In reality, investors are uncertain about the dynamics of risky asset returns. Therefore, investors prefer to make robust investment decisions. In this paper, we propose an α-robust utility maximization problem under uncertain parameters. The investor is allowed to invest in a financial market consisting of a risk-free asset and a risky asset. The uncertainty about the expected return rate is parameterized by a nonempty set. Different from most existing literature on robust utility maximization problems where investors are generally assumed to be extremely ambiguity averse because they tend to consider only expected utility in the worst-case scenario, we pay attention to the investors who are not only ambiguity averse but also ambiguity seeking. Under power utility, we provide the implicit function representations for the precommitted strategy, equilibrium strategy of the open-loop type, and equilibrium strategy of the closed-loop type. Some properties about the optimal trading strategies, the best-case and worst-case parameters under three different kinds of strategies, are provided.Funding: This work was supported by National Natural Science Foundation of China [Grants 12071147, 12171169, 12271171, 12371470, 71721001, 71931004, 72371256], the Shanghai Philosophy Social Science Planning Office Project [Grant 2022ZJB005], Fundamental Research Funds for the Central Universities [Grant 2022QKT001], the Excellent Young Team Project Natural Science Foundation of Guangdong Province of China [Grant 2023B1515040001], the Philosophy and Social Science Programming Foundation of Guangdong Province [Grant GD22CYJ17], the Nature Science Foundation of Guangdong Province of China [Grant 2022A1515011472], and the 111 Project [Grant B14019].
{"title":"Optimal Investment Strategy for α-Robust Utility Maximization Problem","authors":"Zhou Yang, Danping Li, Yan Zeng, Guanting Liu","doi":"10.1287/moor.2023.0076","DOIUrl":"https://doi.org/10.1287/moor.2023.0076","url":null,"abstract":"In reality, investors are uncertain about the dynamics of risky asset returns. Therefore, investors prefer to make robust investment decisions. In this paper, we propose an α-robust utility maximization problem under uncertain parameters. The investor is allowed to invest in a financial market consisting of a risk-free asset and a risky asset. The uncertainty about the expected return rate is parameterized by a nonempty set. Different from most existing literature on robust utility maximization problems where investors are generally assumed to be extremely ambiguity averse because they tend to consider only expected utility in the worst-case scenario, we pay attention to the investors who are not only ambiguity averse but also ambiguity seeking. Under power utility, we provide the implicit function representations for the precommitted strategy, equilibrium strategy of the open-loop type, and equilibrium strategy of the closed-loop type. Some properties about the optimal trading strategies, the best-case and worst-case parameters under three different kinds of strategies, are provided.Funding: This work was supported by National Natural Science Foundation of China [Grants 12071147, 12171169, 12271171, 12371470, 71721001, 71931004, 72371256], the Shanghai Philosophy Social Science Planning Office Project [Grant 2022ZJB005], Fundamental Research Funds for the Central Universities [Grant 2022QKT001], the Excellent Young Team Project Natural Science Foundation of Guangdong Province of China [Grant 2023B1515040001], the Philosophy and Social Science Programming Foundation of Guangdong Province [Grant GD22CYJ17], the Nature Science Foundation of Guangdong Province of China [Grant 2022A1515011472], and the 111 Project [Grant B14019].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"183 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140199553","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 and study a new multilevel method for the numerical approximation of a Gibbs distribution π on [Formula: see text], based on (overdamped) Langevin diffusions. This method relies on a multilevel occupation measure, that is, on an appropriate combination of R occupation measures of (constant-step) Euler schemes with respective steps [Formula: see text]. We first state a quantitative result under general assumptions that guarantees an ε-approximation (in an L2-sense) with a cost of the order [Formula: see text] or [Formula: see text] under less contractive assumptions. We then apply it to overdamped Langevin diffusions with strongly convex potential [Formula: see text] and obtain an ε-complexity of the order [Formula: see text] or [Formula: see text] under additional assumptions on U. More precisely, up to universal constants, an appropriate choice of the parameters leads to a cost controlled by [Formula: see text] (where [Formula: see text] and [Formula: see text] respectively denote the supremum and the infimum of the largest and lowest eigenvalue of [Formula: see text]). We finally complete these theoretical results with some numerical illustrations, including comparisons to other algorithms in Bayesian learning and opening to the non–strongly convex setting.Funding: The authors are grateful to the SIRIC ILIAD Nantes-Angers program, supported by the French National Cancer Institute [INCA-DGOS-Inserm Grant 12558].
我们提出并研究了一种新的多级方法，基于（过阻尼）朗格文扩散，对[公式：见正文]上的吉布斯分布π进行数值逼近。该方法依赖于多级占优度量，即具有各自步长的（恒定步长）欧拉方案的 R 级占优度量的适当组合[公式：见正文]。我们首先给出一个一般假设下的定量结果，它保证了ε 近似（在 L2 意义上），其代价为[公式：见正文]或[公式：见正文]。然后，我们将其应用于具有强凸势的、过阻尼的朗格文扩散[公式：见正文]，并在 U 的额外假设下得到[公式：见正文]或[公式：见正文]阶的ε复杂性。更确切地说，在不超出普遍常数的情况下，参数的适当选择会导致[公式：见正文]所控制的代价（其中[公式：见正文]和[公式：见正文]分别表示[公式：见正文]的最大和最小特征值的上峰和下峰）。最后，我们通过一些数值说明完成了这些理论结果，包括与贝叶斯学习中其他算法的比较，以及向非强凸设置的开放：作者感谢法国国家癌症研究所[INCA-DGOS-Inserm Grant 12558]支持的 SIRIC ILIAD Nantes-Angers 计划。
{"title":"Multilevel Langevin Pathwise Average for Gibbs Approximation","authors":"Maxime Egéa, Fabien Panloup","doi":"10.1287/moor.2021.0243","DOIUrl":"https://doi.org/10.1287/moor.2021.0243","url":null,"abstract":"We propose and study a new multilevel method for the numerical approximation of a Gibbs distribution π on [Formula: see text], based on (overdamped) Langevin diffusions. This method relies on a multilevel occupation measure, that is, on an appropriate combination of R occupation measures of (constant-step) Euler schemes with respective steps [Formula: see text]. We first state a quantitative result under general assumptions that guarantees an ε-approximation (in an L<jats:sup>2</jats:sup>-sense) with a cost of the order [Formula: see text] or [Formula: see text] under less contractive assumptions. We then apply it to overdamped Langevin diffusions with strongly convex potential [Formula: see text] and obtain an ε-complexity of the order [Formula: see text] or [Formula: see text] under additional assumptions on U. More precisely, up to universal constants, an appropriate choice of the parameters leads to a cost controlled by [Formula: see text] (where [Formula: see text] and [Formula: see text] respectively denote the supremum and the infimum of the largest and lowest eigenvalue of [Formula: see text]). We finally complete these theoretical results with some numerical illustrations, including comparisons to other algorithms in Bayesian learning and opening to the non–strongly convex setting.Funding: The authors are grateful to the SIRIC ILIAD Nantes-Angers program, supported by the French National Cancer Institute [INCA-DGOS-Inserm Grant 12558].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"80 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140297973","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}
One of the challenges for multiagent reinforcement learning (MARL) is designing efficient learning algorithms for a large system in which each agent has only limited or partial information of the entire system. Whereas exciting progress has been made to analyze decentralized MARL with the network of agents for social networks and team video games, little is known theoretically for decentralized MARL with the network of states for modeling self-driving vehicles, ride-sharing, and data and traffic routing. This paper proposes a framework of localized training and decentralized execution to study MARL with the network of states. Localized training means that agents only need to collect local information in their neighboring states during the training phase; decentralized execution implies that agents can execute afterward the learned decentralized policies, which depend only on agents’ current states. The theoretical analysis consists of three key components: the first is the reformulation of the MARL system as a networked Markov decision process with teams of agents, enabling updating the associated team Q-function in a localized fashion; the second is the Bellman equation for the value function and the appropriate Q-function on the probability measure space; and the third is the exponential decay property of the team Q-function, facilitating its approximation with efficient sample efficiency and controllable error. The theoretical analysis paves the way for a new algorithm LTDE-Neural-AC, in which the actor–critic approach with overparameterized neural networks is proposed. The convergence and sample complexity are established and shown to be scalable with respect to the sizes of both agents and states. To the best of our knowledge, this is the first neural network–based MARL algorithm with network structure and provable convergence guarantee.Funding: X. Wei is partially supported by NSFC no. 12201343. R. Xu is partially supported by the NSF CAREER award DMS-2339240.
{"title":"Mean-Field Multiagent Reinforcement Learning: A Decentralized Network Approach","authors":"Haotian Gu, Xin Guo, Xiaoli Wei, Renyuan Xu","doi":"10.1287/moor.2022.0055","DOIUrl":"https://doi.org/10.1287/moor.2022.0055","url":null,"abstract":"One of the challenges for multiagent reinforcement learning (MARL) is designing efficient learning algorithms for a large system in which each agent has only limited or partial information of the entire system. Whereas exciting progress has been made to analyze decentralized MARL with the network of agents for social networks and team video games, little is known theoretically for decentralized MARL with the network of states for modeling self-driving vehicles, ride-sharing, and data and traffic routing. This paper proposes a framework of localized training and decentralized execution to study MARL with the network of states. Localized training means that agents only need to collect local information in their neighboring states during the training phase; decentralized execution implies that agents can execute afterward the learned decentralized policies, which depend only on agents’ current states. The theoretical analysis consists of three key components: the first is the reformulation of the MARL system as a networked Markov decision process with teams of agents, enabling updating the associated team Q-function in a localized fashion; the second is the Bellman equation for the value function and the appropriate Q-function on the probability measure space; and the third is the exponential decay property of the team Q-function, facilitating its approximation with efficient sample efficiency and controllable error. The theoretical analysis paves the way for a new algorithm LTDE-Neural-AC, in which the actor–critic approach with overparameterized neural networks is proposed. The convergence and sample complexity are established and shown to be scalable with respect to the sizes of both agents and states. To the best of our knowledge, this is the first neural network–based MARL algorithm with network structure and provable convergence guarantee.Funding: X. Wei is partially supported by NSFC no. 12201343. R. Xu is partially supported by the NSF CAREER award DMS-2339240.","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"30 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140147661","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 investigate some graph parameters dealing with bi-independent pairs (A, B) in a bipartite graph [Formula: see text], that is, pairs (A, B) where [Formula: see text], and [Formula: see text] are independent. These parameters also allow us to study bicliques in general graphs. When maximizing the cardinality [Formula: see text], one finds the stability number [Formula: see text], well-known to be polynomial-time computable. When maximizing the product [Formula: see text], one finds the parameter g(G), shown to be NP-hard by Peeters in 2003, and when maximizing the ratio [Formula: see text], one finds h(G), introduced by Vallentin in 2020 for bounding product-free sets in finite groups. We show that h(G) is an NP-hard parameter and, as a crucial ingredient, that it is NP-complete to decide whether a bipartite graph G has a balanced maximum independent set. These hardness results motivate introducing semidefinite programming (SDP) bounds for g(G), h(G), and [Formula: see text] (the maximum cardinality of a balanced independent set). We show that these bounds can be seen as natural variations of the Lovász ϑ-number, a well-known semidefinite bound on [Formula: see text]. In addition, we formulate closed-form eigenvalue bounds, and we show relationships among them as well as with earlier spectral parameters by Hoffman and Haemers in 2001 and Vallentin in 2020.Funding: This work was supported by H2020 Marie Skłodowska-Curie Actions [Grant 813211 (POEMA)].
{"title":"Semidefinite Approximations for Bicliques and Bi-Independent Pairs","authors":"Monique Laurent, Sven Polak, Luis Felipe Vargas","doi":"10.1287/moor.2023.0046","DOIUrl":"https://doi.org/10.1287/moor.2023.0046","url":null,"abstract":"We investigate some graph parameters dealing with bi-independent pairs (A, B) in a bipartite graph [Formula: see text], that is, pairs (A, B) where [Formula: see text], and [Formula: see text] are independent. These parameters also allow us to study bicliques in general graphs. When maximizing the cardinality [Formula: see text], one finds the stability number [Formula: see text], well-known to be polynomial-time computable. When maximizing the product [Formula: see text], one finds the parameter g(G), shown to be NP-hard by Peeters in 2003, and when maximizing the ratio [Formula: see text], one finds h(G), introduced by Vallentin in 2020 for bounding product-free sets in finite groups. We show that h(G) is an NP-hard parameter and, as a crucial ingredient, that it is NP-complete to decide whether a bipartite graph G has a balanced maximum independent set. These hardness results motivate introducing semidefinite programming (SDP) bounds for g(G), h(G), and [Formula: see text] (the maximum cardinality of a balanced independent set). We show that these bounds can be seen as natural variations of the Lovász ϑ-number, a well-known semidefinite bound on [Formula: see text]. In addition, we formulate closed-form eigenvalue bounds, and we show relationships among them as well as with earlier spectral parameters by Hoffman and Haemers in 2001 and Vallentin in 2020.Funding: This work was supported by H2020 Marie Skłodowska-Curie Actions [Grant 813211 (POEMA)].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"23 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140147775","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}
Zero-sum stochastic games are parameterized by payoffs, transitions, and possibly a discount rate. In this article, we study how the main solution concepts, the discounted and undiscounted values, vary when these parameters are perturbed. We focus on the marginal values, introduced by Mills in 1956 in the context of matrix games—that is, the directional derivatives of the value along any fixed perturbation. We provide a formula for the marginal values of a discounted stochastic game. Further, under mild assumptions on the perturbation, we provide a formula for their limit as the discount rate vanishes and for the marginal values of an undiscounted stochastic game. We also show, via an example, that the two latter differ in general.Funding: This work was supported by Fondation CFM pour la Recherche; the European Research Council [Grant ERC-CoG-863818 (ForM-SMArt)]; and Agence Nationale de la Recherche [Grant ANR-21-CE40-0020].
零和随机博弈的参数包括报酬、转换和可能的贴现率。在本文中，我们将研究当这些参数受到扰动时，主要的解概念（贴现值和未贴现值）是如何变化的。我们重点研究米尔斯于 1956 年在矩阵博弈中引入的边际值--即沿着任何固定扰动的值的方向导数。我们提供了贴现随机博弈的边际值公式。此外，根据对扰动的温和假设，我们还给出了贴现率消失时的极限值公式，以及未贴现随机博弈的边际值公式。我们还通过一个例子说明，后两者在一般情况下是不同的：本研究得到了 Fondation CFM pour la Recherche、欧洲研究理事会 [Grant ERC-CoG-863818 (ForM-SMArt)] 和 Agence Nationale de la Recherche [Grant ANR-21-CE40-0020] 的支持。
{"title":"Marginal Values of a Stochastic Game","authors":"Luc Attia, Miquel Oliu-Barton, Raimundo Saona","doi":"10.1287/moor.2023.0297","DOIUrl":"https://doi.org/10.1287/moor.2023.0297","url":null,"abstract":"Zero-sum stochastic games are parameterized by payoffs, transitions, and possibly a discount rate. In this article, we study how the main solution concepts, the discounted and undiscounted values, vary when these parameters are perturbed. We focus on the marginal values, introduced by Mills in 1956 in the context of matrix games—that is, the directional derivatives of the value along any fixed perturbation. We provide a formula for the marginal values of a discounted stochastic game. Further, under mild assumptions on the perturbation, we provide a formula for their limit as the discount rate vanishes and for the marginal values of an undiscounted stochastic game. We also show, via an example, that the two latter differ in general.Funding: This work was supported by Fondation CFM pour la Recherche; the European Research Council [Grant ERC-CoG-863818 (ForM-SMArt)]; and Agence Nationale de la Recherche [Grant ANR-21-CE40-0020].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"50 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140147603","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 learning dynamics to model how strategic agents repeatedly play a continuous game while relying on an information platform to learn an unknown payoff-relevant parameter. In each time step, the platform updates a belief estimate of the parameter based on players’ strategies and realized payoffs using Bayes’ rule. Then, players adopt a generic learning rule to adjust their strategies based on the updated belief. We present results on the convergence of beliefs and strategies and the properties of convergent fixed points of the dynamics. We obtain sufficient and necessary conditions for the existence of globally stable fixed points. We also provide sufficient conditions for the local stability of fixed points. These results provide an approach to analyzing the long-term outcomes that arise from the interplay between Bayesian belief learning and strategy learning in games and enable us to characterize conditions under which learning leads to a complete information equilibrium.Funding: Financial support from the Air Force Office of Scientific Research [Project Building Attack Resilience into Complex Networks], the Simons Institute [research fellowship], and a Michael Hammer Fellowship is gratefully acknowledged.
{"title":"Convergence and Stability of Coupled Belief-Strategy Learning Dynamics in Continuous Games","authors":"Manxi Wu, Saurabh Amin, Asuman Ozdaglar","doi":"10.1287/moor.2022.0161","DOIUrl":"https://doi.org/10.1287/moor.2022.0161","url":null,"abstract":"We propose a learning dynamics to model how strategic agents repeatedly play a continuous game while relying on an information platform to learn an unknown payoff-relevant parameter. In each time step, the platform updates a belief estimate of the parameter based on players’ strategies and realized payoffs using Bayes’ rule. Then, players adopt a generic learning rule to adjust their strategies based on the updated belief. We present results on the convergence of beliefs and strategies and the properties of convergent fixed points of the dynamics. We obtain sufficient and necessary conditions for the existence of globally stable fixed points. We also provide sufficient conditions for the local stability of fixed points. These results provide an approach to analyzing the long-term outcomes that arise from the interplay between Bayesian belief learning and strategy learning in games and enable us to characterize conditions under which learning leads to a complete information equilibrium.Funding: Financial support from the Air Force Office of Scientific Research [Project Building Attack Resilience into Complex Networks], the Simons Institute [research fellowship], and a Michael Hammer Fellowship is gratefully acknowledged.","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"72 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140147400","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}
Mehrdad Moharrami, Yashaswini Murthy, Arghyadip Roy, R. Srikant
We study the risk-sensitive exponential cost Markov decision process (MDP) formulation and develop a trajectory-based gradient algorithm to find the stationary point of the cost associated with a set of parameterized policies. We derive a formula that can be used to compute the policy gradient from (state, action, cost) information collected from sample paths of the MDP for each fixed parameterized policy. Unlike the traditional average cost problem, standard stochastic approximation theory cannot be used to exploit this formula. To address the issue, we introduce a truncated and smooth version of the risk-sensitive cost and show that this new cost criterion can be used to approximate the risk-sensitive cost and its gradient uniformly under some mild assumptions. We then develop a trajectory-based gradient algorithm to minimize the smooth truncated estimation of the risk-sensitive cost and derive conditions under which a sequence of truncations can be used to solve the original, untruncated cost problem.Funding: This work was supported by the Office of Naval Research Global [Grant N0001419-1-2566], the Division of Computer and Network Systems [Grant 21-06801], the Army Research Office [Grant W911NF-19-1-0379], and the Division of Computing and Communication Foundations [Grants 17-04970 and 19-34986].
{"title":"A Policy Gradient Algorithm for the Risk-Sensitive Exponential Cost MDP","authors":"Mehrdad Moharrami, Yashaswini Murthy, Arghyadip Roy, R. Srikant","doi":"10.1287/moor.2022.0139","DOIUrl":"https://doi.org/10.1287/moor.2022.0139","url":null,"abstract":"We study the risk-sensitive exponential cost Markov decision process (MDP) formulation and develop a trajectory-based gradient algorithm to find the stationary point of the cost associated with a set of parameterized policies. We derive a formula that can be used to compute the policy gradient from (state, action, cost) information collected from sample paths of the MDP for each fixed parameterized policy. Unlike the traditional average cost problem, standard stochastic approximation theory cannot be used to exploit this formula. To address the issue, we introduce a truncated and smooth version of the risk-sensitive cost and show that this new cost criterion can be used to approximate the risk-sensitive cost and its gradient uniformly under some mild assumptions. We then develop a trajectory-based gradient algorithm to minimize the smooth truncated estimation of the risk-sensitive cost and derive conditions under which a sequence of truncations can be used to solve the original, untruncated cost problem.Funding: This work was supported by the Office of Naval Research Global [Grant N0001419-1-2566], the Division of Computer and Network Systems [Grant 21-06801], the Army Research Office [Grant W911NF-19-1-0379], and the Division of Computing and Communication Foundations [Grants 17-04970 and 19-34986].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"7 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140107747","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}
Antonio Bellon, Didier Henrion, Vyacheslav Kungurtsev, Jakub Mareček
In many applications, solutions of convex optimization problems are updated on-line, as functions of time. In this paper, we consider parametric semidefinite programs, which are linear optimization problems in the semidefinite cone whose coefficients (input data) depend on a time parameter. We are interested in the geometry of the solution (output data) trajectory, defined as the set of solutions depending on the parameter. We propose an exhaustive description of the geometry of the solution trajectory. As our main result, we show that only six distinct behaviors can be observed at a neighborhood of a given point along the solution trajectory. Each possible behavior is then illustrated by an example.Funding: This work was supported by OP RDE [Grant CZ.02.1.01/0.0/0.0/16_019/0000765].
在许多应用中，凸优化问题的解作为时间函数进行在线更新。在本文中，我们考虑的是参数半定量程序，它是半定量锥中的线性优化问题，其系数（输入数据）取决于时间参数。我们感兴趣的是解决方案（输出数据）轨迹的几何形状，它被定义为取决于参数的解决方案集合。我们提出了对解法轨迹几何的详尽描述。我们的主要结果表明，在解轨迹上给定点的邻域只能观察到六种不同的行为。然后，我们将通过一个例子来说明每种可能的行为：本研究得到了 RDE OP [Grant CZ.02.1.01/0.0/0.0/16_019/0000765] 的支持。
{"title":"Parametric Semidefinite Programming: Geometry of the Trajectory of Solutions","authors":"Antonio Bellon, Didier Henrion, Vyacheslav Kungurtsev, Jakub Mareček","doi":"10.1287/moor.2021.0097","DOIUrl":"https://doi.org/10.1287/moor.2021.0097","url":null,"abstract":"In many applications, solutions of convex optimization problems are updated on-line, as functions of time. In this paper, we consider parametric semidefinite programs, which are linear optimization problems in the semidefinite cone whose coefficients (input data) depend on a time parameter. We are interested in the geometry of the solution (output data) trajectory, defined as the set of solutions depending on the parameter. We propose an exhaustive description of the geometry of the solution trajectory. As our main result, we show that only six distinct behaviors can be observed at a neighborhood of a given point along the solution trajectory. Each possible behavior is then illustrated by an example.Funding: This work was supported by OP RDE [Grant CZ.02.1.01/0.0/0.0/16_019/0000765].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"86 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140070190","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 random variable is difference-form decomposable (DFD) if it may be written as the difference of two i.i.d. random terms. We show that densities of such variables exhibit a remarkable degree of structure. Specifically, a DFD density can be neither approximately uniform, nor quasiconvex, nor strictly concave. On the other hand, a DFD density need, in general, be neither unimodal nor logconcave. Regarding smoothness, we show that a compactly supported DFD density cannot be analytic and will often exhibit a kink even if its components are smooth. The analysis highlights the risks for model consistency resulting from the strategy widely adopted in the economics literature of imposing assumptions directly on a difference of noise terms rather than on its components.
{"title":"On the (Im-)Possibility of Representing Probability Distributions as a Difference of I.I.D. Noise Terms","authors":"Christian Ewerhart, Marco Serena","doi":"10.1287/moor.2023.0081","DOIUrl":"https://doi.org/10.1287/moor.2023.0081","url":null,"abstract":"A random variable is difference-form decomposable (DFD) if it may be written as the difference of two i.i.d. random terms. We show that densities of such variables exhibit a remarkable degree of structure. Specifically, a DFD density can be neither approximately uniform, nor quasiconvex, nor strictly concave. On the other hand, a DFD density need, in general, be neither unimodal nor logconcave. Regarding smoothness, we show that a compactly supported DFD density cannot be analytic and will often exhibit a kink even if its components are smooth. The analysis highlights the risks for model consistency resulting from the strategy widely adopted in the economics literature of imposing assumptions directly on a difference of noise terms rather than on its components.","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"276 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140070277","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 new dynamic continuous-time model of optimal consumption and investment to include independent stochastic labor income. We reduce the problem of solving the Bellman equation to a problem of solving an integral equation. We then explicitly characterize the optimal consumption and investment strategy as a function of income-to-wealth ratio. We provide some analytical comparative statics associated with the value function and optimal strategies. We also develop a quite general numerical algorithm for control iteration and solve the Bellman equation as a sequence of solutions to ordinary differential equations. This numerical algorithm can be readily applied to many other optimal consumption and investment problems especially with extra nondiversifiable Brownian risks, resulting in nonlinear Bellman equations. Finally, our numerical analysis illustrates how the presence of stochastic labor income affects the optimal consumption and investment strategy.Funding: A. Bensoussan was supported by the National Science Foundation under grant [DMS-2204795]. S. Park was supported by the Ministry of Education of the Republic of Korea and the National Research Foundation of Korea, South Korea [NRF-2022S1A3A2A02089950].
我们建立了一个新的动态连续时间最优消费和投资模型，其中包括独立的随机劳动收入。我们将贝尔曼方程的求解问题简化为积分方程的求解问题。然后，我们将最优消费和投资策略明确表征为收入与财富比率的函数。我们提供了一些与价值函数和最优策略相关的分析比较静态。我们还开发了一种相当通用的控制迭代数值算法，并将贝尔曼方程作为常微分方程的解序列来求解。这种数值算法可以很容易地应用于许多其他最优消费和投资问题，尤其是涉及额外的不可分散布朗风险，从而导致非线性贝尔曼方程的问题。最后，我们的数值分析说明了随机劳动收入的存在如何影响最优消费和投资策略：A. Bensoussan 受美国国家科学基金会资助[DMS-2204795]。S. Park 由大韩民国教育部和韩国国家研究基金会 [NRF-2022S1A3A2A02089950] 资助。
{"title":"Optimal Consumption and Investment with Independent Stochastic Labor Income","authors":"Alain Bensoussan, Seyoung Park","doi":"10.1287/moor.2023.0119","DOIUrl":"https://doi.org/10.1287/moor.2023.0119","url":null,"abstract":"We develop a new dynamic continuous-time model of optimal consumption and investment to include independent stochastic labor income. We reduce the problem of solving the Bellman equation to a problem of solving an integral equation. We then explicitly characterize the optimal consumption and investment strategy as a function of income-to-wealth ratio. We provide some analytical comparative statics associated with the value function and optimal strategies. We also develop a quite general numerical algorithm for control iteration and solve the Bellman equation as a sequence of solutions to ordinary differential equations. This numerical algorithm can be readily applied to many other optimal consumption and investment problems especially with extra nondiversifiable Brownian risks, resulting in nonlinear Bellman equations. Finally, our numerical analysis illustrates how the presence of stochastic labor income affects the optimal consumption and investment strategy.Funding: A. Bensoussan was supported by the National Science Foundation under grant [DMS-2204795]. S. Park was supported by the Ministry of Education of the Republic of Korea and the National Research Foundation of Korea, South Korea [NRF-2022S1A3A2A02089950].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"278 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140055887","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：