Costanza Catalano, Maria Castaldo, Giacomo Como, Fabio Fagnani
We study a network formation game where n players, identified with the nodes of a directed graph to be formed, choose where to wire their outgoing links in order to maximize their PageRank centrality. Specifically, the action of every player i consists in the wiring of a predetermined number di of directed out-links, and her utility is her own PageRank centrality in the network resulting from the actions of all players. We show that this is a potential game and that the best response correspondence always exhibits a local structure in that it is never convenient for a node i to link to other nodes that are at incoming distance more than di from her. We then study the equilibria of this game determining necessary conditions for a graph to be a (strict, recurrent) Nash equilibrium. Moreover, in the homogeneous case, where players all have the same number d of out-links, we characterize the structure of the potential-maximizing equilibria, and in the special cases d = 1 and d = 2, we provide a complete classification of the set of (strict, recurrent) Nash equilibria. Our analysis shows in particular that the considered formation mechanism leads to the emergence of undirected and disconnected or loosely connected networks.Funding: This research was carried out within the framework of the Ministero dell’Università e della Ricerca (MIUR)-funded Progetto di Eccellenza of the Dipartimento di Scienze Matematiche G. L. Lagrange, Politecnico di Torino [CUP: E11G18000350001]. It received partial support from the MIUR-funded project PRIN 2017 “Advanced Network Control of Future Smart Grids” and from the Compagnia di San Paolo.
我们研究的是一个网络形成博弈，在这个博弈中，n 个参与者（与待形成的有向图的节点相对应）选择在哪里连接他们的外向链接，以最大化他们的 PageRank 中心度。具体来说，每个玩家 i 的行动都包括预先确定数量 di 的有向外链的布线，而她的效用就是她自己在所有玩家行动所形成的网络中的 PageRank 中心度。我们证明这是一个潜在博弈，而且最佳响应对应关系总是呈现局部结构，即节点 i 绝不方便链接到与她的传入距离大于 di 的其他节点。然后，我们研究了这个博弈的均衡，确定了一个图成为（严格的、经常性的）纳什均衡的必要条件。此外，在同质情况下，即所有博弈者都有相同数量的外链 d，我们描述了潜在最大化均衡的结构，而在 d = 1 和 d = 2 的特殊情况下，我们提供了（严格、循环）纳什均衡集的完整分类。我们的分析特别表明，所考虑的形成机制会导致出现无向、断开或松散连接的网络：本研究是在都灵理工大学 G. L. Lagrange 数学科学系都灵大学和研究部（MIUR）资助的杰出项目[CUP: E11G18000350001]框架内进行的。它得到了 MIUR 资助的 PRIN 2017 项目 "未来智能电网的高级网络控制 "和 Compagnia di San Paolo 的部分支持。
{"title":"On a Network Centrality Maximization Game","authors":"Costanza Catalano, Maria Castaldo, Giacomo Como, Fabio Fagnani","doi":"10.1287/moor.2022.0251","DOIUrl":"https://doi.org/10.1287/moor.2022.0251","url":null,"abstract":"We study a network formation game where n players, identified with the nodes of a directed graph to be formed, choose where to wire their outgoing links in order to maximize their PageRank centrality. Specifically, the action of every player i consists in the wiring of a predetermined number d<jats:sub>i</jats:sub> of directed out-links, and her utility is her own PageRank centrality in the network resulting from the actions of all players. We show that this is a potential game and that the best response correspondence always exhibits a local structure in that it is never convenient for a node i to link to other nodes that are at incoming distance more than d<jats:sub>i</jats:sub> from her. We then study the equilibria of this game determining necessary conditions for a graph to be a (strict, recurrent) Nash equilibrium. Moreover, in the homogeneous case, where players all have the same number d of out-links, we characterize the structure of the potential-maximizing equilibria, and in the special cases d = 1 and d = 2, we provide a complete classification of the set of (strict, recurrent) Nash equilibria. Our analysis shows in particular that the considered formation mechanism leads to the emergence of undirected and disconnected or loosely connected networks.Funding: This research was carried out within the framework of the Ministero dell’Università e della Ricerca (MIUR)-funded Progetto di Eccellenza of the Dipartimento di Scienze Matematiche G. L. Lagrange, Politecnico di Torino [CUP: E11G18000350001]. It received partial support from the MIUR-funded project PRIN 2017 “Advanced Network Control of Future Smart Grids” and from the Compagnia di San Paolo.","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"2 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142227349","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 investigate the optimal dividend problem with capital injection and ratcheting constraint with nondecreasing dividend payout rate. Capital injections are introduced in order to eliminate the possibility of bankruptcy. Under the Cramér–Lundberg risk model, the problem is formulated as a two-dimensional stochastic control problem. By applying the viscosity theory, we show that the value function is the unique viscosity solution to the associated Hamilton–Jacobi–Bellman equation. In order to obtain analytical results, we further study the problem with finite ratcheting constraint, where the dividend rate takes only a finite number of available values. We show that the value function under general ratcheting can be approximated arbitrarily closely by the one with finite ratcheting. Finally, we derive the expressions of value function when the threshold-type finite ratcheting dividend strategy with capital injection is applied, and we show the optimality of such a strategy under certain conditions of concavity. Numerical examples under various scenarios are provided at the end.Funding W. Wang was supported by the National Natural Science Foundation of China [Grants 12171405, 12271066, and 11661074] and the Fundamental Research Funds for the Central Universities of China [Grant 20720220044]. R. Xu was supported by the National Natural Science Foundation of China [Grants 12201506 and 12371468], the Natural Science Foundation of the Jiangsu Higher Education Institutions of China [Grant 21KJB110024], and Xi’an Jiaotong-Liverpool University Research Development Funding [Grant RDF-20-01-02].
{"title":"Optimal Ratcheting of Dividends with Capital Injection","authors":"Wenyuan Wang, Ran Xu, Kaixin Yan","doi":"10.1287/moor.2023.0102","DOIUrl":"https://doi.org/10.1287/moor.2023.0102","url":null,"abstract":"In this paper, we investigate the optimal dividend problem with capital injection and ratcheting constraint with nondecreasing dividend payout rate. Capital injections are introduced in order to eliminate the possibility of bankruptcy. Under the Cramér–Lundberg risk model, the problem is formulated as a two-dimensional stochastic control problem. By applying the viscosity theory, we show that the value function is the unique viscosity solution to the associated Hamilton–Jacobi–Bellman equation. In order to obtain analytical results, we further study the problem with finite ratcheting constraint, where the dividend rate takes only a finite number of available values. We show that the value function under general ratcheting can be approximated arbitrarily closely by the one with finite ratcheting. Finally, we derive the expressions of value function when the threshold-type finite ratcheting dividend strategy with capital injection is applied, and we show the optimality of such a strategy under certain conditions of concavity. Numerical examples under various scenarios are provided at the end.Funding W. Wang was supported by the National Natural Science Foundation of China [Grants 12171405, 12271066, and 11661074] and the Fundamental Research Funds for the Central Universities of China [Grant 20720220044]. R. Xu was supported by the National Natural Science Foundation of China [Grants 12201506 and 12371468], the Natural Science Foundation of the Jiangsu Higher Education Institutions of China [Grant 21KJB110024], and Xi’an Jiaotong-Liverpool University Research Development Funding [Grant RDF-20-01-02].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"38 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220587","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}
Pedro Garcia-Segador, Michel Grabisch, Pedro Miranda
We study the geometric structure of the set of cooperative transferable utility games having a nonempty core, characterized by Bondareva and Shapley as balanced games. We show that this set is a nonpointed polyhedral cone, and we find the set of its extremal rays and facets. This study is also done for the set of balanced games whose value for the grand coalition is fixed, which yields an affine nonpointed polyhedral cone. Finally, the case of nonnegative balanced games with fixed value for the grand coalition is tackled. This set is a convex polytope, with remarkable properties. We characterize its vertices and facets, study the adjacency structure of vertices, develop an algorithm for generating vertices in a random uniform way, and show that this polytope is combinatorial and its adjacency graph is Hamiltonian. Last, we give a characterization of the set of games having a core reduced to a singleton.Funding: This work was supported by the Spanish Government [Grant PID2021-124933NB-I00].
{"title":"On the Set of Balanced Games","authors":"Pedro Garcia-Segador, Michel Grabisch, Pedro Miranda","doi":"10.1287/moor.2023.0379","DOIUrl":"https://doi.org/10.1287/moor.2023.0379","url":null,"abstract":"We study the geometric structure of the set of cooperative transferable utility games having a nonempty core, characterized by Bondareva and Shapley as balanced games. We show that this set is a nonpointed polyhedral cone, and we find the set of its extremal rays and facets. This study is also done for the set of balanced games whose value for the grand coalition is fixed, which yields an affine nonpointed polyhedral cone. Finally, the case of nonnegative balanced games with fixed value for the grand coalition is tackled. This set is a convex polytope, with remarkable properties. We characterize its vertices and facets, study the adjacency structure of vertices, develop an algorithm for generating vertices in a random uniform way, and show that this polytope is combinatorial and its adjacency graph is Hamiltonian. Last, we give a characterization of the set of games having a core reduced to a singleton.Funding: This work was supported by the Spanish Government [Grant PID2021-124933NB-I00].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"62 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142227350","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 the mixed-integer epigraph of a special class of convex functions with nonconvex indicator constraints, which are often used to impose logical constraints on the support of the solutions. The class of functions we consider are defined as compositions of low-dimensional nonlinear functions with affine functions. Extended formulations describing the convex hull of such sets can easily be constructed via disjunctive programming although a direct application of this method often yields prohibitively large formulations, whose size is exponential in the number of variables. In this paper, we propose a new disjunctive representation of the sets under study, which leads to compact formulations with size exponential in the dimension of the nonlinear function but polynomial in the number of variables. Moreover, we show how to project out the additional variables for the case of dimension one, recovering or generalizing known results for the convex hulls of such sets (in the original space of variables). Our computational results indicate that the proposed approach can significantly improve the performance of solvers in structured problems.Funding: This work was supported by the National Science Foundation Division of Computing and Communication Foundations [Grant 2006762].
{"title":"Compact Extended Formulations for Low-Rank Functions with Indicator Variables","authors":"Shaoning Han, Andrés Gómez","doi":"10.1287/moor.2021.0281","DOIUrl":"https://doi.org/10.1287/moor.2021.0281","url":null,"abstract":"We study the mixed-integer epigraph of a special class of convex functions with nonconvex indicator constraints, which are often used to impose logical constraints on the support of the solutions. The class of functions we consider are defined as compositions of low-dimensional nonlinear functions with affine functions. Extended formulations describing the convex hull of such sets can easily be constructed via disjunctive programming although a direct application of this method often yields prohibitively large formulations, whose size is exponential in the number of variables. In this paper, we propose a new disjunctive representation of the sets under study, which leads to compact formulations with size exponential in the dimension of the nonlinear function but polynomial in the number of variables. Moreover, we show how to project out the additional variables for the case of dimension one, recovering or generalizing known results for the convex hulls of such sets (in the original space of variables). Our computational results indicate that the proposed approach can significantly improve the performance of solvers in structured problems.Funding: This work was supported by the National Science Foundation Division of Computing and Communication Foundations [Grant 2006762].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"5 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220586","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}
This paper is devoted to the study of the second-order variational analysis of spectral functions. It is well-known that spectral functions can be expressed as a composite function of symmetric functions and eigenvalue functions. We establish several second-order properties of spectral functions when their associated symmetric functions enjoy these properties. Our main attention is given to characterize parabolic regularity for this class of functions. It was observed recently that parabolic regularity can play a central rule in ensuring the validity of important second-order variational properties, such as twice epi-differentiability. We demonstrates that for convex spectral functions, their parabolic regularity amounts to that of their symmetric functions. As an important consequence, we calculate the second subderivative of convex spectral functions, which allows us to establish second-order optimality conditions for a class of matrix optimization problems.Funding: The research of A. Mohammadi is funded by a postdoctoral fellowship from Georgetown University. E. Sarabi is partially supported by the U.S. National Science Foundation [Grant DMS 2108546].
{"title":"Parabolic Regularity of Spectral Functions","authors":"Ashkan Mohammadi, Ebrahim Sarabi","doi":"10.1287/moor.2023.0010","DOIUrl":"https://doi.org/10.1287/moor.2023.0010","url":null,"abstract":"This paper is devoted to the study of the second-order variational analysis of spectral functions. It is well-known that spectral functions can be expressed as a composite function of symmetric functions and eigenvalue functions. We establish several second-order properties of spectral functions when their associated symmetric functions enjoy these properties. Our main attention is given to characterize parabolic regularity for this class of functions. It was observed recently that parabolic regularity can play a central rule in ensuring the validity of important second-order variational properties, such as twice epi-differentiability. We demonstrates that for convex spectral functions, their parabolic regularity amounts to that of their symmetric functions. As an important consequence, we calculate the second subderivative of convex spectral functions, which allows us to establish second-order optimality conditions for a class of matrix optimization problems.Funding: The research of A. Mohammadi is funded by a postdoctoral fellowship from Georgetown University. E. Sarabi is partially supported by the U.S. National Science Foundation [Grant DMS 2108546].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"111 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220588","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}
Marlon R. Moresco, Mélina Mailhot, Silvana M. Pesenti
We introduce a framework for quantifying propagation of uncertainty arising in a dynamic setting. Specifically, we define dynamic uncertainty sets designed explicitly for discrete stochastic processes over a finite time horizon. These dynamic uncertainty sets capture the uncertainty surrounding stochastic processes and models, accounting for factors such as distributional ambiguity. Examples of uncertainty sets include those induced by the Wasserstein distance and f-divergences. We further define dynamic robust risk measures as the supremum of all candidates’ risks within the uncertainty set. In an axiomatic way, we discuss conditions on the uncertainty sets that lead to well-known properties of dynamic robust risk measures, such as convexity and coherence. Furthermore, we discuss the necessary and sufficient properties of dynamic uncertainty sets that lead to time-consistencies of dynamic robust risk measures. We find that uncertainty sets stemming from f-divergences lead to strong time-consistency whereas the Wasserstein distance results in a new time-consistent notion of weak recursiveness. Moreover, we show that a dynamic robust risk measure is strong time-consistent or weak recursive if and only if it admits a recursive representation of one-step conditional robust risk measures arising from static uncertainty sets.Funding: M. Mailhot and S. M. Pesenti acknowledge support from the Canadian Statistical Sciences Institute (CANSSI) and from the Natural Sciences and Engineering Research Council of Canada [Grants RGPIN-2015-05447, DGECR-2020-00333, and RGPIN-2020-04289]. M. R. Moresco thanks the Horizon Postdoctoral Fellowship for the support.
我们介绍了一种量化动态环境中不确定性传播的框架。具体来说，我们定义了动态不确定性集，明确用于有限时间范围内的离散随机过程。这些动态不确定性集捕捉了随机过程和模型的不确定性，并考虑了分布模糊性等因素。不确定性集的例子包括由瓦瑟斯坦距离和 f-divergences 引起的不确定性集。我们进一步将动态稳健风险度量定义为不确定性集内所有候选风险的上集。我们以公理的方式讨论了不确定性集的条件，这些条件导致了动态稳健风险度量的众所周知的特性，如凸性和一致性。此外，我们还讨论了动态不确定性集的必要和充分属性，这些属性会导致动态稳健风险度量的时间一致性。我们发现，源于 f-divergences 的不确定性集会导致强时间一致性，而 Wasserstein 距离则会导致一种新的时间一致性概念，即弱递归性。此外，我们还证明了动态稳健风险度量是强时间一致性或弱递归性的，当且仅当它允许静态不确定性集产生的一步条件稳健风险度量的递归表示时：M. Mailhot 和 S. M. Pesenti 感谢加拿大统计科学研究所 (CANSSI) 和加拿大自然科学与工程研究理事会 [Grants RGPIN-2015-05447, DGECR-2020-00333, and RGPIN-2020-04289] 的支持。M. R. Moresco 感谢地平线博士后奖学金的支持。
{"title":"Uncertainty Propagation and Dynamic Robust Risk Measures","authors":"Marlon R. Moresco, Mélina Mailhot, Silvana M. Pesenti","doi":"10.1287/moor.2023.0267","DOIUrl":"https://doi.org/10.1287/moor.2023.0267","url":null,"abstract":"We introduce a framework for quantifying propagation of uncertainty arising in a dynamic setting. Specifically, we define dynamic uncertainty sets designed explicitly for discrete stochastic processes over a finite time horizon. These dynamic uncertainty sets capture the uncertainty surrounding stochastic processes and models, accounting for factors such as distributional ambiguity. Examples of uncertainty sets include those induced by the Wasserstein distance and f-divergences. We further define dynamic robust risk measures as the supremum of all candidates’ risks within the uncertainty set. In an axiomatic way, we discuss conditions on the uncertainty sets that lead to well-known properties of dynamic robust risk measures, such as convexity and coherence. Furthermore, we discuss the necessary and sufficient properties of dynamic uncertainty sets that lead to time-consistencies of dynamic robust risk measures. We find that uncertainty sets stemming from f-divergences lead to strong time-consistency whereas the Wasserstein distance results in a new time-consistent notion of weak recursiveness. Moreover, we show that a dynamic robust risk measure is strong time-consistent or weak recursive if and only if it admits a recursive representation of one-step conditional robust risk measures arising from static uncertainty sets.Funding: M. Mailhot and S. M. Pesenti acknowledge support from the Canadian Statistical Sciences Institute (CANSSI) and from the Natural Sciences and Engineering Research Council of Canada [Grants RGPIN-2015-05447, DGECR-2020-00333, and RGPIN-2020-04289]. M. R. Moresco thanks the Horizon Postdoctoral Fellowship for the support.","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"20 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141939658","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 fair allocation of indivisible goods to n equally entitled agents. Every agent i has a valuation function vi from some given class of valuation functions. A share s is a function that maps [Formula: see text] to a nonnegative value. A share is feasible if for every allocation instance, there is an allocation that gives every agent i a bundle that is acceptable with respect to vi, one of value at least her share value [Formula: see text]. We introduce the following concepts. A share is self-maximizing if reporting the true valuation maximizes the minimum true value of a bundle that is acceptable with respect to the report. A share s ρ-dominates another share [Formula: see text] if [Formula: see text] for every valuation function. We initiate a systematic study of feasible and self-maximizing shares and a systematic study of ρ-domination relation between shares, presenting both positive and negative results.Funding: The research of M. Babaioff is supported in part by a Golda Meir Fellowship. The research of U. Feige is supported in part by the Israel Science Foundation [Grant 1122/22].
我们考虑将不可分割的商品公平分配给 n 个权利平等的代理人。每个代理人 i 都有一个估值函数 vi，该函数来自给定的某类估值函数。份额 s 是一个将[公式：见正文]映射为非负值的函数。如果对每个分配实例来说，有一种分配能给每个代理人 i 提供一个对 vi 来说是可接受的捆绑包，其价值至少等于她的份额值[公式：见正文]，那么这种份额就是可行的。我们引入以下概念。如果报告真实估值能使就报告而言可接受的价值包的最小真实值最大化，那么份额就是自我最大化的。如果对于每个估值函数[公式：见正文]，一个股票 s ρ支配另一个股票[公式：见正文]。我们开始系统地研究可行股份和自我最大化股份，并系统地研究股份间的ρ支配关系，提出了正反两方面的结果：M. Babaioff 的研究得到了 Golda Meir 奖学金的部分资助。U. Feige 的研究部分得到以色列科学基金会 [1122/22 号拨款] 的支持。
{"title":"Fair Shares: Feasibility, Domination, and Incentives","authors":"Moshe Babaioff, Uriel Feige","doi":"10.1287/moor.2022.0257","DOIUrl":"https://doi.org/10.1287/moor.2022.0257","url":null,"abstract":"We consider fair allocation of indivisible goods to n equally entitled agents. Every agent i has a valuation function v<jats:sub>i</jats:sub> from some given class of valuation functions. A share s is a function that maps [Formula: see text] to a nonnegative value. A share is feasible if for every allocation instance, there is an allocation that gives every agent i a bundle that is acceptable with respect to v<jats:sub>i</jats:sub>, one of value at least her share value [Formula: see text]. We introduce the following concepts. A share is self-maximizing if reporting the true valuation maximizes the minimum true value of a bundle that is acceptable with respect to the report. A share s ρ-dominates another share [Formula: see text] if [Formula: see text] for every valuation function. We initiate a systematic study of feasible and self-maximizing shares and a systematic study of ρ-domination relation between shares, presenting both positive and negative results.Funding: The research of M. Babaioff is supported in part by a Golda Meir Fellowship. The research of U. Feige is supported in part by the Israel Science Foundation [Grant 1122/22].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"56 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141873019","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 consider a nonmonotone (mixed) variational inequality (VI) model with (nonlinear) convex conic constraints. Through developing an equivalent Lagrangian function-like primal-dual saddle point system for the VI model in question, we introduce an augmented Lagrangian primal-dual method, called ALAVI (Augmented Lagrangian Approach to Variational Inequality) in the paper, for solving a general constrained VI model. Under an assumption, called the primal-dual variational coherence condition in the paper, we prove the convergence of ALAVI. Next, we show that many existing generalized monotonicity properties are sufficient—though by no means necessary—to imply the abovementioned coherence condition and thus are sufficient to ensure convergence of ALAVI. Under that assumption, we further show that ALAVI has in fact an [Formula: see text] global rate of convergence where k is the iteration count. By introducing a new gap function, this rate further improves to be [Formula: see text] if the mapping is monotone. Finally, we show that under a metric subregularity condition, even if the VI model may be nonmonotone, the local convergence rate of ALAVI improves to be linear. Numerical experiments on some randomly generated highly nonlinear and nonmonotone VI problems show the practical efficacy of the newly proposed method.Funding: L. Zhao and D. Zhu were partially supported by the Major Project of the National Natural Science Foundation of China [Grant 72293582], the National Key R&D Program of China [Grant 2023YFA0915202], and the Fundamental Research Funds for the Central Universities (the Interdisciplinary Program of Shanghai Jiao Tong University) [Grant YG2024QNA36]. L. Zhao was partially supported by the Startup Fund for Young Faculty at SJTU (SFYF at SJTU) [Grant 22X010503839].
在本文中，我们考虑了一个具有（非线性）凸圆锥约束的非单调（混合）变分法不等式（VI）模型。通过为有关 VI 模型开发一个等效的类似拉格朗日函数的初等二元鞍点系统，我们引入了一种用于求解一般约束 VI 模型的增强拉格朗日初等二元方法，本文称之为 ALAVI（增强拉格朗日变分不等式方法）。在本文中称为初等-二元变分一致性条件的假设下，我们证明了 ALAVI 的收敛性。接下来，我们证明了许多现有的广义单调性特性足以隐含上述一致性条件，因此足以确保 ALAVI 的收敛性。在这一假设下，我们进一步证明了 ALAVI 事实上具有[公式：见正文]全局收敛率，其中 k 是迭代次数。通过引入一个新的间隙函数，如果映射是单调的，这个收敛率会进一步提高到[式中：见正文]。最后，我们证明了在度量次规则条件下，即使 VI 模型可能是非单调的，ALAVI 的局部收敛速率也会提高到线性。在一些随机生成的高度非线性和非单调 VI 问题上的数值实验表明，新提出的方法非常实用：国家自然科学基金重大项目[72293582]、国家重点研发计划[2023YFA0915202]和中央高校基本科研业务费（上海交通大学交叉学科项目）[YG2024QNA36]的部分资助。赵立受上海交通大学青年教师创业基金（SFYF at SJTU）[22X010503839]的部分资助。
{"title":"An Augmented Lagrangian Approach to Conically Constrained Nonmonotone Variational Inequality Problems","authors":"Lei Zhao, Daoli Zhu, Shuzhong Zhang","doi":"10.1287/moor.2023.0167","DOIUrl":"https://doi.org/10.1287/moor.2023.0167","url":null,"abstract":"In this paper we consider a nonmonotone (mixed) variational inequality (VI) model with (nonlinear) convex conic constraints. Through developing an equivalent Lagrangian function-like primal-dual saddle point system for the VI model in question, we introduce an augmented Lagrangian primal-dual method, called ALAVI (Augmented Lagrangian Approach to Variational Inequality) in the paper, for solving a general constrained VI model. Under an assumption, called the primal-dual variational coherence condition in the paper, we prove the convergence of ALAVI. Next, we show that many existing generalized monotonicity properties are sufficient—though by no means necessary—to imply the abovementioned coherence condition and thus are sufficient to ensure convergence of ALAVI. Under that assumption, we further show that ALAVI has in fact an [Formula: see text] global rate of convergence where k is the iteration count. By introducing a new gap function, this rate further improves to be [Formula: see text] if the mapping is monotone. Finally, we show that under a metric subregularity condition, even if the VI model may be nonmonotone, the local convergence rate of ALAVI improves to be linear. Numerical experiments on some randomly generated highly nonlinear and nonmonotone VI problems show the practical efficacy of the newly proposed method.Funding: L. Zhao and D. Zhu were partially supported by the Major Project of the National Natural Science Foundation of China [Grant 72293582], the National Key R&D Program of China [Grant 2023YFA0915202], and the Fundamental Research Funds for the Central Universities (the Interdisciplinary Program of Shanghai Jiao Tong University) [Grant YG2024QNA36]. L. Zhao was partially supported by the Startup Fund for Young Faculty at SJTU (SFYF at SJTU) [Grant 22X010503839].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"20 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141873020","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}
José Luis Pérez, Neofytos Rodosthenous, Kazutoshi Yamazaki
We introduce a new nonzero-sum game of optimal stopping with asymmetric exercise opportunities. Given a stochastic process modeling the value of an asset, one player observes and can act on the process continuously, whereas the other player can act on it only periodically at independent Poisson arrival times. The first one to stop receives a reward, different for each player, whereas the other one gets nothing. We study how each player balances the maximization of gains against the maximization of the likelihood of stopping before the opponent. In such a setup driven by a Lévy process with positive jumps, we not only prove the existence but also explicitly construct a Nash equilibrium with values of the game written in terms of the scale function. Numerical illustrations with put-option payoffs are also provided to study the behavior of the players’ strategies as well as the quantification of the value of available exercise opportunities.Funding: K. Yamazaki was partly supported by The Japan Society for the Promotion of Science (JSPS) Grant-in-Aid for Scientific Research (KAKENHI) [Grants 19H01791, 20K03758, and 24K06844], Open Partnership Joint Research Projects [Grant JPJSBP120209921], and the University of Queensland [start-up grant].
{"title":"Nonzero-Sum Optimal Stopping Game with Continuous vs. Periodic Exercise Opportunities","authors":"José Luis Pérez, Neofytos Rodosthenous, Kazutoshi Yamazaki","doi":"10.1287/moor.2023.0123","DOIUrl":"https://doi.org/10.1287/moor.2023.0123","url":null,"abstract":"We introduce a new nonzero-sum game of optimal stopping with asymmetric exercise opportunities. Given a stochastic process modeling the value of an asset, one player observes and can act on the process continuously, whereas the other player can act on it only periodically at independent Poisson arrival times. The first one to stop receives a reward, different for each player, whereas the other one gets nothing. We study how each player balances the maximization of gains against the maximization of the likelihood of stopping before the opponent. In such a setup driven by a Lévy process with positive jumps, we not only prove the existence but also explicitly construct a Nash equilibrium with values of the game written in terms of the scale function. Numerical illustrations with put-option payoffs are also provided to study the behavior of the players’ strategies as well as the quantification of the value of available exercise opportunities.Funding: K. Yamazaki was partly supported by The Japan Society for the Promotion of Science (JSPS) Grant-in-Aid for Scientific Research (KAKENHI) [Grants 19H01791, 20K03758, and 24K06844], Open Partnership Joint Research Projects [Grant JPJSBP120209921], and the University of Queensland [start-up grant].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"48 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141863649","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}
The goal of this paper is to demonstrate that common noise may serve as an exploration noise for learning the solution of a mean field game. This concept is here exemplified through a toy linear-quadratic model, for which a suitable form of common noise has already been proven to restore existence and uniqueness. We here go one step further and prove that the same form of common noise may force the convergence of the learning algorithm called fictitious play, and this without any further potential or monotone structure. Several numerical examples are provided to support our theoretical analysis.Funding: F. Delarue acknowledges the financial support of the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme [AdG ELISA project, Grant 101054746]. A. Vasileiadis acknowledge the financial support of French ANR project ANR-19-P3IA-0002-3IA Côte d'Azur-Nice-Interdisciplinary Institute for Artificial Intelligence.
本文旨在证明普通噪声可以作为学习均值场博弈解的探索噪声。本文通过一个玩具线性二次模型来例证这一概念，对于该模型，一种合适形式的普通噪声已被证明可以恢复其存在性和唯一性。在此，我们更进一步证明，同样形式的共同噪声可以迫使称为虚构博弈的学习算法收敛，而且无需任何进一步的势或单调结构。我们提供了几个数值例子来支持我们的理论分析：F. Delarue 感谢欧洲研究理事会（ERC）在欧盟地平线 2020 研究与创新计划[AdG ELISA 项目，资助金 101054746]下提供的资金支持。A. Vasileiadis 感谢法国国家科学研究署项目 ANR-19-P3IA-0002-3IA Côte d'Azur-Nice-Interdisciplinary Institute for Artificial Intelligence 的资助。
{"title":"Exploration Noise for Learning Linear-Quadratic Mean Field Games","authors":"François Delarue, Athanasios Vasileiadis","doi":"10.1287/moor.2021.0157","DOIUrl":"https://doi.org/10.1287/moor.2021.0157","url":null,"abstract":"The goal of this paper is to demonstrate that common noise may serve as an exploration noise for learning the solution of a mean field game. This concept is here exemplified through a toy linear-quadratic model, for which a suitable form of common noise has already been proven to restore existence and uniqueness. We here go one step further and prove that the same form of common noise may force the convergence of the learning algorithm called fictitious play, and this without any further potential or monotone structure. Several numerical examples are provided to support our theoretical analysis.Funding: F. Delarue acknowledges the financial support of the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme [AdG ELISA project, Grant 101054746]. A. Vasileiadis acknowledge the financial support of French ANR project ANR-19-P3IA-0002-3IA Côte d'Azur-Nice-Interdisciplinary Institute for Artificial Intelligence.","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"1 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141784517","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}
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：