The timing of strategic exit is one of the most important but difficult business decisions, especially under competition and uncertainty. Motivated by this problem, we examine a stochastic game of exit in which players are uncertain about their competitor’s exit value. We construct an equilibrium for a large class of payoff flows driven by a general one-dimensional diffusion. In the equilibrium, the players employ sophisticated exit strategies involving both the state variable and the posterior belief process. These strategies are specified explicitly in terms of the problem data and a solution to an auxiliary optimal stopping problem. The equilibrium that we obtain is further shown to be unique within a wide subclass of symmetric Bayesian equilibria.
{"title":"Exit Game with Private Information","authors":"H. Dharma Kwon, Jan Palczewski","doi":"10.1287/moor.2022.0285","DOIUrl":"https://doi.org/10.1287/moor.2022.0285","url":null,"abstract":"The timing of strategic exit is one of the most important but difficult business decisions, especially under competition and uncertainty. Motivated by this problem, we examine a stochastic game of exit in which players are uncertain about their competitor’s exit value. We construct an equilibrium for a large class of payoff flows driven by a general one-dimensional diffusion. In the equilibrium, the players employ sophisticated exit strategies involving both the state variable and the posterior belief process. These strategies are specified explicitly in terms of the problem data and a solution to an auxiliary optimal stopping problem. The equilibrium that we obtain is further shown to be unique within a wide subclass of symmetric Bayesian equilibria.","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"11 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220717","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 studies duality and optimality conditions for general convex stochastic optimization problems. The main result gives sufficient conditions for the absence of a duality gap and the existence of dual solutions in a locally convex space of random variables. It implies, in particular, the necessity of scenario-wise optimality conditions that are behind many fundamental results in operations research, stochastic optimal control, and financial mathematics. Our analysis builds on the theory of Fréchet spaces of random variables whose topological dual can be identified with the direct sum of another space of random variables and a space of singular functionals. The results are illustrated by deriving sufficient and necessary optimality conditions for several more specific problem classes. We obtain significant extensions to earlier models, for example, on stochastic optimal control, portfolio optimization, and mathematical programming.
{"title":"Dual Solutions in Convex Stochastic Optimization","authors":"Teemu Pennanen, Ari-Pekka Perkkiö","doi":"10.1287/moor.2022.0270","DOIUrl":"https://doi.org/10.1287/moor.2022.0270","url":null,"abstract":"This paper studies duality and optimality conditions for general convex stochastic optimization problems. The main result gives sufficient conditions for the absence of a duality gap and the existence of dual solutions in a locally convex space of random variables. It implies, in particular, the necessity of scenario-wise optimality conditions that are behind many fundamental results in operations research, stochastic optimal control, and financial mathematics. Our analysis builds on the theory of Fréchet spaces of random variables whose topological dual can be identified with the direct sum of another space of random variables and a space of singular functionals. The results are illustrated by deriving sufficient and necessary optimality conditions for several more specific problem classes. We obtain significant extensions to earlier models, for example, on stochastic optimal control, portfolio optimization, and mathematical programming.","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"6 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220711","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}
David Newton, Raghu Bollapragada, Raghu Pasupathy, Nung Kwan Yip
Stochastic Gradient (SG) is the de facto iterative technique to solve stochastic optimization (SO) problems with a smooth (nonconvex) objective f and a stochastic first-order oracle. SG’s attractiveness is due in part to its simplicity of executing a single step along the negative subsampled gradient direction to update the incumbent iterate. In this paper, we question SG’s choice of executing a single step as opposed to multiple steps between subsample updates. Our investigation leads naturally to generalizing SG into Retrospective Approximation (RA), where, during each iteration, a “deterministic solver” executes possibly multiple steps on a subsampled deterministic problem and stops when further solving is deemed unnecessary from the standpoint of statistical efficiency. RA thus formalizes what is appealing for implementation—during each iteration, “plug in” a solver—for example, L-BFGS line search or Newton-CG—as is, and solve only to the extent necessary. We develop a complete theory using relative error of the observed gradients as the principal object, demonstrating that almost sure and L1 consistency of RA are preserved under especially weak conditions when sample sizes are increased at appropriate rates. We also characterize the iteration and oracle complexity (for linear and sublinear solvers) of RA and identify a practical termination criterion leading to optimal complexity rates. To subsume nonconvex f, we present a certain “random central limit theorem” that incorporates the effect of curvature across all first-order critical points, demonstrating that the asymptotic behavior is described by a certain mixture of normals. The message from our numerical experiments is that the ability of RA to incorporate existing second-order deterministic solvers in a strategic manner might be important from the standpoint of dispensing with hyper-parameter tuning.Funding: R. Pasupathy received financial support from the Office of Naval Research [Grants N000141712295 and 13000991]. R. Bollapragada received financial support from the Lawrence Livermore National Laboratory and the National Science Foundation [Grant NSF DMS 2324643].
{"title":"A Retrospective Approximation Approach for Smooth Stochastic Optimization","authors":"David Newton, Raghu Bollapragada, Raghu Pasupathy, Nung Kwan Yip","doi":"10.1287/moor.2022.0136","DOIUrl":"https://doi.org/10.1287/moor.2022.0136","url":null,"abstract":"Stochastic Gradient (SG) is the de facto iterative technique to solve stochastic optimization (SO) problems with a smooth (nonconvex) objective f and a stochastic first-order oracle. SG’s attractiveness is due in part to its simplicity of executing a single step along the negative subsampled gradient direction to update the incumbent iterate. In this paper, we question SG’s choice of executing a single step as opposed to multiple steps between subsample updates. Our investigation leads naturally to generalizing SG into Retrospective Approximation (RA), where, during each iteration, a “deterministic solver” executes possibly multiple steps on a subsampled deterministic problem and stops when further solving is deemed unnecessary from the standpoint of statistical efficiency. RA thus formalizes what is appealing for implementation—during each iteration, “plug in” a solver—for example, L-BFGS line search or Newton-CG—as is, and solve only to the extent necessary. We develop a complete theory using relative error of the observed gradients as the principal object, demonstrating that almost sure and L<jats:sub>1</jats:sub> consistency of RA are preserved under especially weak conditions when sample sizes are increased at appropriate rates. We also characterize the iteration and oracle complexity (for linear and sublinear solvers) of RA and identify a practical termination criterion leading to optimal complexity rates. To subsume nonconvex f, we present a certain “random central limit theorem” that incorporates the effect of curvature across all first-order critical points, demonstrating that the asymptotic behavior is described by a certain mixture of normals. The message from our numerical experiments is that the ability of RA to incorporate existing second-order deterministic solvers in a strategic manner might be important from the standpoint of dispensing with hyper-parameter tuning.Funding: R. Pasupathy received financial support from the Office of Naval Research [Grants N000141712295 and 13000991]. R. Bollapragada received financial support from the Lawrence Livermore National Laboratory and the National Science Foundation [Grant NSF DMS 2324643].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"182 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220712","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 explore a version of the minimax theorem for two-person win-lose games with infinitely many pure strategies. In the countable case, we give a combinatorial condition on the game which implies the minimax property. In the general case, we prove that a game satisfies the minimax property along with all its subgames if and only if none of its subgames is isomorphic to the “larger number game.” This generalizes a recent theorem of Hanneke, Livni, and Moran. We also propose several applications of our results outside of game theory.
{"title":"The Minimax Property in Infinite Two-Person Win-Lose Games","authors":"Ron Holzman","doi":"10.1287/moor.2023.0352","DOIUrl":"https://doi.org/10.1287/moor.2023.0352","url":null,"abstract":"We explore a version of the minimax theorem for two-person win-lose games with infinitely many pure strategies. In the countable case, we give a combinatorial condition on the game which implies the minimax property. In the general case, we prove that a game satisfies the minimax property along with all its subgames if and only if none of its subgames is isomorphic to the “larger number game.” This generalizes a recent theorem of Hanneke, Livni, and Moran. We also propose several applications of our results outside of game theory.","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"38 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220714","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}
Dividing a multilayered cake under nonoverlapping constraints captures several scenarios (e.g., allocating multiple facilities over time where each agent can utilize at most one facility simultaneously). We establish the existence of an envy-free multidivision that is nonoverlapping and contiguous within each layer when the number of agents is a prime power, solving partially an open question by Hosseini et al. [Hosseini H, Igarashi A, Searns A (2020) Fair division of time: Multi-layered cake cutting. Proc. 29th Internat. Joint Conf. Artificial Intelligence (IJCAI), 182–188; Hosseini H, Igarashi A, Searns A (2020) Fair division of time: Multi-layered cake cutting. Preprint, submitted April 28, http://arxiv.org/abs/2004.13397 ]. Our approach follows an idea proposed by Jojić et al. [Jojić D, Panina G, Živaljević R (2021) Splitting necklaces, with constraints. SIAM J. Discrete Math. 35(2):1268–1286] for envy-free divisions, relying on a general fixed-point theorem. We further design a fully polynomial-time approximation scheme for the two-layer, three-agent case, with monotone preferences. All results are actually established for divisions among groups of almost the same size. In the one-layer, three-group case, our algorithm is able to deal with any predetermined sizes, still with monotone preferences. For three groups, this provides an algorithmic version of a recent theorem by Segal-Halevi and Suksompong [Segal-Halevi E, Suksompong W (2021) How to cut a cake fairly: A generalization to groups. Amer. Math. Monthly 128(1):79–83].Funding: This work was partially supported by the Japan Science and Technology Agency [Grant JPMJPR20C], Fusion Oriented REsearch for disruptive Science and Technology [Grant JPMJFR226O], and Exploratory Research for Advanced Technology [Grant JPMJER2301].
在非重叠约束条件下分割多层蛋糕可以捕捉到多种情况（例如，在一段时间内分配多个设施，每个代理最多可以同时使用一个设施）。我们证明了当代理人的数量是质数时，存在一种无嫉妒的多层分割，这种分割在每一层内都是不重叠且连续的，从而部分解决了 Hosseini 等人提出的一个开放性问题[Hosseini H, Igarashi A, Searns A (2020) Fair division of time：多层切蛋糕。Proc.Joint Conf.人工智能（IJCAI），182-188；Hosseini H、Igarashi A、Searns A (2020)：公平分配时间：多层切蛋糕。预印本，4 月 28 日提交，http://arxiv.org/abs/2004.13397 ]。我们的方法沿袭了约吉奇等人提出的想法 [Jojić D, Panina G, Živaljević R (2021) Splitting necklaces, with constraints.SIAM J. Discrete Math.35(2):1268-1286]的无羡分割，依赖于一般定点定理。我们进一步设计了一个完全多项式时间的近似方案，用于具有单调偏好的两层三代理情况。实际上，所有结果都是针对大小几乎相同的组之间的除法建立的。在单层三组的情况下，我们的算法能够处理任何预定的大小，仍然是单调偏好。对于三组，这提供了 Segal-Halevi 和 Suksompong [Segal-Halevi E, Suksompong W (2021) How to cut a cake fairly：A generalization to groups.Amer.Math.Monthly 128(1):79-83].Funding：本研究得到日本科学技术振兴机构[JPMJPR20C]、"面向颠覆性科学技术的融合研究"[JPMJFR226O]和 "先进技术探索研究"[JPMJER2301]的部分资助。
{"title":"Envy-Free Division of Multilayered Cakes","authors":"Ayumi Igarashi, Frédéric Meunier","doi":"10.1287/moor.2022.0350","DOIUrl":"https://doi.org/10.1287/moor.2022.0350","url":null,"abstract":"Dividing a multilayered cake under nonoverlapping constraints captures several scenarios (e.g., allocating multiple facilities over time where each agent can utilize at most one facility simultaneously). We establish the existence of an envy-free multidivision that is nonoverlapping and contiguous within each layer when the number of agents is a prime power, solving partially an open question by Hosseini et al. [Hosseini H, Igarashi A, Searns A (2020) Fair division of time: Multi-layered cake cutting. Proc. 29th Internat. Joint Conf. Artificial Intelligence (IJCAI), 182–188; Hosseini H, Igarashi A, Searns A (2020) Fair division of time: Multi-layered cake cutting. Preprint, submitted April 28, http://arxiv.org/abs/2004.13397 ]. Our approach follows an idea proposed by Jojić et al. [Jojić D, Panina G, Živaljević R (2021) Splitting necklaces, with constraints. SIAM J. Discrete Math. 35(2):1268–1286] for envy-free divisions, relying on a general fixed-point theorem. We further design a fully polynomial-time approximation scheme for the two-layer, three-agent case, with monotone preferences. All results are actually established for divisions among groups of almost the same size. In the one-layer, three-group case, our algorithm is able to deal with any predetermined sizes, still with monotone preferences. For three groups, this provides an algorithmic version of a recent theorem by Segal-Halevi and Suksompong [Segal-Halevi E, Suksompong W (2021) How to cut a cake fairly: A generalization to groups. Amer. Math. Monthly 128(1):79–83].Funding: This work was partially supported by the Japan Science and Technology Agency [Grant JPMJPR20C], Fusion Oriented REsearch for disruptive Science and Technology [Grant JPMJFR226O], and Exploratory Research for Advanced Technology [Grant JPMJER2301].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"51 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220719","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}
Sebastian Perez-Salazar, Mohit Singh, Alejandro Toriello
Online advertising has motivated interest in online selection problems. Displaying ads to the right users benefits both the platform (e.g., via pay-per-click) and the advertisers (by increasing their reach). In practice, not all users click on displayed ads, while the platform’s algorithm may miss the users most disposed to do so. This mismatch decreases the platform’s revenue and the advertiser’s chances to reach the right customers. With this motivation, we propose a secretary problem where a candidate may or may not accept an offer according to a known probability p. Because we do not know the top candidate willing to accept an offer, the goal is to maximize a robust objective defined as the minimum over integers k of the probability of choosing one of the top k candidates, given that one of these candidates will accept an offer. Using Markov decision process theory, we derive a linear program for this max-min objective whose solution encodes an optimal policy. The derivation may be of independent interest, as it is generalizable and can be used to obtain linear programs for many online selection models. We further relax this linear program into an infinite counterpart, which we use to provide bounds for the objective and closed-form policies. For [Formula: see text], an optimal policy is a simple threshold rule that observes the first [Formula: see text] fraction of candidates and subsequently makes offers to the best candidate observed so far.Funding: Financial support from the U.S. National Science Foundation [Grants CCF-2106444, CCF-1910423, and CMMI 1552479] is gratefully acknowledged.
在线广告激发了人们对在线选择问题的兴趣。向合适的用户展示广告既有利于平台（例如通过点击付费），也有利于广告商（通过扩大覆盖面）。实际上，并非所有用户都会点击显示的广告，而平台的算法可能会错过最有意愿点击广告的用户。这种不匹配降低了平台的收入和广告商接触到合适客户的机会。由于我们不知道最愿意接受报价的候选人是谁，因此我们的目标是最大化一个稳健目标，该目标被定义为在前 k 名候选人中有一人接受报价的情况下，选择其中一名候选人的概率在整数 k 上的最小值。利用马尔可夫决策过程理论，我们为这个最大最小目标推导出了一个线性程序，其解法编码了一个最优策略。这一推导可能具有独立的意义，因为它具有通用性，可用于获得许多在线选择模型的线性程序。我们进一步将这个线性程序放宽为无限对应程序，并用它来提供目标和闭式策略的边界。对于[公式：见正文]，最优策略是一个简单的阈值规则，即观察第一批[公式：见正文]候选人，然后向迄今为止观察到的最佳候选人发出邀请：感谢美国国家科学基金会[CCF-2106444、CCF-1910423 和 CMMI 1552479]的资助。
{"title":"Robust Online Selection with Uncertain Offer Acceptance","authors":"Sebastian Perez-Salazar, Mohit Singh, Alejandro Toriello","doi":"10.1287/moor.2023.0210","DOIUrl":"https://doi.org/10.1287/moor.2023.0210","url":null,"abstract":"Online advertising has motivated interest in online selection problems. Displaying ads to the right users benefits both the platform (e.g., via pay-per-click) and the advertisers (by increasing their reach). In practice, not all users click on displayed ads, while the platform’s algorithm may miss the users most disposed to do so. This mismatch decreases the platform’s revenue and the advertiser’s chances to reach the right customers. With this motivation, we propose a secretary problem where a candidate may or may not accept an offer according to a known probability p. Because we do not know the top candidate willing to accept an offer, the goal is to maximize a robust objective defined as the minimum over integers k of the probability of choosing one of the top k candidates, given that one of these candidates will accept an offer. Using Markov decision process theory, we derive a linear program for this max-min objective whose solution encodes an optimal policy. The derivation may be of independent interest, as it is generalizable and can be used to obtain linear programs for many online selection models. We further relax this linear program into an infinite counterpart, which we use to provide bounds for the objective and closed-form policies. For [Formula: see text], an optimal policy is a simple threshold rule that observes the first [Formula: see text] fraction of candidates and subsequently makes offers to the best candidate observed so far.Funding: Financial support from the U.S. National Science Foundation [Grants CCF-2106444, CCF-1910423, and CMMI 1552479] is gratefully acknowledged.","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"8 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220713","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 housing markets as introduced by Shapley and Scarf. We investigate the computational complexity of various questions regarding the situation of an agent a in a housing market H: we show that it is [Formula: see text]-hard to find an allocation in the core of H in which (i) a receives a certain house, (ii) a does not receive a certain house, or (iii) a receives a house other than a’s own. We prove that the core of housing markets respects improvement in the following sense: given an allocation in the core of H in which agent a receives a house h, if the value of the house owned by a increases, then the resulting housing market admits an allocation in its core in which a receives either h or a house that a prefers to h; moreover, such an allocation can be found efficiently. We further show an analogous result in the Stable Roommates setting by proving that stable matchings in a one-sided market also respect improvement.Funding: This work was supported by the Hungarian Scientific Research Fund [Grants K124171, K128611]. I. Schlotter is supported by the Hungarian Academy of Sciences under its Momentum Programme (LP2021-2) and its János Bolyai Research Scholarship. The research reported in this paper and carried out by T. Fleiner at the Budapest University of Technology and Economics was supported by the “TKP2020, National Challenges Program” of the National Research Development and Innovation Office [BME NC TKP2020 and OTKA K143858] and by the Higher Education Excellence Program of the Ministry of Human Capacities in the frame of the Artificial Intelligence research area of the Budapest University of Technology and Economics (BME FIKP-MI/SC). P. Biró gratefully acknowledges financial support from the Hungarian Scientific Research Fund, OTKA [Grant K143858] and the Hungarian Academy of Sciences [Momentum Grant LP2021-2].
我们研究沙普利和斯卡夫提出的住房市场。我们研究了与代理人 a 在住房市场 H 中的处境有关的各种问题的计算复杂性：我们证明，要在 H 的核心中找到一种分配是[公式：见正文]困难的，在这种分配中：(i) a 得到某所房子，(ii) a 没有得到某所房子，或者 (iii) a 得到的房子不是 a 自己的房子。我们证明，住房市场的核心尊重以下意义上的改进：给定 H 核心中的一个分配，其中代理人 a 得到一个房子 h，如果 a 拥有的房子的价值增加，那么由此产生的住房市场在其核心中允许一个分配，其中 a 得到 h 或 a 比 h 更喜欢的房子；此外，这样的分配可以有效地找到。我们进一步证明了单边市场中的稳定匹配也尊重改进，从而在稳定室友设置中展示了类似的结果：本研究得到了匈牙利科学研究基金[Grants K124171, K128611]的支持。I. Schlotter 由匈牙利科学院 Momentum 计划 (LP2021-2) 和 János Bolyai 研究奖学金资助。本文所报道的 T. Fleiner 在布达佩斯技术经济大学进行的研究得到了国家研究发展和创新办公室 "TKP2020，国家挑战计划"[BME NC TKP2020 和 OTKA K143858]以及布达佩斯技术经济大学人工智能研究领域框架内的人力部高等教育卓越计划（BME FIKP-MI/SC）的支持。P. Biró 感谢匈牙利科学研究基金 OTKA [Grant K143858] 和匈牙利科学院 [Momentum Grant LP2021-2] 的资助。
{"title":"The Core of Housing Markets from an Agent’s Perspective: Is It Worth Sprucing up Your Home?","authors":"Ildikó Schlotter, Péter Biró, Tamás Fleiner","doi":"10.1287/moor.2023.0092","DOIUrl":"https://doi.org/10.1287/moor.2023.0092","url":null,"abstract":"We study housing markets as introduced by Shapley and Scarf. We investigate the computational complexity of various questions regarding the situation of an agent a in a housing market H: we show that it is [Formula: see text]-hard to find an allocation in the core of H in which (i) a receives a certain house, (ii) a does not receive a certain house, or (iii) a receives a house other than a’s own. We prove that the core of housing markets respects improvement in the following sense: given an allocation in the core of H in which agent a receives a house h, if the value of the house owned by a increases, then the resulting housing market admits an allocation in its core in which a receives either h or a house that a prefers to h; moreover, such an allocation can be found efficiently. We further show an analogous result in the Stable Roommates setting by proving that stable matchings in a one-sided market also respect improvement.Funding: This work was supported by the Hungarian Scientific Research Fund [Grants K124171, K128611]. I. Schlotter is supported by the Hungarian Academy of Sciences under its Momentum Programme (LP2021-2) and its János Bolyai Research Scholarship. The research reported in this paper and carried out by T. Fleiner at the Budapest University of Technology and Economics was supported by the “TKP2020, National Challenges Program” of the National Research Development and Innovation Office [BME NC TKP2020 and OTKA K143858] and by the Higher Education Excellence Program of the Ministry of Human Capacities in the frame of the Artificial Intelligence research area of the Budapest University of Technology and Economics (BME FIKP-MI/SC). P. Biró gratefully acknowledges financial support from the Hungarian Scientific Research Fund, OTKA [Grant K143858] and the Hungarian Academy of Sciences [Momentum Grant LP2021-2].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"5 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220784","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 minimization of nonlower semicontinuous functions is a difficult topic that has been minimally studied. Among such functions is a Heaviside composite function that is the composition of a Heaviside function with a possibly nonsmooth multivariate function. Unifying a statistical estimation problem with hierarchical selection of variables and a sample average approximation of composite chance constrained stochastic programs, a Heaviside composite optimization problem is one whose objective and constraints are defined by sums of possibly nonlinear multiples of such composite functions. Via a pulled-out formulation, a pseudostationarity concept for a feasible point was introduced in an earlier work as a necessary condition for a local minimizer of a Heaviside composite optimization problem. The present paper extends this previous study in several directions: (a) showing that pseudostationarity is implied by (and thus, weaker than) a sharper subdifferential-based stationarity condition that we term epistationarity; (b) introducing a set-theoretic sufficient condition, which we term a local convexity-like property, under which an epistationary point of a possibly nonlower semicontinuous optimization problem is a local minimizer; (c) providing several classes of Heaviside composite functions satisfying this local convexity-like property; (d) extending the epigraphical formulation of a nonnegative multiple of a Heaviside composite function to a lifted formulation for arbitrarily signed multiples of the Heaviside composite function, based on which we show that an epistationary solution of the given Heaviside composite program with broad classes of B-differentiable component functions can in principle be approximately computed by surrogation methods.Funding: The work of Y. Cui was based on research supported by the National Science Foundation [Grants CCF-2153352, DMS-2309729, and CCF-2416172] and the National Institutes of Health [Grant 1R01CA287413-01]. The work of J.-S. Pang was based on research supported by the Air Force Office of Scientific Research [Grant FA9550-22-1-0045].
{"title":"Analysis of a Class of Minimization Problems Lacking Lower Semicontinuity","authors":"Shaoning Han, Ying Cui, Jong-Shi Pang","doi":"10.1287/moor.2023.0295","DOIUrl":"https://doi.org/10.1287/moor.2023.0295","url":null,"abstract":"The minimization of nonlower semicontinuous functions is a difficult topic that has been minimally studied. Among such functions is a Heaviside composite function that is the composition of a Heaviside function with a possibly nonsmooth multivariate function. Unifying a statistical estimation problem with hierarchical selection of variables and a sample average approximation of composite chance constrained stochastic programs, a Heaviside composite optimization problem is one whose objective and constraints are defined by sums of possibly nonlinear multiples of such composite functions. Via a pulled-out formulation, a pseudostationarity concept for a feasible point was introduced in an earlier work as a necessary condition for a local minimizer of a Heaviside composite optimization problem. The present paper extends this previous study in several directions: (a) showing that pseudostationarity is implied by (and thus, weaker than) a sharper subdifferential-based stationarity condition that we term epistationarity; (b) introducing a set-theoretic sufficient condition, which we term a local convexity-like property, under which an epistationary point of a possibly nonlower semicontinuous optimization problem is a local minimizer; (c) providing several classes of Heaviside composite functions satisfying this local convexity-like property; (d) extending the epigraphical formulation of a nonnegative multiple of a Heaviside composite function to a lifted formulation for arbitrarily signed multiples of the Heaviside composite function, based on which we show that an epistationary solution of the given Heaviside composite program with broad classes of B-differentiable component functions can in principle be approximately computed by surrogation methods.Funding: The work of Y. Cui was based on research supported by the National Science Foundation [Grants CCF-2153352, DMS-2309729, and CCF-2416172] and the National Institutes of Health [Grant 1R01CA287413-01]. The work of J.-S. Pang was based on research supported by the Air Force Office of Scientific Research [Grant FA9550-22-1-0045].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"9 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220715","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 multipopulation Bayesian games with a large number of players. Each player aims at minimizing a cost function that depends on this player’s own action, the distribution of players’ actions in all populations, and an unknown state parameter. We study the nonatomic limit versions of these games and introduce the concept of Bayes correlated Wardrop equilibrium, which extends the concept of Bayes correlated equilibrium to nonatomic games. We prove that Bayes correlated Wardrop equilibria are limits of action flows induced by Bayes correlated equilibria of the game with a large finite set of small players. For nonatomic games with complete information admitting a convex potential, we prove that the set of correlated and of coarse correlated Wardrop equilibria coincide with the set of probability distributions over Wardrop equilibria and that all equilibrium outcomes have the same costs. We get the following consequences. First, all flow distributions of (coarse) correlated equilibria in convex potential games with finitely many players converge to mixtures of Wardrop equilibria when the weight of each player tends to zero. Second, for any sequence of flows satisfying a no-regret property, its empirical distribution converges to the set of distributions over Wardrop equilibria, and the average cost converges to the unique Wardrop cost.Funding: This work was partially supported by European Cooperation in Science and Technology Action 16228 GAMENET. F. Koessler acknowledges the support of the Agence Nationale de la Recherche [Grant StratCom ANR-19-CE26-0010-01]. M. Scarsini acknowledges the support of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni project [Grant CUP_E53C22001930001], the Ministero dell’Università e della Ricerca Progetti di Rilevante Interesse Nazionale [Grant 2022EKNE5K], and the European Union-Next Generation EU, component M4C2, investment 1.1 (Ministero dell’Università e della Ricerca Progetti di Rilevante Interesse Nazionale Piano Nazionale di Ripresa e Resilienza) [Grant P2022XT8C8]. T. Tomala gratefully acknowledges the support of the HEC foundation and Agence Nationale de la Recherche/Investissements d’Avenir [Grant ANR-11-IDEX-0003/Labex Ecodec/ANR-11-LABX-0047].
我们考虑的是有大量参与者的多种群贝叶斯博弈。每个博弈者的目标都是最小化成本函数，该函数取决于博弈者自身的行动、博弈者在所有种群中的行动分布以及一个未知的状态参数。我们研究了这些博弈的非原子极限版本，并引入了贝叶斯相关沃德洛普均衡的概念，它将贝叶斯相关均衡的概念扩展到了非原子博弈。我们证明，贝叶斯相关沃德洛普均衡是有大量有限小博弈者的博弈的贝叶斯相关均衡所诱导的行动流的极限。对于具有完整信息且允许凸势能的非原子博弈，我们证明相关和粗相关的沃德洛普均衡集与沃德洛普均衡的概率分布集重合，并且所有均衡结果的代价相同。我们会得到以下结果。首先，当每个博弈者的权重趋于零时，有限多博弈者的凸势能博弈中（粗）相关均衡的所有流量分布都会趋近于沃德罗普均衡的混合物。其次，对于任何满足无悔属性的流量序列，其经验分布都会收敛到沃德罗普均衡状态的分布集合，平均成本也会收敛到唯一的沃德罗普成本：这项工作得到了欧洲科技合作行动 16228 GAMENET 的部分支持。F. Koessler 感谢 Agence Nationale de la Recherche [Grant StratCom ANR-19-CE26-0010-01] 的支持。M. Scarsini 感谢 Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni 项目[CUP_E53C22001930001]、Ministero dell'Università e della Ricerca Progetti di Rilevante Interesse Nazionale [2022EKNE5K]和欧盟-下一代欧盟 M4C2 部分投资 1 的支持。1 (Ministero dell'Università e della Ricerca Progetti di Rilevante Interesse Nazionale Piano Nazionale di Ripresa e Resilienza) [Grant P2022XT8C8]。T. Tomala 感谢 HEC 基金会和 Agence Nationale de la Recherche/Investissements d'Avenir [Grant ANR-11-IDEX-0003/Labex Ecodec/ANR-11-LABX-0047] 的支持。
{"title":"Correlated Equilibria in Large Anonymous Bayesian Games","authors":"Frédéric Koessler, Marco Scarsini, Tristan Tomala","doi":"10.1287/moor.2023.0278","DOIUrl":"https://doi.org/10.1287/moor.2023.0278","url":null,"abstract":"We consider multipopulation Bayesian games with a large number of players. Each player aims at minimizing a cost function that depends on this player’s own action, the distribution of players’ actions in all populations, and an unknown state parameter. We study the nonatomic limit versions of these games and introduce the concept of Bayes correlated Wardrop equilibrium, which extends the concept of Bayes correlated equilibrium to nonatomic games. We prove that Bayes correlated Wardrop equilibria are limits of action flows induced by Bayes correlated equilibria of the game with a large finite set of small players. For nonatomic games with complete information admitting a convex potential, we prove that the set of correlated and of coarse correlated Wardrop equilibria coincide with the set of probability distributions over Wardrop equilibria and that all equilibrium outcomes have the same costs. We get the following consequences. First, all flow distributions of (coarse) correlated equilibria in convex potential games with finitely many players converge to mixtures of Wardrop equilibria when the weight of each player tends to zero. Second, for any sequence of flows satisfying a no-regret property, its empirical distribution converges to the set of distributions over Wardrop equilibria, and the average cost converges to the unique Wardrop cost.Funding: This work was partially supported by European Cooperation in Science and Technology Action 16228 GAMENET. F. Koessler acknowledges the support of the Agence Nationale de la Recherche [Grant StratCom ANR-19-CE26-0010-01]. M. Scarsini acknowledges the support of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni project [Grant CUP_E53C22001930001], the Ministero dell’Università e della Ricerca Progetti di Rilevante Interesse Nazionale [Grant 2022EKNE5K], and the European Union-Next Generation EU, component M4C2, investment 1.1 (Ministero dell’Università e della Ricerca Progetti di Rilevante Interesse Nazionale Piano Nazionale di Ripresa e Resilienza) [Grant P2022XT8C8]. T. Tomala gratefully acknowledges the support of the HEC foundation and Agence Nationale de la Recherche/Investissements d’Avenir [Grant ANR-11-IDEX-0003/Labex Ecodec/ANR-11-LABX-0047].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"21 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142227389","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}
Friedrich Eisenbrand, Christoph Hunkenschröder, Kim-Manuel Klein, Martin Koutecký, Asaf Levin, Shmuel Onn
We study the general integer programming problem where the number of variables n is a variable part of the input. We consider two natural parameters of the constraint matrix A: its numeric measure a and its sparsity measure d. We present an algorithm for solving integer programming in time [Formula: see text], where g is some computable function of the parameters a and d, and L is the binary encoding length of the input. In particular, integer programming is fixed-parameter tractable parameterized by a and d, and is solvable in polynomial time for every fixed a and d. Our results also extend to nonlinear separable convex objective functions.Funding: F. Eisenbrand, C. Hunkenschröder, and K.-M. Klein were supported by the Swiss National Science Foundation (SNSF) within the project “Convexity, geometry of numbers, and the complexity of integer programming” [Grant 163071]. A. Levin and S. Onn are partially supported by the Israel Science Foundation [Grant 308/18]. A. Levin is also partially supported by the Israel Science Foundation [Grant 1467/22]. S. Onn is also partially supported by the Dresner Chair at the Technion. M. Koutecký is partially supported by Charles University project UNCE 24/SCI/008, and by the project 22-22997S of the Grantová Agentura České Republiky (GA ČR).
我们研究的是一般整数编程问题，其中变量数 n 是输入变量的一部分。我们考虑了约束矩阵 A 的两个自然参数：其数值度量 a 和其稀疏度量 d。我们提出了一种在时间内求解整数编程的算法[公式：见正文]，其中 g 是参数 a 和 d 的某个可计算函数，L 是输入的二进制编码长度。我们的结果还扩展到非线性可分离凸目标函数：F. Eisenbrand、C. Hunkenschröder和K.-M. Klein得到了瑞士国家科学院的资助。克莱因得到了瑞士国家科学基金会 (SNSF) 在 "凸性、数字几何和整数编程的复杂性 "项目 [163071] 中的支持。A. Levin 和 S. Onn 得到了以色列科学基金会 [308/18 号拨款] 的部分资助。A. Levin 还得到以色列科学基金会 [1467/22 号拨款] 的部分资助。S. Onn 也得到了以色列理工学院 Dresner 讲座的部分资助。M. Koutecký 由查尔斯大学项目 UNCE 24/SCI/008 和 Grantová Agentura České Republiky (GA ČR) 项目 22-22997S 提供部分资助。
{"title":"Sparse Integer Programming Is Fixed-Parameter Tractable","authors":"Friedrich Eisenbrand, Christoph Hunkenschröder, Kim-Manuel Klein, Martin Koutecký, Asaf Levin, Shmuel Onn","doi":"10.1287/moor.2023.0162","DOIUrl":"https://doi.org/10.1287/moor.2023.0162","url":null,"abstract":"We study the general integer programming problem where the number of variables n is a variable part of the input. We consider two natural parameters of the constraint matrix A: its numeric measure a and its sparsity measure d. We present an algorithm for solving integer programming in time [Formula: see text], where g is some computable function of the parameters a and d, and L is the binary encoding length of the input. In particular, integer programming is fixed-parameter tractable parameterized by a and d, and is solvable in polynomial time for every fixed a and d. Our results also extend to nonlinear separable convex objective functions.Funding: F. Eisenbrand, C. Hunkenschröder, and K.-M. Klein were supported by the Swiss National Science Foundation (SNSF) within the project “Convexity, geometry of numbers, and the complexity of integer programming” [Grant 163071]. A. Levin and S. Onn are partially supported by the Israel Science Foundation [Grant 308/18]. A. Levin is also partially supported by the Israel Science Foundation [Grant 1467/22]. S. Onn is also partially supported by the Dresner Chair at the Technion. M. Koutecký is partially supported by Charles University project UNCE 24/SCI/008, and by the project 22-22997S of the Grantová Agentura České Republiky (GA ČR).","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"7 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142227392","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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