We consider the problem of minimizing the sum of a smooth function and a composition of a zero-one loss function with a linear operator, namely the zero-one composite optimization problem (0/1-COP). It has a vast body of applications, including the support vector machine (SVM), calcium dynamics fitting (CDF), one-bit compressive sensing (1-bCS), and so on. However, it remains challenging to design a globally convergent algorithm for the original model of 0/1-COP because of the nonconvex and discontinuous zero-one loss function. This paper aims to develop an inexact augmented Lagrangian method (IALM), in which the generated whole sequence converges to a local minimizer of 0/1-COP under reasonable assumptions. In the iteration process, IALM performs minimization on a Lyapunov function with an adaptively adjusted multiplier. The involved Lyapunov penalty subproblem is shown to admit the exact penalty theorem for 0/1-COP, provided that the multiplier is optimal in the sense of the proximal-type stationarity. An efficient zero-one Bregman alternating linearized minimization algorithm is also designed to achieve an approximate solution of the underlying subproblem in finite steps. Numerical experiments for handling SVM, CDF, and 1-bCS demonstrate the satisfactory performance of the proposed method in terms of solution accuracy and time efficiency. Funding: This work was supported by the Fundamental Research Funds for the Central Universities [Grant 2022YJS099] and the National Natural Science Foundation of China [Grants 12131004 and 12271022].
{"title":"Zero-One Composite Optimization: Lyapunov Exact Penalty and a Globally Convergent Inexact Augmented Lagrangian Method","authors":"Penghe Zhang, Naihua Xiu, Ziyan Luo","doi":"10.1287/moor.2021.0320","DOIUrl":"https://doi.org/10.1287/moor.2021.0320","url":null,"abstract":"We consider the problem of minimizing the sum of a smooth function and a composition of a zero-one loss function with a linear operator, namely the zero-one composite optimization problem (0/1-COP). It has a vast body of applications, including the support vector machine (SVM), calcium dynamics fitting (CDF), one-bit compressive sensing (1-bCS), and so on. However, it remains challenging to design a globally convergent algorithm for the original model of 0/1-COP because of the nonconvex and discontinuous zero-one loss function. This paper aims to develop an inexact augmented Lagrangian method (IALM), in which the generated whole sequence converges to a local minimizer of 0/1-COP under reasonable assumptions. In the iteration process, IALM performs minimization on a Lyapunov function with an adaptively adjusted multiplier. The involved Lyapunov penalty subproblem is shown to admit the exact penalty theorem for 0/1-COP, provided that the multiplier is optimal in the sense of the proximal-type stationarity. An efficient zero-one Bregman alternating linearized minimization algorithm is also designed to achieve an approximate solution of the underlying subproblem in finite steps. Numerical experiments for handling SVM, CDF, and 1-bCS demonstrate the satisfactory performance of the proposed method in terms of solution accuracy and time efficiency. Funding: This work was supported by the Fundamental Research Funds for the Central Universities [Grant 2022YJS099] and the National Natural Science Foundation of China [Grants 12131004 and 12271022].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"26 5","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138947219","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}
Calvin Beideman, Karthekeyan Chandrasekaran, Weihang Wang
We consider the problem of enumerating optimal solutions for two hypergraph k-partitioning problems, namely, Hypergraph-k-Cut and Minmax-Hypergraph-k-Partition. The input in hypergraph k-partitioning problems is a hypergraph [Formula: see text] with positive hyperedge costs along with a fixed positive integer k. The goal is to find a partition of V into k nonempty parts [Formula: see text]—known as a k-partition—so as to minimize an objective of interest. (1) If the objective of interest is the maximum cut value of the parts, then the problem is known as Minmax-Hypergraph-k-Partition. A subset of hyperedges is a minmax-k-cut-set if it is the subset of hyperedges crossing an optimum k-partition for Minmax-Hypergraph-k-Partition. (2) If the objective of interest is the total cost of hyperedges crossing the k-partition, then the problem is known as Hypergraph-k-Cut. A subset of hyperedges is a min-k-cut-set if it is the subset of hyperedges crossing an optimum k-partition for Hypergraph-k-Cut. We give the first polynomial bound on the number of minmax-k-cut-sets and a polynomial-time algorithm to enumerate all of them in hypergraphs for every fixed k. Our technique is strong enough to also enable an [Formula: see text]-time deterministic algorithm to enumerate all min-k-cut-sets in hypergraphs, thus improving on the previously known [Formula: see text]-time deterministic algorithm, in which n is the number of vertices and p is the size of the hypergraph. The correctness analysis of our enumeration approach relies on a structural result that is a strong and unifying generalization of known structural results for Hypergraph-k-Cut and Minmax-Hypergraph-k-Partition. We believe that our structural result is likely to be of independent interest in the theory of hypergraphs (and graphs).Funding: All authors were supported by NSF AF 1814613 and 1907937.
我们考虑了两个超图k-划分问题的最优解枚举问题，即hypergraph -k- cut和Minmax-Hypergraph-k-Partition。超图k-分区问题的输入是一个超图[公式:见文本]，其超边代价为正，且k为固定正整数。目标是将V划分为k个非空部分[公式:见文本]，即k分区，从而最小化感兴趣的目标。(1)如果感兴趣的目标是各部分的最大切割值，则该问题称为Minmax-Hypergraph-k-Partition。如果一个超边的子集通过一个最优的k分区，那么这个超边子集就是一个最小最大k切割集。(2)如果目标是超边穿过k分区的总代价，则该问题称为Hypergraph-k-Cut。如果超边的子集是超图-k-cut的最优k划分的超边子集，那么它就是最小k-cut集。我们给第一个多项式绑定minmax-k-cut-sets的数量和一个多项式时间算法枚举所有的超图每一个固定的k。我们的技术还足以使一个[公式:看到文本]-确定性算法,列举所有min-k-cut-sets超图,从而提高在之前所知(公式:看到文本)-确定性算法,其中n是顶点的数量和p是超图的大小。我们的枚举方法的正确性分析依赖于一个结构结果，该结果是Hypergraph-k-Cut和Minmax-Hypergraph-k-Partition已知结构结果的一个强大而统一的推广。我们相信我们的结构结果很可能在超图(和图)理论中具有独立的兴趣。所有作者均获得NSF AF 1814613和1907937的资助。
{"title":"Counting and Enumerating Optimum Cut Sets for Hypergraph k-Partitioning Problems for Fixed k","authors":"Calvin Beideman, Karthekeyan Chandrasekaran, Weihang Wang","doi":"10.1287/moor.2022.0259","DOIUrl":"https://doi.org/10.1287/moor.2022.0259","url":null,"abstract":"We consider the problem of enumerating optimal solutions for two hypergraph k-partitioning problems, namely, Hypergraph-k-Cut and Minmax-Hypergraph-k-Partition. The input in hypergraph k-partitioning problems is a hypergraph [Formula: see text] with positive hyperedge costs along with a fixed positive integer k. The goal is to find a partition of V into k nonempty parts [Formula: see text]—known as a k-partition—so as to minimize an objective of interest. (1) If the objective of interest is the maximum cut value of the parts, then the problem is known as Minmax-Hypergraph-k-Partition. A subset of hyperedges is a minmax-k-cut-set if it is the subset of hyperedges crossing an optimum k-partition for Minmax-Hypergraph-k-Partition. (2) If the objective of interest is the total cost of hyperedges crossing the k-partition, then the problem is known as Hypergraph-k-Cut. A subset of hyperedges is a min-k-cut-set if it is the subset of hyperedges crossing an optimum k-partition for Hypergraph-k-Cut. We give the first polynomial bound on the number of minmax-k-cut-sets and a polynomial-time algorithm to enumerate all of them in hypergraphs for every fixed k. Our technique is strong enough to also enable an [Formula: see text]-time deterministic algorithm to enumerate all min-k-cut-sets in hypergraphs, thus improving on the previously known [Formula: see text]-time deterministic algorithm, in which n is the number of vertices and p is the size of the hypergraph. The correctness analysis of our enumeration approach relies on a structural result that is a strong and unifying generalization of known structural results for Hypergraph-k-Cut and Minmax-Hypergraph-k-Partition. We believe that our structural result is likely to be of independent interest in the theory of hypergraphs (and graphs).Funding: All authors were supported by NSF AF 1814613 and 1907937.","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"16 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138630708","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}
Jérôme Bolte, Cyrille W. Combettes, Edouard Pauwels
The Frank–Wolfe algorithm is a popular method for minimizing a smooth convex function f over a compact convex set [Formula: see text]. Whereas many convergence results have been derived in terms of function values, almost nothing is known about the convergence behavior of the sequence of iterates [Formula: see text]. Under the usual assumptions, we design several counterexamples to the convergence of [Formula: see text], where f is d-time continuously differentiable, [Formula: see text], and [Formula: see text]. Our counterexamples cover the cases of open-loop, closed-loop, and line-search step-size strategies and work for any choice of the linear minimization oracle, thus demonstrating the fundamental pathologies in the convergence behavior of [Formula: see text].Funding: The authors acknowledge the support of the AI Interdisciplinary Institute ANITI funding through the French “Investments for the Future – PIA3” program under the Agence Nationale de la Recherche (ANR) agreement [Grant ANR-19-PI3A0004], the Air Force Office of Scientific Research, Air Force Material Command, U.S. Air Force [Grants FA866-22-1-7012 and ANR MaSDOL 19-CE23-0017-0], ANR Chess [Grant ANR-17-EURE-0010], ANR Regulia, and Centre Lagrange.
弗兰克-沃尔夫算法是在紧凑凸集上最小化光滑凸函数 f 的常用方法[公式：见正文]。虽然许多收敛结果都是根据函数值推导出来的，但对迭代序列的收敛行为几乎一无所知[公式：见正文]。在通常的假设条件下，我们设计了几个反例来证明 f 为 d 时连续可微分的[公式：见正文]、[公式：见正文]和[公式：见正文]的收敛性。我们的反例涵盖了开环、闭环和线性搜索步长策略的情况，并且适用于任何线性最小化神谕的选择，从而证明了[公式：见正文]收敛行为的基本病理：作者感谢人工智能跨学科研究所ANITI通过法国国家研究署（ANR）协议下的 "未来投资-PIA3 "计划[赠款ANR-19-PI3A0004]、美国空军材料司令部空军科学研究办公室[赠款FA866-22-1-7012和ANR MaSDOL 19-CE23-0017-0]、ANR国际象棋[赠款ANR-17-EURE-0010]、ANR Regulia和拉格朗日中心提供的资助。
{"title":"The Iterates of the Frank–Wolfe Algorithm May Not Converge","authors":"Jérôme Bolte, Cyrille W. Combettes, Edouard Pauwels","doi":"10.1287/moor.2022.0057","DOIUrl":"https://doi.org/10.1287/moor.2022.0057","url":null,"abstract":"The Frank–Wolfe algorithm is a popular method for minimizing a smooth convex function f over a compact convex set [Formula: see text]. Whereas many convergence results have been derived in terms of function values, almost nothing is known about the convergence behavior of the sequence of iterates [Formula: see text]. Under the usual assumptions, we design several counterexamples to the convergence of [Formula: see text], where f is d-time continuously differentiable, [Formula: see text], and [Formula: see text]. Our counterexamples cover the cases of open-loop, closed-loop, and line-search step-size strategies and work for any choice of the linear minimization oracle, thus demonstrating the fundamental pathologies in the convergence behavior of [Formula: see text].Funding: The authors acknowledge the support of the AI Interdisciplinary Institute ANITI funding through the French “Investments for the Future – PIA3” program under the Agence Nationale de la Recherche (ANR) agreement [Grant ANR-19-PI3A0004], the Air Force Office of Scientific Research, Air Force Material Command, U.S. Air Force [Grants FA866-22-1-7012 and ANR MaSDOL 19-CE23-0017-0], ANR Chess [Grant ANR-17-EURE-0010], ANR Regulia, and Centre Lagrange.","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"4 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138563425","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the convergence of accelerated stochastic gradient descent (SGD) for strongly convex objectives under the growth condition, which states that the variance of stochastic gradient is bounded by a multiplicative part that grows with the full gradient and a constant additive part. Through the lens of the growth condition, we investigate four widely used accelerated methods: Nesterov’s accelerated method (NAM), robust momentum method (RMM), accelerated dual averaging method (DAM+), and implicit DAM+ (iDAM+). Although these methods are known to improve the convergence rate of SGD under the condition that the stochastic gradient has bounded variance, it is not well understood how their convergence rates are affected by the multiplicative noise. In this paper, we show that these methods all converge to a neighborhood of the optimum with accelerated convergence rates (compared with SGD), even under the growth condition. In particular, NAM, RMM, and iDAM+ enjoy acceleration only with a mild multiplicative noise, whereas DAM+ enjoys acceleration, even with a large multiplicative noise. Furthermore, we propose a generic tail-averaged scheme that allows the accelerated rates of DAM+ and iDAM+ to nearly attain the theoretical lower bound (up to a logarithmic factor in the variance term). We conduct numerical experiments to support our theoretical conclusions.
{"title":"Convergence Analysis of Accelerated Stochastic Gradient Descent Under the Growth Condition","authors":"You-Lin Chen, Sen Na, Mladen Kolar","doi":"10.1287/moor.2021.0293","DOIUrl":"https://doi.org/10.1287/moor.2021.0293","url":null,"abstract":"We study the convergence of accelerated stochastic gradient descent (SGD) for strongly convex objectives under the growth condition, which states that the variance of stochastic gradient is bounded by a multiplicative part that grows with the full gradient and a constant additive part. Through the lens of the growth condition, we investigate four widely used accelerated methods: Nesterov’s accelerated method (NAM), robust momentum method (RMM), accelerated dual averaging method (DAM+), and implicit DAM+ (iDAM+). Although these methods are known to improve the convergence rate of SGD under the condition that the stochastic gradient has bounded variance, it is not well understood how their convergence rates are affected by the multiplicative noise. In this paper, we show that these methods all converge to a neighborhood of the optimum with accelerated convergence rates (compared with SGD), even under the growth condition. In particular, NAM, RMM, and iDAM+ enjoy acceleration only with a mild multiplicative noise, whereas DAM+ enjoys acceleration, even with a large multiplicative noise. Furthermore, we propose a generic tail-averaged scheme that allows the accelerated rates of DAM+ and iDAM+ to nearly attain the theoretical lower bound (up to a logarithmic factor in the variance term). We conduct numerical experiments to support our theoretical conclusions.","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"24 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138580818","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the convergence problem for mean field games with common noise and controlled volatility. We adopt the strategy recently put forth by Laurière and the second author, using the maximum principle to recast the convergence problem as a question of “forward-backward propagation of chaos” (i.e., (conditional) propagation of chaos for systems of particles evolving forward and backward in time). Our main results show that displacement monotonicity can be used to obtain this propagation of chaos, which leads to quantitative convergence results for open-loop Nash equilibria for a class of mean field games. Our results seem to be the first (quantitative or qualitative) that apply to games in which the common noise is controlled. The proofs are relatively simple and rely on a well-known technique for proving wellposedness of forward-backward stochastic differential equations, which is combined with displacement monotonicity in a novel way. To demonstrate the flexibility of the approach, we also use the same arguments to obtain convergence results for a class of infinite horizon discounted mean field games.Funding: J. Jackson is supported by the National Science Foundation [Grant DGE1610403]. L. Tangpi is partially supported by the National Science Foundation [Grants DMS-2005832 and DMS-2143861].
{"title":"Quantitative Convergence for Displacement Monotone Mean Field Games with Controlled Volatility","authors":"Joe Jackson, Ludovic Tangpi","doi":"10.1287/moor.2023.0106","DOIUrl":"https://doi.org/10.1287/moor.2023.0106","url":null,"abstract":"We study the convergence problem for mean field games with common noise and controlled volatility. We adopt the strategy recently put forth by Laurière and the second author, using the maximum principle to recast the convergence problem as a question of “forward-backward propagation of chaos” (i.e., (conditional) propagation of chaos for systems of particles evolving forward and backward in time). Our main results show that displacement monotonicity can be used to obtain this propagation of chaos, which leads to quantitative convergence results for open-loop Nash equilibria for a class of mean field games. Our results seem to be the first (quantitative or qualitative) that apply to games in which the common noise is controlled. The proofs are relatively simple and rely on a well-known technique for proving wellposedness of forward-backward stochastic differential equations, which is combined with displacement monotonicity in a novel way. To demonstrate the flexibility of the approach, we also use the same arguments to obtain convergence results for a class of infinite horizon discounted mean field games.Funding: J. Jackson is supported by the National Science Foundation [Grant DGE1610403]. L. Tangpi is partially supported by the National Science Foundation [Grants DMS-2005832 and DMS-2143861].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"17 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138580879","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 provide a framework to analyze control policies for the restless Markovian bandit model under both finite and infinite time horizons. We show that when the population of arms goes to infinity, the value of the optimal control policy converges to the solution of a linear program (LP). We provide necessary and sufficient conditions for a generic control policy to be (i) asymptotically optimal, (ii) asymptotically optimal with square root convergence rate, and (iii) asymptotically optimal with exponential rate. We then construct the LP-index policy that is asymptotically optimal with square root convergence rate on all models and with exponential rate if the model is nondegenerate in finite horizon and satisfies a uniform global attractor property in infinite horizon. We next define the LP-update policy, which is essentially a repeated LP-index policy that solves a new LP at each decision epoch. We conclude by providing numerical experiments to compare the efficiency of different LP-based policies.Funding: This work was supported by Agence Nationale de la Recherche [Grant ANR-19-CE23-0015].
我们提供了一个框架来分析不宁马尔可夫强盗模型在有限和无限时间范围下的控制策略。我们证明了当武器数量趋于无穷时，最优控制策略的值收敛于线性规划(LP)的解。给出了一般控制策略(i)渐近最优，(ii)以平方根收敛速率渐近最优，(iii)以指数速率渐近最优的充要条件。然后构造了在所有模型上具有平方根收敛率的渐近最优的lp -指数策略，当模型在有限视界上非退化且在无限视界上满足一致全局吸引子性质时具有指数收敛率。接下来我们定义LP更新策略，它本质上是一个重复的LP索引策略，在每个决策时期解决一个新的LP。最后，我们通过提供数值实验来比较不同基于lp的策略的效率。本研究由法国国家研究机构(Agence Nationale de la Recherche)资助[Grant ANR-19-CE23-0015]。
{"title":"Linear Program-Based Policies for Restless Bandits: Necessary and Sufficient Conditions for (Exponentially Fast) Asymptotic Optimality","authors":"Nicolas Gast, Bruno Gaujal, Chen Yan","doi":"10.1287/moor.2022.0101","DOIUrl":"https://doi.org/10.1287/moor.2022.0101","url":null,"abstract":"We provide a framework to analyze control policies for the restless Markovian bandit model under both finite and infinite time horizons. We show that when the population of arms goes to infinity, the value of the optimal control policy converges to the solution of a linear program (LP). We provide necessary and sufficient conditions for a generic control policy to be (i) asymptotically optimal, (ii) asymptotically optimal with square root convergence rate, and (iii) asymptotically optimal with exponential rate. We then construct the LP-index policy that is asymptotically optimal with square root convergence rate on all models and with exponential rate if the model is nondegenerate in finite horizon and satisfies a uniform global attractor property in infinite horizon. We next define the LP-update policy, which is essentially a repeated LP-index policy that solves a new LP at each decision epoch. We conclude by providing numerical experiments to compare the efficiency of different LP-based policies.Funding: This work was supported by Agence Nationale de la Recherche [Grant ANR-19-CE23-0015].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"232 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138508635","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}
Marcel Celaya, Stefan Kuhlmann, Joseph Paat, Robert Weismantel
This paper deals with linear integer optimization. We develop a technique that can be applied to provide improved upper bounds for two important questions in linear integer optimization. Given an optimal vertex solution for the linear relaxation, how far away is the nearest optimal integer solution (if one exists; proximity bounds)? If a polyhedron contains no integer point, what is the smallest number of integer parallel hyperplanes defined by an integral, nonzero, normal vector that intersect the polyhedron (flatness bounds)? This paper presents a link between these two questions by refining a proof technique that has been recently introduced by the authors. A key technical lemma underlying our technique concerns the areas of certain convex polygons in the plane; if a polygon [Formula: see text] satisfies [Formula: see text], where τ denotes [Formula: see text] counterclockwise rotation and [Formula: see text] denotes the polar of K, then the area of [Formula: see text] is at least three.Funding: J. Paat was supported by the Natural Sciences and Engineering Research Council of Canada [Grant RGPIN-2021-02475]. R. Weismantel was supported by the Einstein Stiftung Berlin.
{"title":"Proximity and Flatness Bounds for Linear Integer Optimization","authors":"Marcel Celaya, Stefan Kuhlmann, Joseph Paat, Robert Weismantel","doi":"10.1287/moor.2022.0335","DOIUrl":"https://doi.org/10.1287/moor.2022.0335","url":null,"abstract":"This paper deals with linear integer optimization. We develop a technique that can be applied to provide improved upper bounds for two important questions in linear integer optimization. Given an optimal vertex solution for the linear relaxation, how far away is the nearest optimal integer solution (if one exists; proximity bounds)? If a polyhedron contains no integer point, what is the smallest number of integer parallel hyperplanes defined by an integral, nonzero, normal vector that intersect the polyhedron (flatness bounds)? This paper presents a link between these two questions by refining a proof technique that has been recently introduced by the authors. A key technical lemma underlying our technique concerns the areas of certain convex polygons in the plane; if a polygon [Formula: see text] satisfies [Formula: see text], where τ denotes [Formula: see text] counterclockwise rotation and [Formula: see text] denotes the polar of K, then the area of [Formula: see text] is at least three.Funding: J. Paat was supported by the Natural Sciences and Engineering Research Council of Canada [Grant RGPIN-2021-02475]. R. Weismantel was supported by the Einstein Stiftung Berlin.","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"225 2","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138508674","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 problem of fairly allocating a set of indivisible goods to a set of strategic agents with additive valuation functions. We assume no monetary transfers, and therefore, a mechanism in our setting is an algorithm that takes as input the reported—rather than the true—values of the agents. Our main goal is to explore whether there exist mechanisms that have pure Nash equilibria for every instance and, at the same time, provide fairness guarantees for the allocations that correspond to these equilibria. We focus on two relaxations of envy-freeness, namely, envy-freeness up to one good (EF1) and envy-freeness up to any good (EFX), and we positively answer the preceding question. In particular, we study two algorithms that are known to produce such allocations in the nonstrategic setting: round-robin (EF1 allocations for any number of agents) and a cut-and-choose algorithm of Plaut and Roughgarden (EFX allocations for two agents). For round-robin, we show that all of its pure Nash equilibria induce allocations that are EF1 with respect to the underlying true values, whereas for the algorithm of Plaut and Roughgarden, we show that the corresponding allocations not only are EFX, but also satisfy maximin share fairness, something that is not true for this algorithm in the nonstrategic setting! Further, we show that a weaker version of the latter result holds for any mechanism for two agents that always has pure Nash equilibria, which all induce EFX allocations.Funding: This work was supported by the Horizon 2020 European Research Council Advanced “Algorithmic and Mechanism Design Research in Online Markets” [Grant 788893], the Ministero dell’Università e della Ricerca Research project of national interest (PRIN) “Algorithms, Games, and Digital Markets,” the Future Artificial Intelligence Research project funded by the NextGenerationEU program within the National Recovery and Resilience Plan (PNRR-PE-AI) scheme [M4C2, investment 1.3, line on Artificial Intelligence], the National Recovery and Resilience Plan-Ministero dell’Università e della Ricerca (PNRR-MUR) project IR0000013-SoBigData.it, the Nederlandse Organisatie voor Wetenschappelijk Onderzoek Veni Project [Grant VI.Veni.192.153], and the National Recovery and Resilience Plan Greece 2.0 funded by the European Union under the NextGenerationEU Program [Grant MIS 5154714].
研究了一组不可分割的商品公平分配给一组具有可加性估价函数的战略代理的问题。我们假设没有货币转移，因此，我们设置中的机制是一种算法，它将代理的报告值(而不是真实值)作为输入。我们的主要目标是探索是否存在一种机制，在每种情况下都有纯纳什均衡，同时，为这些均衡对应的分配提供公平保证。我们着重讨论了嫉妒自由的两种松弛状态，即对一种善的嫉妒自由(EF1)和对任何善的嫉妒自由(EFX)，并积极地回答了前面的问题。特别是，我们研究了两种已知在非策略设置中产生这种分配的算法:round-robin(对任意数量的代理进行EF1分配)和Plaut和Roughgarden的cut-and-choose算法(对两个代理进行EFX分配)。对于round-robin，我们表明其所有的纯纳什均衡诱导分配是EF1相对于潜在的真值，而对于Plaut和Roughgarden的算法，我们表明，相应的分配不仅是EFX，而且还满足最大份额公平，这在非战略设置的算法中是不正确的!进一步，我们证明了后一种结果的较弱版本适用于总是具有纯纳什均衡的两个代理的任何机制，它们都诱导EFX分配。资助:这项工作得到了地平线2020欧洲研究委员会高级“在线市场的算法和机制设计研究”[Grant 788893]、墨西哥大学部长国家利益研究项目(PRIN)“算法、游戏和数字市场”的支持。未来人工智能研究项目由国家恢复和恢复计划(PNRR-PE-AI)计划[M4C2，投资1.3，人工智能线]下的下一代欧盟计划资助，国家恢复和恢复计划-墨西哥德拉里卡多大学部长(PNRR-MUR)项目IR0000013-SoBigData。它，荷兰组织voor Wetenschappelijk Onderzoek Veni项目[Grant VI.Veni.192.153]，以及欧盟在下一代欧盟计划下资助的希腊国家恢复和弹性计划2.0 [Grant MIS 5154714]。
{"title":"Allocating Indivisible Goods to Strategic Agents: Pure Nash Equilibria and Fairness","authors":"Georgios Amanatidis, Georgios Birmpas, Federico Fusco, Philip Lazos, Stefano Leonardi, Rebecca Reiffenhäuser","doi":"10.1287/moor.2022.0058","DOIUrl":"https://doi.org/10.1287/moor.2022.0058","url":null,"abstract":"We consider the problem of fairly allocating a set of indivisible goods to a set of strategic agents with additive valuation functions. We assume no monetary transfers, and therefore, a mechanism in our setting is an algorithm that takes as input the reported—rather than the true—values of the agents. Our main goal is to explore whether there exist mechanisms that have pure Nash equilibria for every instance and, at the same time, provide fairness guarantees for the allocations that correspond to these equilibria. We focus on two relaxations of envy-freeness, namely, envy-freeness up to one good (EF1) and envy-freeness up to any good (EFX), and we positively answer the preceding question. In particular, we study two algorithms that are known to produce such allocations in the nonstrategic setting: round-robin (EF1 allocations for any number of agents) and a cut-and-choose algorithm of Plaut and Roughgarden (EFX allocations for two agents). For round-robin, we show that all of its pure Nash equilibria induce allocations that are EF1 with respect to the underlying true values, whereas for the algorithm of Plaut and Roughgarden, we show that the corresponding allocations not only are EFX, but also satisfy maximin share fairness, something that is not true for this algorithm in the nonstrategic setting! Further, we show that a weaker version of the latter result holds for any mechanism for two agents that always has pure Nash equilibria, which all induce EFX allocations.Funding: This work was supported by the Horizon 2020 European Research Council Advanced “Algorithmic and Mechanism Design Research in Online Markets” [Grant 788893], the Ministero dell’Università e della Ricerca Research project of national interest (PRIN) “Algorithms, Games, and Digital Markets,” the Future Artificial Intelligence Research project funded by the NextGenerationEU program within the National Recovery and Resilience Plan (PNRR-PE-AI) scheme [M4C2, investment 1.3, line on Artificial Intelligence], the National Recovery and Resilience Plan-Ministero dell’Università e della Ricerca (PNRR-MUR) project IR0000013-SoBigData.it, the Nederlandse Organisatie voor Wetenschappelijk Onderzoek Veni Project [Grant VI.Veni.192.153], and the National Recovery and Resilience Plan Greece 2.0 funded by the European Union under the NextGenerationEU Program [Grant MIS 5154714].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"228 5","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138508652","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the impact of nonconvexity on the complexity of nonsmooth optimization, emphasizing objectives such as piecewise linear functions, which may not be weakly convex. We focus on a dimension-independent analysis, slightly modifying a 2020 black-box algorithm of Zhang-Lin-Jegelka-Sra-Jadbabaie that approximates an ϵ-stationary point of any directionally differentiable Lipschitz objective using [Formula: see text] calls to a specialized subgradient oracle and a randomized line search. Seeking by contrast a deterministic method, we present a simple black-box version that achieves [Formula: see text] for any difference-of-convex objective and [Formula: see text] for the weakly convex case. Our complexity bound depends on a natural nonconvexity modulus that is related, intriguingly, to the negative part of directional second derivatives of the objective, understood in the distributional sense.Funding: This work was supported by the National Science Foundation [Grant DMS-2006990].
{"title":"The Cost of Nonconvexity in Deterministic Nonsmooth Optimization","authors":"Siyu Kong, A. S. Lewis","doi":"10.1287/moor.2022.0289","DOIUrl":"https://doi.org/10.1287/moor.2022.0289","url":null,"abstract":"We study the impact of nonconvexity on the complexity of nonsmooth optimization, emphasizing objectives such as piecewise linear functions, which may not be weakly convex. We focus on a dimension-independent analysis, slightly modifying a 2020 black-box algorithm of Zhang-Lin-Jegelka-Sra-Jadbabaie that approximates an ϵ-stationary point of any directionally differentiable Lipschitz objective using [Formula: see text] calls to a specialized subgradient oracle and a randomized line search. Seeking by contrast a deterministic method, we present a simple black-box version that achieves [Formula: see text] for any difference-of-convex objective and [Formula: see text] for the weakly convex case. Our complexity bound depends on a natural nonconvexity modulus that is related, intriguingly, to the negative part of directional second derivatives of the objective, understood in the distributional sense.Funding: This work was supported by the National Science Foundation [Grant DMS-2006990].","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"231 5","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138508637","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}
Interior point methods (IPMs) that handle nonconvex constraints such as IPOPT, KNITRO and LOQO have had enormous practical success. We consider IPMs in the setting where the objective and constraints are thrice differentiable, and have Lipschitz first and second derivatives on the feasible region. We provide an IPM that, starting from a strictly feasible point, finds a μ-approximate Fritz John point by solving [Formula: see text] trust-region subproblems. For IPMs that handle nonlinear constraints, this result represents the first iteration bound with a polynomial dependence on [Formula: see text]. We also show how to use our method to find scaled-KKT points starting from an infeasible solution and improve on existing complexity bounds.Funding: This work was supported by Air Force Office of Scientific Research [9550-23-1-0242]. A significant portion of this work was done at Stanford where O. Hinder was supported by the PACCAR, Inc., Stanford Graduate Fellowship and the Dantzig-Lieberman fellowship.
{"title":"Worst-Case Iteration Bounds for Log Barrier Methods on Problems with Nonconvex Constraints","authors":"Oliver Hinder, Yinyu Ye","doi":"10.1287/moor.2020.0274","DOIUrl":"https://doi.org/10.1287/moor.2020.0274","url":null,"abstract":"Interior point methods (IPMs) that handle nonconvex constraints such as IPOPT, KNITRO and LOQO have had enormous practical success. We consider IPMs in the setting where the objective and constraints are thrice differentiable, and have Lipschitz first and second derivatives on the feasible region. We provide an IPM that, starting from a strictly feasible point, finds a μ-approximate Fritz John point by solving [Formula: see text] trust-region subproblems. For IPMs that handle nonlinear constraints, this result represents the first iteration bound with a polynomial dependence on [Formula: see text]. We also show how to use our method to find scaled-KKT points starting from an infeasible solution and improve on existing complexity bounds.Funding: This work was supported by Air Force Office of Scientific Research [9550-23-1-0242]. A significant portion of this work was done at Stanford where O. Hinder was supported by the PACCAR, Inc., Stanford Graduate Fellowship and the Dantzig-Lieberman fellowship.","PeriodicalId":49852,"journal":{"name":"Mathematics of Operations Research","volume":"231 6","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138508636","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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