges the support of NSF [Grants CCF-1755705 and CMMI-1762744]. A. Tewari acknowledges the support of the NSF [CAREER grant IIS-1452099] and a Sloan Research Fellowship.","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139648200","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}

引用次数: 0

Flow Allocation Games 流量分配游戏

IF 1.7 3区数学

Mathematics of Operations Research Pub Date : 2024-01-22 DOI: 10.1287/moor.2022.0355

Nils Bertschinger, Martin Hoefer, Daniel Schmand

引用次数: 0