Mathematical Programming最新文献_第10页

A 2-approximation for the bounded treewidth sparsest cut problem in $$textsf{FPT}$$ Time $$textsf{FPT}$$时间内有界树宽稀疏切割问题的2次近似值

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-01-04 DOI: 10.1007/s10107-023-02044-1

Vincent Cohen-Addad, Tobias Mömke, Victor Verdugo

{"title":"A 2-approximation for the bounded treewidth sparsest cut problem in $$textsf{FPT}$$ Time","authors":"Vincent Cohen-Addad, Tobias Mömke, Victor Verdugo","doi":"10.1007/s10107-023-02044-1","DOIUrl":"https://doi.org/10.1007/s10107-023-02044-1","url":null,"abstract":"In the non-uniform sparsest cut problem, we are given a supply graph G and a demand graph D, both with the same set of nodes V. The goal is to find a cut of V that minimizes the ratio of the total capacity on the edges of G crossing the cut over the total demand of the crossing edges of D. In this work, we study the non-uniform sparsest cut problem for supply graphs with bounded treewidth k. For this case, Gupta et al. (ACM STOC, 2013) obtained a 2-approximation with polynomial running time for fixed k, and it remained open the question of whether there exists a c-approximation algorithm for a constant c independent of k, that runs in (textsf{FPT}) time. We answer this question in the affirmative. We design a 2-approximation algorithm for the non-uniform sparsest cut with bounded treewidth supply graphs that runs in (textsf{FPT}) time, when parameterized by the treewidth. Our algorithm is based on rounding the optimal solution of a linear programming relaxation inspired by the Sherali-Adams hierarchy. In contrast to the classic Sherali-Adams approach, we construct a relaxation driven by a tree decomposition of the supply graph by including a carefully chosen set of lifting variables and constraints to encode information of subsets of nodes with super-constant size, and at the same time we have a sufficiently small linear program that can be solved in (textsf{FPT}) time.\u0000","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"21 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139105285","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

FPT algorithms for a special block-structured integer program with applications in scheduling 一种特殊块结构整数程序的 FPT 算法及其在调度中的应用

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-01-04 DOI: 10.1007/s10107-023-02046-z

{"title":"FPT algorithms for a special block-structured integer program with applications in scheduling","authors":"","doi":"10.1007/s10107-023-02046-z","DOIUrl":"https://doi.org/10.1007/s10107-023-02046-z","url":null,"abstract":"<h3>Abstract</h3> In this paper, a special case of the generalized 4-block n-fold IPs is investigated, where (B_i=B) and B has a rank at most 1. Such IPs, called almost combinatorial 4-block n-fold IPs, include the generalized n-fold IPs as a subcase. We are interested in fixed parameter tractable (FPT) algorithms by taking as parameters the dimensions of the blocks and the largest coefficient. For almost combinatorial 4-block n-fold IPs, we first show that there exists some (lambda le g(gamma )) such that for any nonzero kernel element ({textbf{g}}) , (lambda {textbf{g}}) can always be decomposed into kernel elements in the same orthant whose (ell _{infty }) -norm is bounded by (g(gamma )) (while ({textbf{g}}) itself might not admit such a decomposition), where g is a computable function and (gamma ) is an upper bound on the dimensions of the blocks and the largest coefficient. Based on this, we are able to bound the (ell _{infty }) -norm of Graver basis elements by ({mathcal {O}}(g(gamma )n)) and develop an ({mathcal {O}}(g(gamma )n^{3+o(1)}hat{L}^2)) -time algorithm (here (hat{L}) denotes the logarithm of the largest absolute value occurring in the input). Additionally, we show that the (ell _{infty }) -norm of Graver basis elements is (varOmega (n)) . As applications, almost combinatorial 4-block n-fold IPs can be used to model generalizations of classical problems, including scheduling with rejection, bi-criteria scheduling, and a generalized delivery problem. Therefore, our FPT algorithm establishes a general framework to settle these problems.","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"34 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139105010","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

High-order methods beyond the classical complexity bounds: inexact high-order proximal-point methods 超越经典复杂性界限的高阶方法：非精确高阶近点法

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-01-04 DOI: 10.1007/s10107-023-02041-4

Masoud Ahookhosh, Yurii Nesterov

{"title":"High-order methods beyond the classical complexity bounds: inexact high-order proximal-point methods","authors":"Masoud Ahookhosh, Yurii Nesterov","doi":"10.1007/s10107-023-02041-4","DOIUrl":"https://doi.org/10.1007/s10107-023-02041-4","url":null,"abstract":"We introduce a Bi-level OPTimization (BiOPT) framework for minimizing the sum of two convex functions, where one of them is smooth enough. The BiOPT framework offers three levels of freedom: (i) choosing the order p of the proximal term; (ii) designing an inexact pth-order proximal-point method in the upper level; (iii) solving the auxiliary problem with a lower-level non-Euclidean method in the lower level. We here regularize the objective by a ((p+1))th-order proximal term (for arbitrary integer (pge 1)) and then develop the generic inexact high-order proximal-point scheme and its acceleration using the standard estimating sequence technique at the upper level. This follows at the lower level with solving the corresponding pth-order proximal auxiliary problem inexactly either by one iteration of the pth-order tensor method or by a lower-order non-Euclidean composite gradient scheme. Ultimately, it is shown that applying the accelerated inexact pth-order proximal-point method at the upper level and handling the auxiliary problem by the non-Euclidean composite gradient scheme lead to a 2q-order method with the convergence rate ({mathcal {O}}(k^{-(p+1)})) (for (q=lfloor p/2rfloor ) and the iteration counter k), which can result to a superfast method for some specific class of problems.\u0000","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"65 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139105015","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Asymmetry in the complexity of the multi-commodity network pricing problem 多商品网络定价问题复杂性的不对称性

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-01-03 DOI: 10.1007/s10107-023-02043-2

Quang Minh Bui, Margarida Carvalho, José Neto

引用次数: 0

Gradient regularization of Newton method with Bregman distances. 具有Bregman距离的牛顿法梯度正则化

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-01-01 Epub Date: 2023-03-24 DOI: 10.1007/s10107-023-01943-7

Nikita Doikov, Yurii Nesterov

{"title":"Gradient regularization of Newton method with Bregman distances.","authors":"Nikita Doikov, Yurii Nesterov","doi":"10.1007/s10107-023-01943-7","DOIUrl":"10.1007/s10107-023-01943-7","url":null,"abstract":"In this paper, we propose a first second-order scheme based on arbitrary non-Euclidean norms, incorporated by Bregman distances. They are introduced directly in the Newton iterate with regularization parameter proportional to the square root of the norm of the current gradient. For the basic scheme, as applied to the composite convex optimization problem, we establish the global convergence rate of the order <math><mrow><mi>O</mi><mo>(</mo><msup><mi>k</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup><mo>)</mo></mrow></math> both in terms of the functional residual and in the norm of subgradients. Our main assumption on the smooth part of the objective is Lipschitz continuity of its Hessian. For uniformly convex functions of degree three, we justify global linear rate, and for strongly convex function we prove the local superlinear rate of convergence. Our approach can be seen as a relaxation of the Cubic Regularization of the Newton method (Nesterov and Polyak in Math Program 108(1):177-205, 2006) for convex minimization problems. This relaxation preserves the convergence properties and global complexities of the Cubic Newton in convex case, while the auxiliary subproblem at each iteration is simpler. We equip our method with adaptive search procedure for choosing the regularization parameter. We propose also an accelerated scheme with convergence rate <math><mrow><mi>O</mi><mo>(</mo><msup><mi>k</mi><mrow><mo>-</mo><mn>3</mn></mrow></msup><mo>)</mo></mrow></math>, where k is the iteration counter.","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"1 1","pages":"1-25"},"PeriodicalIF":2.7,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10869408/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47300576","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

First order asymptotics of the sample average approximation method to solve risk averse stochastic programs 解决风险厌恶随机程序的样本平均近似法的一阶渐近性

IF 2.7 2区数学

Mathematical Programming Pub Date : 2023-12-26 DOI: 10.1007/s10107-023-02036-1

Volker Krätschmer

{"title":"First order asymptotics of the sample average approximation method to solve risk averse stochastic programs","authors":"Volker Krätschmer","doi":"10.1007/s10107-023-02036-1","DOIUrl":"https://doi.org/10.1007/s10107-023-02036-1","url":null,"abstract":"We investigate statistical properties of the optimal value of the Sample Average Approximation of stochastic programs, continuing the study (Krätschmer in Nonasymptotic upper estimates for errors of the sample average approximation method to solve risk averse stochastic programs, 2023. Forthcoming in SIAM J. Optim.). Central Limit Theorem type results are derived for the optimal value. As a crucial point the investigations are based on a new type of conditions from the theory of empirical processes which do not rely on pathwise analytical properties of the goal functions. In particular, continuity or convexity in the parameter is not imposed in advance as usual in the literature on the Sample Average Approximation method. It is also shown that the new condition is satisfied if the paths of the goal functions are Hölder continuous so that the main results carry over in this case. Moreover, the main results are applied to goal functions whose paths are piecewise Hölder continuous as e.g. in two stage mixed-integer programs. The main results are shown for classical risk neutral stochastic programs, but we also demonstrate how to apply them to the Sample Average Approximation of risk averse stochastic programs. In this respect we consider stochastic programs expressed in terms of absolute semideviations and divergence risk measures.","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"15 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2023-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139052819","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

The exact worst-case convergence rate of the alternating direction method of multipliers 交替方向乘法的精确最坏收敛率

IF 2.7 2区数学

Mathematical Programming Pub Date : 2023-12-26 DOI: 10.1007/s10107-023-02037-0

Moslem Zamani, Hadi Abbaszadehpeivasti, Etienne de Klerk

引用次数: 0

Beyond symmetry: best submatrix selection for the sparse truncated SVD 超越对称：稀疏截断 SVD 的最佳子矩阵选择

IF 2.7 2区数学

Mathematical Programming Pub Date : 2023-12-21 DOI: 10.1007/s10107-023-02030-7

{"title":"Beyond symmetry: best submatrix selection for the sparse truncated SVD","authors":"","doi":"10.1007/s10107-023-02030-7","DOIUrl":"https://doi.org/10.1007/s10107-023-02030-7","url":null,"abstract":"<h3>Abstract</h3> The truncated singular value decomposition (SVD), also known as the best low-rank matrix approximation with minimum error measured by a unitarily invariant norm, has been applied to many domains such as biology, healthcare, among others, where high-dimensional datasets are prevalent. To extract interpretable information from the high-dimensional data, sparse truncated SVD (SSVD) has been used to select a handful of rows and columns of the original matrix along with the best low-rank approximation. Different from the literature on SSVD focusing on the top singular value or compromising the sparsity for the seek of computational efficiency, this paper presents a novel SSVD formulation that can select the best submatrix precisely up to a given size to maximize its truncated Ky Fan norm. The fact that the proposed SSVD problem is NP-hard motivates us to study effective algorithms with provable performance guarantees. To do so, we first reformulate SSVD as a mixed-integer semidefinite program, which can be solved exactly for small- or medium-sized instances within a branch-and-cut algorithm framework with closed-form cuts and is extremely useful for evaluating the solution quality of approximation algorithms. We next develop three selection algorithms based on different selection criteria and two searching algorithms, greedy and local search. We prove the approximation ratios for all the approximation algorithms and show that all the ratios are tight when the number of rows or columns of the selected submatrix is no larger than half of the data matrix, i.e., our derived approximation ratios are unimprovable. Our numerical study demonstrates the high solution quality and computational efficiency of the proposed algorithms. Finally, all our analysis can be extended to row-sparse PCA.","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"13 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2023-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138825723","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Intersecting and dense restrictions of clutters in polynomial time 在多项式时间内对杂乱无章的相交和密集进行限制

IF 2.7 2区数学

Mathematical Programming Pub Date : 2023-12-18 DOI: 10.1007/s10107-023-02034-3

Martin Drees

引用次数: 0

Hardness and approximation of submodular minimum linear ordering problems 亚模态最小线性排序问题的难度和近似性

IF 2.7 2区数学

Mathematical Programming Pub Date : 2023-12-14 DOI: 10.1007/s10107-023-02038-z

Majid Farhadi, Swati Gupta, Shengding Sun, Prasad Tetali, Michael C. Wigal

{"title":"Hardness and approximation of submodular minimum linear ordering problems","authors":"Majid Farhadi, Swati Gupta, Shengding Sun, Prasad Tetali, Michael C. Wigal","doi":"10.1007/s10107-023-02038-z","DOIUrl":"https://doi.org/10.1007/s10107-023-02038-z","url":null,"abstract":"The minimum linear ordering problem (MLOP) generalizes well-known combinatorial optimization problems such as minimum linear arrangement and minimum sum set cover. MLOP seeks to minimize an aggregated cost (f(cdot )) due to an ordering (sigma ) of the items (say [n]), i.e., (min _{sigma } sum _{iin [n]} f(E_{i,sigma })), where (E_{i,sigma }) is the set of items mapped by (sigma ) to indices [i]. Despite an extensive literature on MLOP variants and approximations for these, it was unclear whether the graphic matroid MLOP was NP-hard. We settle this question through non-trivial reductions from mininimum latency vertex cover and minimum sum vertex cover problems. We further propose a new combinatorial algorithm for approximating monotone submodular MLOP, using the theory of principal partitions. This is in contrast to the rounding algorithm by Iwata et al. (in: APPROX, 2012), using Lovász extension of submodular functions. We show a ((2-frac{1+ell _{f}}{1+|E|}))-approximation for monotone submodular MLOP where (ell _{f}=frac{f(E)}{max _{xin E}f({x})}) satisfies (1 le ell _f le |E|). Our theory provides new approximation bounds for special cases of the problem, in particular a ((2-frac{1+r(E)}{1+|E|}))-approximation for the matroid MLOP, where (f = r) is the rank function of a matroid. We further show that minimum latency vertex cover is (frac{4}{3})-approximable, by which we also lower bound the integrality gap of its natural LP relaxation, which might be of independent interest.","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"200 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2023-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138690214","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0