Mathematical Programming最新文献_第10页

Counterexample and an additional revealing poll step for a result of “analysis of direct searches for discontinuous functions” 反例和 "不连续函数直接搜索分析 "结果的额外揭示性投票步骤

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-01-08 DOI: 10.1007/s10107-023-02042-3

{"title":"Counterexample and an additional revealing poll step for a result of “analysis of direct searches for discontinuous functions”","authors":"","doi":"10.1007/s10107-023-02042-3","DOIUrl":"https://doi.org/10.1007/s10107-023-02042-3","url":null,"abstract":"<h3>Abstract</h3> This note provides a counterexample to a theorem announced in the last part of the paper (Vicente and Custódio Math Program 133:299–325, 2012). The counterexample involves an objective function (f: mathbb {R}rightarrow mathbb {R}) which satisfies all the assumptions required by the theorem but contradicts some of its conclusions. A corollary of this theorem is also affected by this counterexample. The main flaw revealed by the counterexample is the possibility that a directional direct search method (dDSM) generates a sequence of trial points ((x_k)_{k in mathbb {N}}) converging to a point (x_*) where f is discontinuous, lower semicontinuous and whose objective function value (f(x_*)) is strictly less than (lim _{krightarrow infty } f(x_k)) . Moreover the dDSM generates trial points in only one of the continuity sets of f near (x_*) . This note also investigates the proof of the theorem to highlight the inexact statements in the original paper. Finally this work introduces a modification of the dDSM that allows, in usual cases, to recover the properties broken by the counterexample. ","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"1 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2024-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139410419","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

No dimension-free deterministic algorithm computes approximate stationarities of Lipschitzians 没有一种无维度确定性算法能计算 Lipschitzians 的近似静止性

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-01-06 DOI: 10.1007/s10107-023-02031-6

Lai Tian, Anthony Man-Cho So

引用次数: 0

Complementary composite minimization, small gradients in general norms, and applications 互补复合最小化、一般规范中的小梯度及其应用

IF 2.7 2区数学

Mathematical Programming Pub Date : 2024-01-05 DOI: 10.1007/s10107-023-02040-5

Jelena Diakonikolas, Cristóbal Guzmán

{"title":"Complementary composite minimization, small gradients in general norms, and applications","authors":"Jelena Diakonikolas, Cristóbal Guzmán","doi":"10.1007/s10107-023-02040-5","DOIUrl":"https://doi.org/10.1007/s10107-023-02040-5","url":null,"abstract":"Composite minimization is a powerful framework in large-scale convex optimization, based on decoupling of the objective function into terms with structurally different properties and allowing for more flexible algorithmic design. We introduce a new algorithmic framework for complementary composite minimization, where the objective function decouples into a (weakly) smooth and a uniformly convex term. This particular form of decoupling is pervasive in statistics and machine learning, due to its link to regularization. The main contributions of our work are summarized as follows. First, we introduce the problem of complementary composite minimization in general normed spaces; second, we provide a unified accelerated algorithmic framework to address broad classes of complementary composite minimization problems; and third, we prove that the algorithms resulting from our framework are near-optimal in most of the standard optimization settings. Additionally, we show that our algorithmic framework can be used to address the problem of making the gradients small in general normed spaces. As a concrete example, we obtain a nearly-optimal method for the standard (ell _1) setup (small gradients in the (ell _infty ) norm), essentially matching the bound of Nesterov (Optima Math Optim Soc Newsl 88:10–11, 2012) that was previously known only for the Euclidean setup. Finally, we show that our composite methods are broadly applicable to a number of regression and other classes of optimization problems, where regularization plays a key role. Our methods lead to complexity bounds that are either new or match the best existing ones.","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"20 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139373803","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

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