Mathematics of Operations Research最新文献

{"title":"Is There a Golden Parachute in Sannikov’s Principal–Agent Problem?","authors":"Dylan Possamaï, Nizar Touzi","doi":"10.1287/moor.2022.0305","DOIUrl":"https://doi.org/10.1287/moor.2022.0305","url":null,"abstract":"This paper provides a complete review of the continuous-time optimal contracting problem introduced by Sannikov in the extended context allowing for possibly different discount rates for both parties. The agent’s problem is to seek for optimal effort given the compensation scheme proposed by the principal over a random horizon. Then, given the optimal agent’s response, the principal determines the best compensation scheme in terms of running payment, retirement, and lump-sum payment at retirement. A golden parachute is a situation where the agent ceases any effort at some positive stopping time and receives a payment afterward, possibly under the form of a lump-sum payment or of a continuous stream of payments. We show that a golden parachute only exists in certain specific circumstances. This is in contrast with the results claimed by Sannikov, where the only requirement is a positive agent’s marginal cost of effort at zero. In the general case, we prove that an agent with positive reservation utility is either never retired by the principal or retired above some given threshold (as in Sannikov’s solution). We show that different discount factors induce a facelifted utility function, which allows us to reduce the analysis to a setting similar to the equal-discount rates one. Finally, we also confirm that an agent with small reservation utility does have an informational rent, meaning that the principal optimally offers him a contract with strictly higher utility than his participation value.","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140887228","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}

{"title":"Learning and Balancing Unknown Loads in Large-Scale Systems","authors":"Diego Goldsztajn, Sem C. Borst, Johan S. H. van Leeuwaarden","doi":"10.1287/moor.2021.0212","DOIUrl":"https://doi.org/10.1287/moor.2021.0212","url":null,"abstract":"Consider a system of identical server pools where tasks with exponentially distributed service times arrive as a time-inhomogeneous Poisson process. An admission threshold is used in an inner control loop to assign incoming tasks to server pools, while in an outer control loop, a learning scheme adjusts this threshold over time to keep it aligned with the unknown offered load of the system. In a many-server regime, we prove that the learning scheme reaches an equilibrium along intervals of time when the normalized offered load per server pool is suitably bounded and that this results in a balanced distribution of the load. Furthermore, we establish a similar result when tasks with Coxian distributed service times arrive at a constant rate and the threshold is adjusted using only the total number of tasks in the system. The novel proof technique developed in this paper, which differs from a traditional fluid limit analysis, allows us to handle rapid variations of the first learning scheme, triggered by excursions of the occupancy process that have vanishing size. Moreover, our approach allows us to characterize the asymptotic behavior of the system with Coxian distributed service times without relying on a fluid limit of a detailed state descriptor.Funding: The work in this paper was supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek [Gravitation Grant NETWORKS-024.002.003 and Vici Grant 202.068].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140832251","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}

{"title":"Estimating a Function and Its Derivatives Under a Smoothness Condition","authors":"Eunji Lim","doi":"10.1287/moor.2020.0161","DOIUrl":"https://doi.org/10.1287/moor.2020.0161","url":null,"abstract":"We consider the problem of estimating an unknown function [Formula: see text] and its partial derivatives from a noisy data set of n observations, where we make no assumptions about [Formula: see text] except that it is smooth in the sense that it has square integrable partial derivatives of order m. A natural candidate for the estimator of [Formula: see text] in such a case is the best fit to the data set that satisfies a certain smoothness condition. This estimator can be seen as a least squares estimator subject to an upper bound on some measure of smoothness. Another useful estimator is the one that minimizes the degree of smoothness subject to an upper bound on the average of squared errors. We prove that these two estimators are computable as solutions to quadratic programs, establish the consistency of these estimators and their partial derivatives, and study the convergence rate as [Formula: see text]. The effectiveness of the estimators is illustrated numerically in a setting where the value of a stock option and its second derivative are estimated as functions of the underlying stock price.","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140842178","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}

{"title":"Correlated Equilibria for Mean Field Games with Progressive Strategies","authors":"Ofelia Bonesini, Luciano Campi, Markus Fischer","doi":"10.1287/moor.2022.0357","DOIUrl":"https://doi.org/10.1287/moor.2022.0357","url":null,"abstract":"In a discrete space and time framework, we study the mean field game limit for a class of symmetric N-player games based on the notion of correlated equilibrium. We give a definition of correlated solution that allows us to construct approximate N-player correlated equilibria that are robust with respect to progressive deviations. We illustrate our definition by way of an example with explicit solutions.Funding: O. Bonesini acknowledges financial support from Engineering and Physical Sciences Research Council [Grant EP/T032146/1]. M. Fischer acknowledges partial support through the University of Padua [Research Project BIRD229791 “Stochastic mean field control and the Schrödinger problem”].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140832372","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}

{"title":"Convexification of Bilinear Terms over Network Polytopes","authors":"Erfan Khademnia, Danial Davarnia","doi":"10.1287/moor.2023.0001","DOIUrl":"https://doi.org/10.1287/moor.2023.0001","url":null,"abstract":"It is well-known that the McCormick relaxation for the bilinear constraint z = xy gives the convex hull over the box domains for x and y. In network applications where the domain of bilinear variables is described by a network polytope, the McCormick relaxation, also referred to as linearization, fails to provide the convex hull and often leads to poor dual bounds. We study the convex hull of the set containing bilinear constraints [Formula: see text] where x<jats:sub>i</jats:sub> represents the arc-flow variable in a network polytope, and y<jats:sub>j</jats:sub> is in a simplex. For the case where the simplex contains a single y variable, we introduce a systematic procedure to obtain the convex hull of the above set in the original space of variables, and show that all facet-defining inequalities of the convex hull can be obtained explicitly through identifying a special tree structure in the underlying network. For the generalization where the simplex contains multiple y variables, we design a constructive procedure to obtain an important class of facet-defining inequalities for the convex hull of the underlying bilinear set that is characterized by a special forest structure in the underlying network. Computational experiments conducted on different applications show the effectiveness of the proposed methods in improving the dual bounds obtained from alternative techniques.Funding: This work was supported by Air Force Office of Scientific Research [Grant FA9550-23-1-0183]; National Science Foundation, Division of Civil, Mechanical and Manufacturing Innovation [Grant 2338641].Supplemental Material: The online appendix is available at https://doi.org/10.1287/moor.2023.0001 .","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140798174","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}

{"title":"Finite-Time High-Probability Bounds for Polyak–Ruppert Averaged Iterates of Linear Stochastic Approximation","authors":"Alain Durmus, Eric Moulines, Alexey Naumov, Sergey Samsonov","doi":"10.1287/moor.2022.0179","DOIUrl":"https://doi.org/10.1287/moor.2022.0179","url":null,"abstract":"This paper provides a finite-time analysis of linear stochastic approximation (LSA) algorithms with fixed step size, a core method in statistics and machine learning. LSA is used to compute approximate solutions of a d-dimensional linear system [Formula: see text] for which [Formula: see text] can only be estimated by (asymptotically) unbiased observations [Formula: see text]. We consider here the case where [Formula: see text] is an a sequence of independent and identically distributed random variables sequence or a uniformly geometrically ergodic Markov chain. We derive pth moment and high-probability deviation bounds for the iterates defined by LSA and its Polyak–Ruppert-averaged version. Our finite-time instance-dependent bounds for the averaged LSA iterates are sharp in the sense that the leading term we obtain coincides with the local asymptotic minimax limit. Moreover, the remainder terms of our bounds admit a tight dependence on the mixing time [Formula: see text] of the underlying chain and the norm of the noise variables. We emphasize that our result requires the LSA step size to scale only with logarithm of the problem dimension d.Funding: The work of A. Durmus and E. Moulines was partly supported by [Grant ANR-19-CHIA-0002]. This project received funding from the European Research Council [ERC-SyG OCEAN Grant 101071601]. The research of A. Naumov and S. Samsonov was prepared within the framework of the HSE University Basic Research Program.","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140612440","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}

{"title":"Robust-to-Dynamics Optimization","authors":"Amir Ali Ahmadi, Oktay Günlük","doi":"10.1287/moor.2023.0116","DOIUrl":"https://doi.org/10.1287/moor.2023.0116","url":null,"abstract":"A robust-to-dynamics optimization (RDO) problem is an optimization problem specified by two pieces of input: (i) a mathematical program (an objective function [Formula: see text] and a feasible set [Formula: see text]) and (ii) a dynamical system (a map [Formula: see text]). Its goal is to minimize f over the set [Formula: see text] of initial conditions that forever remain in [Formula: see text] under g. The focus of this paper is on the case where the mathematical program is a linear program and where the dynamical system is either a known linear map or an uncertain linear map that can change over time. In both cases, we study a converging sequence of polyhedral outer approximations and (lifted) spectrahedral inner approximations to [Formula: see text]. Our inner approximations are optimized with respect to the objective function f, and their semidefinite characterization—which has a semidefinite constraint of fixed size—is obtained by applying polar duality to convex sets that are invariant under (multiple) linear maps. We characterize three barriers that can stop convergence of the outer approximations to [Formula: see text] from being finite. We prove that once these barriers are removed, our inner and outer approximating procedures find an optimal solution and a certificate of optimality for the RDO problem in a finite number of steps. Moreover, in the case where the dynamics are linear, we show that this phenomenon occurs in a number of steps that can be computed in time polynomial in the bit size of the input data. Our analysis also leads to a polynomial-time algorithm for RDO instances where the spectral radius of the linear map is bounded above by any constant less than one. Finally, in our concluding section, we propose a broader research agenda for studying optimization problems with dynamical systems constraints, of which RDO is a special case.Funding: O. Günlük was partially supported by the Office of Naval Research [Grant N00014-21-1-2575]. This work was partially funded by the Alfred P. Sloan Foundation, the Air Force Office of Scientific Research, Defense Advanced Research Projects Agency [Young Faculty Award], the National Science Foundation [Faculty Early Career Development Program Award], and Google [Faculty Award].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140612441","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}

{"title":"Efficiency of Stochastic Coordinate Proximal Gradient Methods on Nonseparable Composite Optimization","authors":"Ion Necoara, Flavia Chorobura","doi":"10.1287/moor.2023.0044","DOIUrl":"https://doi.org/10.1287/moor.2023.0044","url":null,"abstract":"This paper deals with composite optimization problems having the objective function formed as the sum of two terms; one has a Lipschitz continuous gradient along random subspaces and may be nonconvex, and the second term is simple and differentiable but possibly nonconvex and nonseparable. Under these settings, we design a stochastic coordinate proximal gradient method that takes into account the nonseparable composite form of the objective function. This algorithm achieves scalability by constructing at each iteration a local approximation model of the whole nonseparable objective function along a random subspace with user-determined dimension. We outline efficient techniques for selecting the random subspace, yielding an implementation that has low cost per iteration, also achieving fast convergence rates. We present a probabilistic worst case complexity analysis for our stochastic coordinate proximal gradient method in convex and nonconvex settings; in particular, we prove high-probability bounds on the number of iterations before a given optimality is achieved. Extensive numerical results also confirm the efficiency of our algorithm.Funding: This work was supported by Norway Grants 2014-2021 [Grant ELO-Hyp 24/2020]; Unitatea Executiva pentru Finantarea Invatamantului Superior, a Cercetarii, Dezvoltarii si Inovarii [Grants PN-III-P4-PCE-2021-0720, L2O-MOC, nr 70/2022]; and the ITN-ETN project TraDE-OPT funded by the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Skłodowska-Curie grant agreement [Grant 861137].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140612454","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}

{"title":"Practical Algorithms with Guaranteed Approximation Ratio for Traveling Tournament Problem with Maximum Tour Length 2","authors":"Jingyang Zhao, Mingyu Xiao","doi":"10.1287/moor.2022.0356","DOIUrl":"https://doi.org/10.1287/moor.2022.0356","url":null,"abstract":"The traveling tournament problem (TTP) is a hard but interesting sports scheduling problem inspired by Major League Baseball, which is to design a double round-robin schedule such that each pair of teams plays one game in each other’s home venue, minimizing the total distance traveled by all n teams (n is even). In this paper, we consider TTP-2 (i.e., TTP under the constraint that at most two consecutive home games or away games are allowed for each team). In this paper, we propose practical algorithms for TTP-2 with improved approximation ratios. Because of the different structural properties of the problem, all known algorithms for TTP-2 are different for n/2 being odd and even, and our algorithms are also different for these two cases. For even n/2, our approximation ratio is [Formula: see text], improving the previous result of [Formula: see text]. For odd n/2, our approximation ratio is [Formula: see text], improving the previous result of [Formula: see text]. In practice, our algorithms are easy to implement. Experiments on well-known benchmark sets show that our algorithms beat previously known solutions for all instances with an average improvement of 5.66%.Funding: This work was supported by the National Natural Science Foundation of China [Grants 62372095 and 62172077] and the Sichuan Natural Science Foundation [Grant 2023NSFSC0059].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140582291","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}

{"title":"On Constrained Mixed-Integer DR-Submodular Minimization","authors":"Qimeng Yu, Simge Küçükyavuz","doi":"10.1287/moor.2022.0320","DOIUrl":"https://doi.org/10.1287/moor.2022.0320","url":null,"abstract":"Diminishing returns (DR)–submodular functions encompass a broad class of functions that are generally nonconvex and nonconcave. We study the problem of minimizing any DR-submodular function with continuous and general integer variables under box constraints and, possibly, additional monotonicity constraints. We propose valid linear inequalities for the epigraph of any DR-submodular function under the constraints. We further provide the complete convex hull of such an epigraph, which, surprisingly, turns out to be polyhedral. We propose a polynomial-time exact separation algorithm for our proposed valid inequalities with which we first establish the polynomial-time solvability of this class of mixed-integer nonlinear optimization problems.Funding: This work was supported by the Office of Naval Research Global [Grant N00014-22-1-2602].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140582194","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}