By adopting a distributional viewpoint on law-invariant convex risk measures, we construct dynamic risk measures (DRMs) at the distributional level. We then apply these DRMs to investigate Markov decision processes, incorporating latent costs, random actions, and weakly continuous transition kernels. Furthermore, the proposed DRMs allow risk aversion to change dynamically. Under mild assumptions, we derive a dynamic programming principle and show the existence of an optimal policy in both finite and infinite time horizons. Moreover, we provide a sufficient condition for the optimality of deterministic actions. For illustration, we conclude the paper with examples from optimal liquidation with limit order books and autonomous driving.Funding: This work was supported by Natural Sciences and Engineering Research Council of Canada [Grants RGPAS-2018-522715 and RGPIN-2018-05705].
{"title":"Risk-Averse Markov Decision Processes Through a Distributional Lens","authors":"Ziteng Cheng, Sebastian Jaimungal","doi":"10.1287/moor.2023.0211","DOIUrl":"https://doi.org/10.1287/moor.2023.0211","url":null,"abstract":"By adopting a distributional viewpoint on law-invariant convex risk measures, we construct dynamic risk measures (DRMs) at the distributional level. We then apply these DRMs to investigate Markov decision processes, incorporating latent costs, random actions, and weakly continuous transition kernels. Furthermore, the proposed DRMs allow risk aversion to change dynamically. Under mild assumptions, we derive a dynamic programming principle and show the existence of an optimal policy in both finite and infinite time horizons. Moreover, we provide a sufficient condition for the optimality of deterministic actions. For illustration, we conclude the paper with examples from optimal liquidation with limit order books and autonomous driving.Funding: This work was supported by Natural Sciences and Engineering Research Council of Canada [Grants RGPAS-2018-522715 and RGPIN-2018-05705].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"42 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141738361","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}
Robust Markov decision processes (MDPs) are used for applications of dynamic optimization in uncertain environments and have been studied extensively. Many of the main properties and algorithms of MDPs, such as value iteration and policy iteration, extend directly to RMDPs. Surprisingly, there is no known analog of the MDP convex optimization formulation for solving RMDPs. This work describes the first convex optimization formulation of RMDPs under the classical sa-rectangularity and s-rectangularity assumptions. By using entropic regularization and exponential change of variables, we derive a convex formulation with a number of variables and constraints polynomial in the number of states and actions, but with large coefficients in the constraints. We further simplify the formulation for RMDPs with polyhedral, ellipsoidal, or entropy-based uncertainty sets, showing that, in these cases, RMDPs can be reformulated as conic programs based on exponential cones, quadratic cones, and nonnegative orthants. Our work opens a new research direction for RMDPs and can serve as a first step toward obtaining a tractable convex formulation of RMDPs.Funding: The work in the paper was supported, in part, by NSF [Grants 2144601 and 1815275]; and Agence Nationale de la Recherche [Grant 11-LABX-0047].
鲁棒马尔可夫决策过程（MDP）用于不确定环境中的动态优化应用，并已得到广泛研究。MDP 的许多主要特性和算法，如值迭代和策略迭代，都直接扩展到了 RMDP。令人惊讶的是，目前还没有已知的用于求解 RMDP 的 MDP 凸优化公式。本研究首次描述了在经典 sa-rectangularity 和 s-rectangularity 假设下的 RMDPs 凸优化公式。通过使用熵正则化和变量指数变化，我们推导出了一种变量和约束条件数量与状态和行动数量成多项式关系，但约束条件系数较大的凸优化公式。我们进一步简化了具有多面体、椭圆形或基于熵的不确定性集的 RMDPs 的表述，表明在这些情况下，RMDPs 可以重新表述为基于指数锥、二次锥和非负正交的圆锥程序。我们的工作为 RMDPs 开辟了一个新的研究方向，并为获得 RMDPs 的可控凸表述迈出了第一步：本文的部分研究工作得到了国家自然科学基金[Grants 2144601 and 1815275]和Agence Nationale de la Recherche [Grant 11-LABX-0047]的资助。
{"title":"On the Convex Formulations of Robust Markov Decision Processes","authors":"Julien Grand-Clément, Marek Petrik","doi":"10.1287/moor.2022.0284","DOIUrl":"https://doi.org/10.1287/moor.2022.0284","url":null,"abstract":"Robust Markov decision processes (MDPs) are used for applications of dynamic optimization in uncertain environments and have been studied extensively. Many of the main properties and algorithms of MDPs, such as value iteration and policy iteration, extend directly to RMDPs. Surprisingly, there is no known analog of the MDP convex optimization formulation for solving RMDPs. This work describes the first convex optimization formulation of RMDPs under the classical sa-rectangularity and s-rectangularity assumptions. By using entropic regularization and exponential change of variables, we derive a convex formulation with a number of variables and constraints polynomial in the number of states and actions, but with large coefficients in the constraints. We further simplify the formulation for RMDPs with polyhedral, ellipsoidal, or entropy-based uncertainty sets, showing that, in these cases, RMDPs can be reformulated as conic programs based on exponential cones, quadratic cones, and nonnegative orthants. Our work opens a new research direction for RMDPs and can serve as a first step toward obtaining a tractable convex formulation of RMDPs.Funding: The work in the paper was supported, in part, by NSF [Grants 2144601 and 1815275]; and Agence Nationale de la Recherche [Grant 11-LABX-0047].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"34 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141721356","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 discuss optimality conditions for optimization problems involving random state constraints, which are modeled in probabilistic or almost sure form. Although the latter can be understood as the limiting case of the former, the derivation of optimality conditions requires substantially different approaches. We apply them to a linear elliptic partial differential equation with random inputs. In the probabilistic case, we rely on the spherical-radial decomposition of Gaussian random vectors in order to formulate fully explicit optimality conditions involving a spherical integral. In the almost sure case, we derive optimality conditions and compare them with a model based on robust constraints with respect to the (compact) support of the given distribution.Funding: The authors thank the Deutsche Forschungsgemeinschaft [Projects B02 and B04 in the “Sonderforschungsbereich/Transregio 154 Mathematical Modelling, Simulation and Optimization Using the Example of Gas Networks”] for support. C. Geiersbach acknowledges support from the Deutsche Forschungsgemeinschaft [Germany’s Excellence Strategy–the Berlin Mathematics Research Center MATH+ Grant EXC-2046/1, Project 390685689]. R. Henrion acknowledges support from the Fondation Mathématique Jacques Hadamard [Program Gaspard Monge in Optimization and Operations Research, including support to this program by Electricité de France].
{"title":"Optimality Conditions in Control Problems with Random State Constraints in Probabilistic or Almost Sure Form","authors":"Caroline Geiersbach, René Henrion","doi":"10.1287/moor.2023.0177","DOIUrl":"https://doi.org/10.1287/moor.2023.0177","url":null,"abstract":"In this paper, we discuss optimality conditions for optimization problems involving random state constraints, which are modeled in probabilistic or almost sure form. Although the latter can be understood as the limiting case of the former, the derivation of optimality conditions requires substantially different approaches. We apply them to a linear elliptic partial differential equation with random inputs. In the probabilistic case, we rely on the spherical-radial decomposition of Gaussian random vectors in order to formulate fully explicit optimality conditions involving a spherical integral. In the almost sure case, we derive optimality conditions and compare them with a model based on robust constraints with respect to the (compact) support of the given distribution.Funding: The authors thank the Deutsche Forschungsgemeinschaft [Projects B02 and B04 in the “Sonderforschungsbereich/Transregio 154 Mathematical Modelling, Simulation and Optimization Using the Example of Gas Networks”] for support. C. Geiersbach acknowledges support from the Deutsche Forschungsgemeinschaft [Germany’s Excellence Strategy–the Berlin Mathematics Research Center MATH+ Grant EXC-2046/1, Project 390685689]. R. Henrion acknowledges support from the Fondation Mathématique Jacques Hadamard [Program Gaspard Monge in Optimization and Operations Research, including support to this program by Electricité de France].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"308 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141721355","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}
Computing the maximum size of an independent set in a graph is a famously hard combinatorial problem that has been well studied for various classes of graphs. When it comes to random graphs, the classic Erdős–Rényi–Gilbert random graph [Formula: see text] has been analyzed and shown to have the largest independent sets of size [Formula: see text] with high probability (w.h.p.) This classic model does not capture any dependency structure between edges that can appear in real-world networks. We define random graphs [Formula: see text] whose existence of edges is determined by a Markov process that is also governed by a decay parameter [Formula: see text]. We prove that w.h.p. [Formula: see text] has independent sets of size [Formula: see text] for arbitrary [Formula: see text]. This is derived using bounds on the terms of a harmonic series, a Turán bound on a stability number, and a concentration analysis for a certain sequence of dependent Bernoulli variables that may also be of independent interest. Because [Formula: see text] collapses to [Formula: see text] when there is no decay, it follows that having even the slightest bit of dependency (any [Formula: see text]) in the random graph construction leads to the presence of large independent sets, and thus, our random model has a phase transition at its boundary value of r = 1. This implies that there are large matchings in the line graph of [Formula: see text], which is a Markov random field. For the maximal independent set output by a greedy algorithm, we deduce that it has a performance ratio of at most [Formula: see text] w.h.p. when the lowest degree vertex is picked at each iteration and also show that, under any other permutation of vertices, the algorithm outputs a set of size [Formula: see text], where [Formula: see text] and, hence, has a performance ratio of [Formula: see text].Funding: The initial phase of this research was supported by the National Science Foundation [Grant DMS-1913294].
{"title":"Large Independent Sets in Recursive Markov Random Graphs","authors":"Akshay Gupte, Yiran Zhu","doi":"10.1287/moor.2022.0215","DOIUrl":"https://doi.org/10.1287/moor.2022.0215","url":null,"abstract":"Computing the maximum size of an independent set in a graph is a famously hard combinatorial problem that has been well studied for various classes of graphs. When it comes to random graphs, the classic Erdős–Rényi–Gilbert random graph [Formula: see text] has been analyzed and shown to have the largest independent sets of size [Formula: see text] with high probability (w.h.p.) This classic model does not capture any dependency structure between edges that can appear in real-world networks. We define random graphs [Formula: see text] whose existence of edges is determined by a Markov process that is also governed by a decay parameter [Formula: see text]. We prove that w.h.p. [Formula: see text] has independent sets of size [Formula: see text] for arbitrary [Formula: see text]. This is derived using bounds on the terms of a harmonic series, a Turán bound on a stability number, and a concentration analysis for a certain sequence of dependent Bernoulli variables that may also be of independent interest. Because [Formula: see text] collapses to [Formula: see text] when there is no decay, it follows that having even the slightest bit of dependency (any [Formula: see text]) in the random graph construction leads to the presence of large independent sets, and thus, our random model has a phase transition at its boundary value of r = 1. This implies that there are large matchings in the line graph of [Formula: see text], which is a Markov random field. For the maximal independent set output by a greedy algorithm, we deduce that it has a performance ratio of at most [Formula: see text] w.h.p. when the lowest degree vertex is picked at each iteration and also show that, under any other permutation of vertices, the algorithm outputs a set of size [Formula: see text], where [Formula: see text] and, hence, has a performance ratio of [Formula: see text].Funding: The initial phase of this research was supported by the National Science Foundation [Grant DMS-1913294].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"17 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141609401","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 when the voting outcome is independent of the order of issues put up for vote in a spatial multidimensional voting model. Agents equipped with norm-based preferences that use a norm to measure the distance from their ideal policy vote sequentially and issue by issue via simple majority. If the underlying norm is generated by an inner product—such as the Euclidean norm—then the voting outcome is order independent if and only if the issues are orthogonal. If the underlying norm is a general one, then the outcome is order independent if the basis defining the issues to be voted upon satisfies the following property; for any vector in the basis, any linear combination of the other vectors is Birkhoff–James orthogonal to it. We prove a partial converse in the case of two dimensions; if the underlying basis fails this property, then the voting order matters. Finally, despite existence results for the two-dimensional case and for the general lp case, we show that nonexistence of bases with this property is generic.Funding: The research of A. Gershkov is supported by the Israel Science Foundation [Grant 1118/22]. The research of B. Moldovanu is supported by the German Science Foundation through the Hausdorff Center for Mathematics and The Collaborative Research Center Transregio 224. The research of X. Shi is supported by the Social Sciences and Humanities Research Council of Canada.
{"title":"Order Independence in Sequential, Issue-by-Issue Voting","authors":"Alex Gershkov, Benny Moldovanu, Xianwen Shi","doi":"10.1287/moor.2022.0342","DOIUrl":"https://doi.org/10.1287/moor.2022.0342","url":null,"abstract":"We study when the voting outcome is independent of the order of issues put up for vote in a spatial multidimensional voting model. Agents equipped with norm-based preferences that use a norm to measure the distance from their ideal policy vote sequentially and issue by issue via simple majority. If the underlying norm is generated by an inner product—such as the Euclidean norm—then the voting outcome is order independent if and only if the issues are orthogonal. If the underlying norm is a general one, then the outcome is order independent if the basis defining the issues to be voted upon satisfies the following property; for any vector in the basis, any linear combination of the other vectors is Birkhoff–James orthogonal to it. We prove a partial converse in the case of two dimensions; if the underlying basis fails this property, then the voting order matters. Finally, despite existence results for the two-dimensional case and for the general l<jats:sub>p</jats:sub> case, we show that nonexistence of bases with this property is generic.Funding: The research of A. Gershkov is supported by the Israel Science Foundation [Grant 1118/22]. The research of B. Moldovanu is supported by the German Science Foundation through the Hausdorff Center for Mathematics and The Collaborative Research Center Transregio 224. The research of X. Shi is supported by the Social Sciences and Humanities Research Council of Canada.","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"17 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141614395","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 practice, optimization models are often prone to unavoidable inaccuracies because of dubious assumptions and corrupted data. Traditionally, this placed special emphasis on risk-based and robust formulations, and their focus on “conservative” decisions. We develop, in contrast, an “optimistic” framework based on Rockafellian relaxations in which optimization is conducted not only over the original decision space but also jointly with a choice of model perturbation. The framework enables us to address challenging problems with ambiguous probability distributions from the areas of two-stage stochastic optimization without relatively complete recourse, probability functions lacking continuity properties, expectation constraints, and outlier analysis. We are also able to circumvent the fundamental difficulty in stochastic optimization that convergence of distributions fails to guarantee convergence of expectations. The framework centers on the novel concepts of exact and limit-exact Rockafellians, with interpretations of “negative” regularization emerging in certain settings. We illustrate the role of Phi-divergence, examine rates of convergence under changing distributions, and explore extensions to first-order optimality conditions. The main development is free of assumptions about convexity, smoothness, and even continuity of objective functions. Numerical results in the setting of computer vision and text analytics with label noise illustrate the framework.Funding: This work was supported by the Air Force Office of Scientific Research (Mathematical Optimization Program) under the grant: “Optimal Decision Making under Tight Performance Requirements in Adversarial and Uncertain Environments: Insight from Rockafellian Functions.”
{"title":"Rockafellian Relaxation and Stochastic Optimization Under Perturbations","authors":"Johannes O. Royset, Louis L. Chen, Eric Eckstrand","doi":"10.1287/moor.2022.0122","DOIUrl":"https://doi.org/10.1287/moor.2022.0122","url":null,"abstract":"In practice, optimization models are often prone to unavoidable inaccuracies because of dubious assumptions and corrupted data. Traditionally, this placed special emphasis on risk-based and robust formulations, and their focus on “conservative” decisions. We develop, in contrast, an “optimistic” framework based on Rockafellian relaxations in which optimization is conducted not only over the original decision space but also jointly with a choice of model perturbation. The framework enables us to address challenging problems with ambiguous probability distributions from the areas of two-stage stochastic optimization without relatively complete recourse, probability functions lacking continuity properties, expectation constraints, and outlier analysis. We are also able to circumvent the fundamental difficulty in stochastic optimization that convergence of distributions fails to guarantee convergence of expectations. The framework centers on the novel concepts of exact and limit-exact Rockafellians, with interpretations of “negative” regularization emerging in certain settings. We illustrate the role of Phi-divergence, examine rates of convergence under changing distributions, and explore extensions to first-order optimality conditions. The main development is free of assumptions about convexity, smoothness, and even continuity of objective functions. Numerical results in the setting of computer vision and text analytics with label noise illustrate the framework.Funding: This work was supported by the Air Force Office of Scientific Research (Mathematical Optimization Program) under the grant: “Optimal Decision Making under Tight Performance Requirements in Adversarial and Uncertain Environments: Insight from Rockafellian Functions.”","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"14 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141614397","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}
Jean-Paul Décamps, Fabien Gensbittel, Thomas Mariotti
We study the optimal investment policy of a firm facing both technological and cash-flow uncertainty. At any point in time, the firm can irreversibly invest in a stand-alone technology or wait for a technological breakthrough. Breakthroughs occur when market conditions become favorable enough, exceeding a threshold value that is ex ante unknown to the firm. The Markov state variables for the optimal investment policy are the current market conditions and their historic maximum, and the firm optimally invests in the stand-alone technology only when market conditions deteriorate enough after reaching a maximum. The path-dependent return required for investing in the stand-alone technology is always higher than if no technological breakthroughs could occur and can take arbitrarily large values following certain histories. Decreases in development costs or increases in the value of the new technology make the firm more prone to bearing downside risk and delaying investment in the stand-alone technology.Funding: This research has benefited from financial support of the ANR [Programmes d’Investissements d’Avenir CHESS ANR-17-EURE-0010 and ANITI ANR-19-PI3A-0004] and the research foundation TSE-Partnership [Chaire Marchés des Risques et Création de Valeur, Fondation du Risque/SCOR].
我们研究的是一家同时面临技术和现金流不确定性的公司的最优投资政策。在任何时间点，企业都可以不可逆转地投资于一项独立技术或等待技术突破。当市场条件变得足够有利，超过企业事先未知的临界值时，突破就会发生。最优投资政策的马尔可夫状态变量是当前市场条件及其历史最大值，只有当市场条件在达到最大值后恶化到足够严重时，企业才会对独立技术进行最优投资。投资独立技术所需的路径依赖回报率总是高于不发生技术突破的情况，并且在某些历史条件下可以任意取大值。开发成本的降低或新技术价值的增加会使企业更容易承担下行风险，并推迟对独立技术的投资：本研究得到了法国国家科学研究署（ANR）[Programmes d'Investissements d'Avenir CHESS ANR-17-EURE-0010 和 ANITI ANR-19-PI3A-0004]以及研究基金会 TSE-Partnership [Chaire Marchés des Risques et Création de Valeur, Fondation du Risque/SCOR] 的资助。
{"title":"Investment Timing and Technological Breakthroughs","authors":"Jean-Paul Décamps, Fabien Gensbittel, Thomas Mariotti","doi":"10.1287/moor.2022.0022","DOIUrl":"https://doi.org/10.1287/moor.2022.0022","url":null,"abstract":"We study the optimal investment policy of a firm facing both technological and cash-flow uncertainty. At any point in time, the firm can irreversibly invest in a stand-alone technology or wait for a technological breakthrough. Breakthroughs occur when market conditions become favorable enough, exceeding a threshold value that is ex ante unknown to the firm. The Markov state variables for the optimal investment policy are the current market conditions and their historic maximum, and the firm optimally invests in the stand-alone technology only when market conditions deteriorate enough after reaching a maximum. The path-dependent return required for investing in the stand-alone technology is always higher than if no technological breakthroughs could occur and can take arbitrarily large values following certain histories. Decreases in development costs or increases in the value of the new technology make the firm more prone to bearing downside risk and delaying investment in the stand-alone technology.Funding: This research has benefited from financial support of the ANR [Programmes d’Investissements d’Avenir CHESS ANR-17-EURE-0010 and ANITI ANR-19-PI3A-0004] and the research foundation TSE-Partnership [Chaire Marchés des Risques et Création de Valeur, Fondation du Risque/SCOR].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"40 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141568637","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 revisits mean-risk portfolio selection in a one-period financial market, where risk is quantified by a star-shaped risk measure ρ. We make three contributions. First, we introduce the new axiom of sensitivity to large expected losses and show that it is key to ensure the existence of optimal portfolios. Second, we give primal and dual characterizations of (strong) ρ-arbitrage. Finally, we use our conditions for the absence of (strong) ρ-arbitrage to explicitly derive the (strong) ρ-consistent price interval for an external financial contract.
{"title":"ρ-Arbitrage and ρ-Consistent Pricing for Star-Shaped Risk Measures","authors":"Martin Herdegen, Nazem Khan","doi":"10.1287/moor.2023.0173","DOIUrl":"https://doi.org/10.1287/moor.2023.0173","url":null,"abstract":"This paper revisits mean-risk portfolio selection in a one-period financial market, where risk is quantified by a star-shaped risk measure ρ. We make three contributions. First, we introduce the new axiom of sensitivity to large expected losses and show that it is key to ensure the existence of optimal portfolios. Second, we give primal and dual characterizations of (strong) ρ-arbitrage. Finally, we use our conditions for the absence of (strong) ρ-arbitrage to explicitly derive the (strong) ρ-consistent price interval for an external financial contract.","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"8 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141568638","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the NP-hard problem of finding the closest rank-one binary tensor to a given binary tensor, which we refer to as the rank-one Boolean tensor factorization (BTF) problem. This optimization problem can be used to recover a planted rank-one tensor from noisy observations. We formulate rank-one BTF as the problem of minimizing a linear function over a highly structured multilinear set. Leveraging on our prior results regarding the facial structure of multilinear polytopes, we propose novel linear programming relaxations for rank-one BTF. We then establish deterministic sufficient conditions under which our proposed linear programs recover a planted rank-one tensor. To analyze the effectiveness of these deterministic conditions, we consider a semirandom model for the noisy tensor and obtain high probability recovery guarantees for the linear programs. Our theoretical results as well as numerical simulations indicate that certain facets of the multilinear polytope significantly improve the recovery properties of linear programming relaxations for rank-one BTF.Funding: A. Del Pia is partially funded by the Air Force Office of Scientific Research [Grant FA9550-23-1-0433]. A. Khajavirad is partially funded by the Air Force Office of Scientific Research [Grant FA9550-23-1-0123].
我们考虑的是寻找与给定二进制张量最接近的秩一二进制张量的 NP 难问题，我们称之为秩一布尔张量因式分解（BTF）问题。这个优化问题可用于从噪声观测中恢复一个种植的秩一张量。我们将 rank-one BTF 问题表述为在高度结构化的多线性集合上最小化线性函数的问题。利用我们之前关于多线性多面体面结构的研究成果，我们提出了针对秩一 BTF 的新颖线性规划松弛方法。然后，我们建立了确定性充分条件，在这些充分条件下，我们提出的线性规划可以恢复一个种植的秩一张量。为了分析这些确定性条件的有效性，我们考虑了噪声张量的半随机模型，并获得了线性规划的高概率恢复保证。我们的理论结果和数值模拟表明，多线性多面体的某些面显著改善了秩一 BTF 线性编程松弛的恢复特性：A. Del Pia 的部分研究经费来自空军科学研究办公室 [拨款 FA9550-23-1-0433]。A. Khajavirad 由空军科学研究办公室[FA9550-23-1-0123 号拨款]提供部分资助。
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This paper studies hidden convexity properties associated with constrained optimization problems over the set of rotation matrices [Formula: see text]. Such problems are nonconvex because of the constraint [Formula: see text]. Nonetheless, we show that certain linear images of [Formula: see text] are convex, opening up the possibility for convex optimization algorithms with provable guarantees for these problems. Our main technical contributions show that any two-dimensional image of [Formula: see text] is convex and that the projection of [Formula: see text] onto its strict upper triangular entries is convex. These results allow us to construct exact convex reformulations for constrained optimization problems over [Formula: see text] with a single constraint or with constraints defined by low-rank matrices. Both of these results are maximal in a formal sense.Funding: A. Ramachandran was supported by the H2020 program of the European Research Council [Grant 805241-QIP]. A. L. Wang was supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek [Grant OCENW.GROOT.2019.015 (OPTIMAL)]. K. Shu was supported by the Georgia Institute of Technology (ACO-ARC fellowship).
本文研究与旋转矩阵集合上的约束优化问题相关的隐凸性质[公式：见正文]。由于[公式：见正文]的约束，这类问题是非凸的。然而，我们证明了[公式：见正文]的某些线性图像是凸的，从而为这些问题提供了可证明的凸优化算法的可能性。我们的主要技术贡献表明，[公式：见正文]的任何二维图像都是凸的，而且[公式：见正文]对其严格上三角项的投影也是凸的。这些结果使我们能够为[公式：见正文]上的约束优化问题构建精确的凸重构，这些问题具有单一约束或由低阶矩阵定义的约束。这两个结果在形式意义上都是最大的：A. Ramachandran 得到了欧洲研究理事会 H2020 计划 [805241-QIP] 的资助。A. L. Wang 得到了 Nederlandse Organisatie voor Wetenschappelijk Onderzoek [Grant OCENW.GROOT.2019.015 (OPTIMAL)] 的资助。K. Shu得到了佐治亚理工学院（ACO-ARC奖学金）的资助。
{"title":"Hidden Convexity, Optimization, and Algorithms on Rotation Matrices","authors":"Akshay Ramachandran, Kevin Shu, Alex L. Wang","doi":"10.1287/moor.2023.0114","DOIUrl":"https://doi.org/10.1287/moor.2023.0114","url":null,"abstract":"This paper studies hidden convexity properties associated with constrained optimization problems over the set of rotation matrices [Formula: see text]. Such problems are nonconvex because of the constraint [Formula: see text]. Nonetheless, we show that certain linear images of [Formula: see text] are convex, opening up the possibility for convex optimization algorithms with provable guarantees for these problems. Our main technical contributions show that any two-dimensional image of [Formula: see text] is convex and that the projection of [Formula: see text] onto its strict upper triangular entries is convex. These results allow us to construct exact convex reformulations for constrained optimization problems over [Formula: see text] with a single constraint or with constraints defined by low-rank matrices. Both of these results are maximal in a formal sense.Funding: A. Ramachandran was supported by the H2020 program of the European Research Council [Grant 805241-QIP]. A. L. Wang was supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek [Grant OCENW.GROOT.2019.015 (OPTIMAL)]. K. Shu was supported by the Georgia Institute of Technology (ACO-ARC fellowship).","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"21 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141505983","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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