Pub Date : 2024-07-05DOI: 10.1007/s10107-024-02089-w
Matúš Benko, Patrick Mehlitz
As a starting point of our research, we show that, for a fixed order (gamma ge 1), each local minimizer of a rather general nonsmooth optimization problem in Euclidean spaces is either M-stationary in the classical sense (corresponding to stationarity of order 1), satisfies stationarity conditions in terms of a coderivative construction of order (gamma ), or is asymptotically stationary with respect to a critical direction as well as order (gamma ) in a certain sense. By ruling out the latter case with a constraint qualification not stronger than directional metric subregularity, we end up with new necessary optimality conditions comprising a mixture of limiting variational tools of orders 1 and (gamma ). These abstract findings are carved out for the broad class of geometric constraints and (gamma :=2), and visualized by examples from complementarity-constrained and nonlinear semidefinite optimization. As a byproduct of the particular setting (gamma :=1), our general approach yields new so-called directional asymptotic regularity conditions which serve as constraint qualifications guaranteeing M-stationarity of local minimizers. We compare these new regularity conditions with standard constraint qualifications from nonsmooth optimization. Further, we extend directional concepts of pseudo- and quasi-normality to arbitrary set-valued mappings. It is shown that these properties provide sufficient conditions for the validity of directional asymptotic regularity. Finally, a novel coderivative-like variational tool is used to construct sufficient conditions for the presence of directional asymptotic regularity. For geometric constraints, it is illustrated that all appearing objects can be calculated in terms of initial problem data.
{"title":"On the directional asymptotic approach in optimization theory","authors":"Matúš Benko, Patrick Mehlitz","doi":"10.1007/s10107-024-02089-w","DOIUrl":"https://doi.org/10.1007/s10107-024-02089-w","url":null,"abstract":"<p>As a starting point of our research, we show that, for a fixed order <span>(gamma ge 1)</span>, each local minimizer of a rather general nonsmooth optimization problem in Euclidean spaces is either M-stationary in the classical sense (corresponding to stationarity of order 1), satisfies stationarity conditions in terms of a coderivative construction of order <span>(gamma )</span>, or is asymptotically stationary with respect to a critical direction as well as order <span>(gamma )</span> in a certain sense. By ruling out the latter case with a constraint qualification not stronger than directional metric subregularity, we end up with new necessary optimality conditions comprising a mixture of limiting variational tools of orders 1 and <span>(gamma )</span>. These abstract findings are carved out for the broad class of geometric constraints and <span>(gamma :=2)</span>, and visualized by examples from complementarity-constrained and nonlinear semidefinite optimization. As a byproduct of the particular setting <span>(gamma :=1)</span>, our general approach yields new so-called directional asymptotic regularity conditions which serve as constraint qualifications guaranteeing M-stationarity of local minimizers. We compare these new regularity conditions with standard constraint qualifications from nonsmooth optimization. Further, we extend directional concepts of pseudo- and quasi-normality to arbitrary set-valued mappings. It is shown that these properties provide sufficient conditions for the validity of directional asymptotic regularity. Finally, a novel coderivative-like variational tool is used to construct sufficient conditions for the presence of directional asymptotic regularity. For geometric constraints, it is illustrated that all appearing objects can be calculated in terms of initial problem data.</p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"30 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141547944","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}
Pub Date : 2024-07-05DOI: 10.1007/s10107-024-02117-9
Benny Sudakov, István Tomon
Given an (mtimes n) binary matrix M with (|M|=pcdot mn) (where |M| denotes the number of 1 entries), define the discrepancy of M as ({{,textrm{disc},}}(M)=displaystyle max nolimits _{Xsubset [m], Ysubset [n]}big ||M[Xtimes Y]|-p|X|cdot |Y|big |). Using semidefinite programming and spectral techniques, we prove that if ({{,textrm{rank},}}(M)le r) and (ple 1/2), then
$$begin{aligned}{{,textrm{disc},}}(M)ge Omega (mn)cdot min left{ p,frac{p^{1/2}}{sqrt{r}}right} .end{aligned}$$
We use this result to obtain a modest improvement of Lovett’s best known upper bound on the log-rank conjecture. We prove that any (mtimes n) binary matrix M of rank at most r contains an ((mcdot 2^{-O(sqrt{r})})times (ncdot 2^{-O(sqrt{r})})) sized all-1 or all-0 submatrix, which implies that the deterministic communication complexity of any Boolean function of rank r is at most (O(sqrt{r})).
给定一个 (mtimes n) 二进制矩阵 M,其 (|M|=pcdot mn) (其中 |M| 表示 1 条目的数量),定义 M 的差异为 ({{、(M)=displaystyle max nolimits _{Xsubset [m], Ysubset [n]}big |||M[Xtimes Y]|-p|X|cdot |Y|big |)。利用半定量编程和谱技术,我们证明如果({{,textrm{rank},}}(M)le r) and(ple 1/2)、then $$begin{aligned}{{,textrm{disc},}}(M)ge Omega (mn)cdot min left{ p,frac{p^{1/2}}{sqrt{r}}right} .end{aligned}$$我们利用这个结果对洛维特最著名的对数秩猜想的上界进行了适度的改进。我们证明了任何秩为 r 的二进制矩阵 M 都包含一个 ((mcdot 2^{-O(sqrt{r})})times (ncdot 2^{-O(sqrt{r})})) 大小的 all-1 或 all-0 子矩阵,这意味着任何秩为 r 的布尔函数的确定性通信复杂度最多为 (O(sqrt{r}))。
{"title":"Matrix discrepancy and the log-rank conjecture","authors":"Benny Sudakov, István Tomon","doi":"10.1007/s10107-024-02117-9","DOIUrl":"https://doi.org/10.1007/s10107-024-02117-9","url":null,"abstract":"<p>Given an <span>(mtimes n)</span> binary matrix <i>M</i> with <span>(|M|=pcdot mn)</span> (where |<i>M</i>| denotes the number of 1 entries), define the <i>discrepancy</i> of <i>M</i> as <span>({{,textrm{disc},}}(M)=displaystyle max nolimits _{Xsubset [m], Ysubset [n]}big ||M[Xtimes Y]|-p|X|cdot |Y|big |)</span>. Using semidefinite programming and spectral techniques, we prove that if <span>({{,textrm{rank},}}(M)le r)</span> and <span>(ple 1/2)</span>, then </p><span>$$begin{aligned}{{,textrm{disc},}}(M)ge Omega (mn)cdot min left{ p,frac{p^{1/2}}{sqrt{r}}right} .end{aligned}$$</span><p>We use this result to obtain a modest improvement of Lovett’s best known upper bound on the log-rank conjecture. We prove that any <span>(mtimes n)</span> binary matrix <i>M</i> of rank at most <i>r</i> contains an <span>((mcdot 2^{-O(sqrt{r})})times (ncdot 2^{-O(sqrt{r})}))</span> sized all-1 or all-0 submatrix, which implies that the deterministic communication complexity of any Boolean function of rank <i>r</i> is at most <span>(O(sqrt{r}))</span>.\u0000</p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"5 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141552588","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}
Pub Date : 2024-07-03DOI: 10.1007/s10107-024-02103-1
Wenjing Li, Wei Bian, Kim-Chuan Toh
Rank regularized minimization problem is an ideal model for the low-rank matrix completion/recovery problem. The matrix factorization approach can transform the high-dimensional rank regularized problem to a low-dimensional factorized column-sparse regularized problem. The latter can greatly facilitate fast computations in applicable algorithms, but needs to overcome the simultaneous non-convexity of the loss and regularization functions. In this paper, we consider the factorized column-sparse regularized model. Firstly, we optimize this model with bound constraints, and establish a certain equivalence between the optimized factorization problem and rank regularized problem. Further, we strengthen the optimality condition for stationary points of the factorization problem and define the notion of strong stationary point. Moreover, we establish the equivalence between the factorization problem and its nonconvex relaxation in the sense of global minimizers and strong stationary points. To solve the factorization problem, we design two types of algorithms and give an adaptive method to reduce their computation. The first algorithm is from the relaxation point of view and its iterates own some properties from global minimizers of the factorization problem after finite iterations. We give some analysis on the convergence of its iterates to a strong stationary point. The second algorithm is designed for directly solving the factorization problem. We improve the PALM algorithm introduced by Bolte et al. (Math Program Ser A 146:459–494, 2014) for the factorization problem and give its improved convergence results. Finally, we conduct numerical experiments to show the promising performance of the proposed model and algorithms for low-rank matrix completion.
秩正则化最小化问题是低秩矩阵补全/恢复问题的理想模型。矩阵因子化方法可以将高维秩正则化问题转化为低维因子化列稀疏正则化问题。后者可以极大地促进适用算法的快速计算,但需要同时克服损失函数和正则化函数的非凸性。本文考虑因子化列稀疏正则化模型。首先,我们对该模型进行了带约束条件的优化,并在优化后的因子化问题和秩正则化问题之间建立了一定的等价关系。此外,我们还强化了因式分解问题静止点的最优性条件,并定义了强静止点的概念。此外,我们还在全局最小值和强驻点的意义上建立了因式分解问题与其非凸松弛之间的等价性。为了解决因式分解问题,我们设计了两类算法,并给出了减少其计算量的自适应方法。第一种算法是从松弛的角度出发,它的迭代拥有有限迭代后因式分解问题全局最小值的一些特性。我们对其迭代数收敛到强静止点进行了一些分析。第二种算法是为直接求解因式分解问题而设计的。我们针对因式分解问题改进了 Bolte 等人(Math Program Ser A 146:459-494, 2014)介绍的 PALM 算法,并给出了其改进后的收敛结果。最后,我们进行了数值实验,展示了所提出的低秩矩阵补全模型和算法的良好性能。
{"title":"On solving a rank regularized minimization problem via equivalent factorized column-sparse regularized models","authors":"Wenjing Li, Wei Bian, Kim-Chuan Toh","doi":"10.1007/s10107-024-02103-1","DOIUrl":"https://doi.org/10.1007/s10107-024-02103-1","url":null,"abstract":"<p>Rank regularized minimization problem is an ideal model for the low-rank matrix completion/recovery problem. The matrix factorization approach can transform the high-dimensional rank regularized problem to a low-dimensional factorized column-sparse regularized problem. The latter can greatly facilitate fast computations in applicable algorithms, but needs to overcome the simultaneous non-convexity of the loss and regularization functions. In this paper, we consider the factorized column-sparse regularized model. Firstly, we optimize this model with bound constraints, and establish a certain equivalence between the optimized factorization problem and rank regularized problem. Further, we strengthen the optimality condition for stationary points of the factorization problem and define the notion of strong stationary point. Moreover, we establish the equivalence between the factorization problem and its nonconvex relaxation in the sense of global minimizers and strong stationary points. To solve the factorization problem, we design two types of algorithms and give an adaptive method to reduce their computation. The first algorithm is from the relaxation point of view and its iterates own some properties from global minimizers of the factorization problem after finite iterations. We give some analysis on the convergence of its iterates to a strong stationary point. The second algorithm is designed for directly solving the factorization problem. We improve the PALM algorithm introduced by Bolte et al. (Math Program Ser A 146:459–494, 2014) for the factorization problem and give its improved convergence results. Finally, we conduct numerical experiments to show the promising performance of the proposed model and algorithms for low-rank matrix completion.</p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"46 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141526139","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}
Pub Date : 2024-06-21DOI: 10.1007/s10107-024-02105-z
Wenbao Ai, Wei Liang, Jianhua Yuan
In this paper, we consider the problem of minimizing a general homogeneous quadratic function, subject to three real or four complex homogeneous quadratic inequality or equality constraints. For this problem, we present a sufficient and necessary test condition to detect whether its standard semi-definite programming (SDP) relaxation is tight or not. This test condition is based on only an optimal solution pair of the SDP relaxation and its dual. When the tightness is confirmed, a global optimal solution of the original problem is found simultaneously in polynomial-time. While the tightness does not hold, the SDP relaxation and its dual are proved to have the unique optimal solutions. Moreover, the Lagrangian version of such the test condition is specified for non-homogeneous cases. Based on the Lagrangian version, it is proved that several latest sufficient conditions to test the SDP tightness are contained by our test condition under the situation of two constraints. Thirdly, as an application of the test condition, S-lemma and Yuan’s lemma are generalized to three real and four complex quadratic forms first under certain exact conditions, which improves some classical results in literature. Finally, a counterexample is presented to show that the test condition cannot be simply extended to four real or five complex homogeneous quadratic constraints.
{"title":"On the tightness of an SDP relaxation for homogeneous QCQP with three real or four complex homogeneous constraints","authors":"Wenbao Ai, Wei Liang, Jianhua Yuan","doi":"10.1007/s10107-024-02105-z","DOIUrl":"https://doi.org/10.1007/s10107-024-02105-z","url":null,"abstract":"<p>In this paper, we consider the problem of minimizing a general homogeneous quadratic function, subject to three real or four complex homogeneous quadratic inequality or equality constraints. For this problem, we present a sufficient and necessary test condition to detect whether its standard semi-definite programming (SDP) relaxation is tight or not. This test condition is based on only an optimal solution pair of the SDP relaxation and its dual. When the tightness is confirmed, a global optimal solution of the original problem is found simultaneously in polynomial-time. While the tightness does not hold, the SDP relaxation and its dual are proved to have the unique optimal solutions. Moreover, the Lagrangian version of such the test condition is specified for non-homogeneous cases. Based on the Lagrangian version, it is proved that several latest sufficient conditions to test the SDP tightness are contained by our test condition under the situation of two constraints. Thirdly, as an application of the test condition, S-lemma and Yuan’s lemma are generalized to three real and four complex quadratic forms first under certain exact conditions, which improves some classical results in literature. Finally, a counterexample is presented to show that the test condition cannot be simply extended to four real or five complex homogeneous quadratic constraints.</p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"57 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141503531","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}
Pub Date : 2024-06-20DOI: 10.1007/s10107-024-02108-w
Aris Daniilidis, David Salas, Sebastián Tapia-García
A classical result of variational analysis, known as Attouch theorem, establishes an equivalence between epigraphical convergence of a sequence of proper convex lower semicontinuous functions and graphical convergence of the corresponding subdifferential maps up to a normalization condition which fixes the integration constant. In this work, we show that in finite dimensions and under a mild boundedness assumption, we can replace subdifferentials (sets of vectors) by slopes (scalars, corresponding to the distance of the subdifferentials to zero) and still obtain the same characterization: namely, the epigraphical convergence of functions is equivalent to the epigraphical convergence of their slopes. This surprising result goes in line with recent developments on slope determination (Boulmezaoud et al. in SIAM J Optim 28(3):2049–2066, 2018; Pérez-Aros et al. in Math Program 190(1–2):561-583, 2021) and slope sensitivity (Daniilidis and Drusvyatskiy in Proc Am Math Soc 151(11):4751-4756, 2023) for convex functions.
变分分析的一个经典结果,即阿图什(Attouch)定理,确定了适当凸下半连续函数序列的表观收敛性与相应次微分映射的图形收敛性之间的等价性,但须满足一个固定积分常数的归一化条件。在这项工作中,我们证明了在有限维度和温和的有界性假设下,我们可以用斜率(标量,对应于子微分到零的距离)替换子微分(向量集),并仍然得到相同的特征:即函数的图解收敛等同于其斜率的图解收敛。这一令人惊讶的结果与凸函数的斜率确定(Boulmezaoud 等人,发表于 SIAM J Optim 28(3):2049-2066, 2018;Pérez-Aros 等人,发表于 Math Program 190(1-2):561-583, 2021)和斜率敏感性(Daniilidis 和 Drusvyatskiy,发表于 Proc Am Math Soc 151(11):4751-4756, 2023)的最新进展一致。
{"title":"A slope generalization of Attouch theorem","authors":"Aris Daniilidis, David Salas, Sebastián Tapia-García","doi":"10.1007/s10107-024-02108-w","DOIUrl":"https://doi.org/10.1007/s10107-024-02108-w","url":null,"abstract":"<p>A classical result of variational analysis, known as Attouch theorem, establishes an equivalence between epigraphical convergence of a sequence of proper convex lower semicontinuous functions and graphical convergence of the corresponding subdifferential maps up to a normalization condition which fixes the integration constant. In this work, we show that in finite dimensions and under a mild boundedness assumption, we can replace subdifferentials (sets of vectors) by slopes (scalars, corresponding to the distance of the subdifferentials to zero) and still obtain the same characterization: namely, the epigraphical convergence of functions is equivalent to the epigraphical convergence of their slopes. This surprising result goes in line with recent developments on slope determination (Boulmezaoud et al. in SIAM J Optim 28(3):2049–2066, 2018; Pérez-Aros et al. in Math Program 190(1–2):561-583, 2021) and slope sensitivity (Daniilidis and Drusvyatskiy in Proc Am Math Soc 151(11):4751-4756, 2023) for convex functions.</p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"21 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2024-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141503532","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}
Pub Date : 2024-06-20DOI: 10.1007/s10107-024-02101-3
Zhongzhu Chen, Marcia Fampa, Jon Lee
The best practical techniques for exact solution of instances of the constrained maximum-entropy sampling problem, a discrete-optimization problem arising in the design of experiments, are via a branch-and-bound framework, working with a variety of concave continuous relaxations of the objective function. A standard and computationally-important bound-enhancement technique in this context is (ordinary) scaling, via a single positive parameter. Scaling adjusts the shape of continuous relaxations to reduce the gaps between the upper bounds and the optimal value. We extend this technique to generalized scaling, employing a positive vector of parameters, which allows much more flexibility and thus potentially reduces the gaps further. We give mathematical results aimed at supporting algorithmic methods for computing optimal generalized scalings, and we give computational results demonstrating the performance of generalized scaling on benchmark problem instances.
{"title":"Generalized scaling for the constrained maximum-entropy sampling problem","authors":"Zhongzhu Chen, Marcia Fampa, Jon Lee","doi":"10.1007/s10107-024-02101-3","DOIUrl":"https://doi.org/10.1007/s10107-024-02101-3","url":null,"abstract":"<p>The best practical techniques for exact solution of instances of the constrained maximum-entropy sampling problem, a discrete-optimization problem arising in the design of experiments, are via a branch-and-bound framework, working with a variety of concave continuous relaxations of the objective function. A standard and computationally-important bound-enhancement technique in this context is <i>(ordinary) scaling</i>, via a single positive parameter. Scaling adjusts the shape of continuous relaxations to reduce the gaps between the upper bounds and the optimal value. We extend this technique to <i>generalized scaling</i>, employing a positive vector of parameters, which allows much more flexibility and thus potentially reduces the gaps further. We give mathematical results aimed at supporting algorithmic methods for computing optimal generalized scalings, and we give computational results demonstrating the performance of generalized scaling on benchmark problem instances.</p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"25 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2024-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141503533","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}
Pub Date : 2024-06-13DOI: 10.1007/s10107-024-02106-y
Arindam Khan, Aditya Subramanian, Andreas Wiese
We study rectangle stabbing problems in which we are given n axis-aligned rectangles in the plane that we want to stab, that is, we want to select line segments such that for each given rectangle there is a line segment that intersects two opposite edges of it. In the horizontal rectangle stabbing problem (Stabbing), the goal is to find a set of horizontal line segments of minimum total length such that all rectangles are stabbed. In the horizontal–vertical stabbing problem (HV-Stabbing), the goal is to find a set of rectilinear (that is, either vertical or horizontal) line segments of minimum total length such that all rectangles are stabbed. Both variants are NP-hard. Chan et al. (ISAAC, 2018) initiated the study of these problems by providing constant approximation algorithms. Recently, Eisenbrand et al. (A QPTAS for stabbing rectangles, 2021) have presented a QPTAS and a polynomial-time 8-approximation algorithm for Stabbing, but it was open whether the problem admits a PTAS. In this paper, we obtain a PTAS for Stabbing, settling this question. For HV-Stabbing, we obtain a ((2+varepsilon ))-approximation. We also obtain PTASs for special cases of HV-Stabbing: (i) when all rectangles are squares, (ii) when each rectangle’s width is at most its height, and (iii) when all rectangles are (delta )-large, that is, have at least one edge whose length is at least (delta ), while all edge lengths are at most 1. Our result also implies improved approximations for other problems such as generalized minimum Manhattan network.
{"title":"A PTAS for the horizontal rectangle stabbing problem","authors":"Arindam Khan, Aditya Subramanian, Andreas Wiese","doi":"10.1007/s10107-024-02106-y","DOIUrl":"https://doi.org/10.1007/s10107-024-02106-y","url":null,"abstract":"<p>We study rectangle stabbing problems in which we are given <i>n</i> axis-aligned rectangles in the plane that we want to <i>stab</i>, that is, we want to select line segments such that for each given rectangle there is a line segment that intersects two opposite edges of it. In the <i>horizontal rectangle stabbing problem</i> (<span>Stabbing</span>), the goal is to find a set of horizontal line segments of minimum total length such that all rectangles are stabbed. In the <i>horizontal–vertical stabbing problem</i> (<span>HV-Stabbing</span>), the goal is to find a set of rectilinear (that is, either vertical or horizontal) line segments of minimum total length such that all rectangles are stabbed. Both variants are NP-hard. Chan et al. (ISAAC, 2018) initiated the study of these problems by providing constant approximation algorithms. Recently, Eisenbrand et al. (A QPTAS for stabbing rectangles, 2021) have presented a QPTAS and a polynomial-time 8-approximation algorithm for <span>Stabbing</span>, but it was open whether the problem admits a PTAS. In this paper, we obtain a PTAS for <span>Stabbing</span>, settling this question. For <span>HV-Stabbing</span>, we obtain a <span>((2+varepsilon ))</span>-approximation. We also obtain PTASs for special cases of <span>HV-Stabbing</span>: (i) when all rectangles are squares, (ii) when each rectangle’s width is at most its height, and (iii) when all rectangles are <span>(delta )</span>-large, that is, have at least one edge whose length is at least <span>(delta )</span>, while all edge lengths are at most 1. Our result also implies improved approximations for other problems such as <i>generalized minimum Manhattan network</i>.</p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"46 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141503476","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}
Pub Date : 2024-06-04DOI: 10.1007/s10107-024-02088-x
Frank Fischer
We develop a fully asynchronous proximal bundle method for solving non-smooth, convex optimization problems. The algorithm can be used as a drop-in replacement for classic bundle methods, i.e., the function must be given by a first-order oracle for computing function values and subgradients. The algorithm allows for an arbitrary number of master problem processes computing new candidate points and oracle processes evaluating functions at those candidate points. These processes share information by communication with a single supervisor process that resembles the main loop of a classic bundle method. All processes run in parallel and no explicit synchronization step is required. Instead, the asynchronous and possibly outdated results of the oracle computations can be seen as an inexact function oracle. Hence, we show the convergence of our method under weak assumptions very similar to inexact and incremental bundle methods. In particular, we show how the algorithm learns important structural properties of the functions to control the inaccuracy induced by the asynchronicity automatically such that overall convergence can be guaranteed.
{"title":"An asynchronous proximal bundle method","authors":"Frank Fischer","doi":"10.1007/s10107-024-02088-x","DOIUrl":"https://doi.org/10.1007/s10107-024-02088-x","url":null,"abstract":"<p>We develop a fully asynchronous proximal bundle method for solving non-smooth, convex optimization problems. The algorithm can be used as a drop-in replacement for classic bundle methods, i.e., the function must be given by a first-order oracle for computing function values and subgradients. The algorithm allows for an arbitrary number of master problem processes computing new candidate points and oracle processes evaluating functions at those candidate points. These processes share information by communication with a single supervisor process that resembles the main loop of a classic bundle method. All processes run in parallel and no explicit synchronization step is required. Instead, the asynchronous and possibly outdated results of the oracle computations can be seen as an inexact function oracle. Hence, we show the convergence of our method under weak assumptions very similar to inexact and incremental bundle methods. In particular, we show how the algorithm learns important structural properties of the functions to control the inaccuracy induced by the asynchronicity automatically such that overall convergence can be guaranteed.</p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"13 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141251916","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}
Pub Date : 2024-06-04DOI: 10.1007/s10107-024-02102-2
Jasper van Doornmalen, Christopher Hojny
Handling symmetries in optimization problems is essential for devising efficient solution methods. In this article, we present a general framework that captures many of the already existing symmetry handling methods. While these methods are mostly discussed independently from each other, our framework allows to apply different methods simultaneously and thus outperforming their individual effect. Moreover, most existing symmetry handling methods only apply to binary variables. Our framework allows to easily generalize these methods to general variable types. Numerical experiments confirm that our novel framework is superior to the state-of-the-art symmetry handling methods as implemented in the solver SCIP on a broad set of instances.
{"title":"A unified framework for symmetry handling","authors":"Jasper van Doornmalen, Christopher Hojny","doi":"10.1007/s10107-024-02102-2","DOIUrl":"https://doi.org/10.1007/s10107-024-02102-2","url":null,"abstract":"<p>Handling symmetries in optimization problems is essential for devising efficient solution methods. In this article, we present a general framework that captures many of the already existing symmetry handling methods. While these methods are mostly discussed independently from each other, our framework allows to apply different methods simultaneously and thus outperforming their individual effect. Moreover, most existing symmetry handling methods only apply to binary variables. Our framework allows to easily generalize these methods to general variable types. Numerical experiments confirm that our novel framework is superior to the state-of-the-art symmetry handling methods as implemented in the solver <span>SCIP</span> on a broad set of instances.\u0000</p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"33 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141251921","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}
Pub Date : 2024-06-04DOI: 10.1007/s10107-024-02100-4
Naoki Marumo, Akiko Takeda
We propose a new first-order method for minimizing nonconvex functions with Lipschitz continuous gradients and Hölder continuous Hessians. The proposed algorithm is a heavy-ball method equipped with two particular restart mechanisms. It finds a solution where the gradient norm is less than (varepsilon ) in (O(H_{nu }^{frac{1}{2 + 2 nu }} varepsilon ^{- frac{4 + 3 nu }{2 + 2 nu }})) function and gradient evaluations, where (nu in [0, 1]) and (H_{nu }) are the Hölder exponent and constant, respectively. This complexity result covers the classical bound of (O(varepsilon ^{-2})) for (nu = 0) and the state-of-the-art bound of (O(varepsilon ^{-7/4})) for (nu = 1). Our algorithm is (nu )-independent and thus universal; it automatically achieves the above complexity bound with the optimal (nu in [0, 1]) without knowledge of (H_{nu }). In addition, the algorithm does not require other problem-dependent parameters as input, including the gradient’s Lipschitz constant or the target accuracy (varepsilon ). Numerical results illustrate that the proposed method is promising.
我们提出了一种新的一阶方法,用于最小化具有 Lipschitz 连续梯度和 Hölder 连续 Hessians 的非凸函数。所提出的算法是一种重球方法,配备了两种特殊的重启机制。它能在(O(H_{nu }^{frac{1}{2 + 2 nu }} 内找到梯度规范小于(varepsilon )的解。varepsilon ^{- frac{4 + 3 nu }{2 + 2 nu }})函数和梯度评估,其中 (nu in [0, 1]) 和 (H_{nu }) 分别是霍尔德指数和常数。这个复杂度结果涵盖了 (nu = 0) 的经典边界(O(varepsilon ^{-2}))和 (nu = 1) 的最新边界(O(varepsilon ^{-7/4}))。我们的算法与 (nu )无关,因此是通用的;它可以在不知道 (H_{nu }) 的情况下,以最优的 (nu in [0, 1]) 自动实现上述复杂度约束。此外,该算法不需要其他与问题相关的参数作为输入,包括梯度的 Lipschitz 常量或目标精度 (varepsilon )。数值结果表明,所提出的方法很有前途。
{"title":"Universal heavy-ball method for nonconvex optimization under Hölder continuous Hessians","authors":"Naoki Marumo, Akiko Takeda","doi":"10.1007/s10107-024-02100-4","DOIUrl":"https://doi.org/10.1007/s10107-024-02100-4","url":null,"abstract":"<p>We propose a new first-order method for minimizing nonconvex functions with Lipschitz continuous gradients and Hölder continuous Hessians. The proposed algorithm is a heavy-ball method equipped with two particular restart mechanisms. It finds a solution where the gradient norm is less than <span>(varepsilon )</span> in <span>(O(H_{nu }^{frac{1}{2 + 2 nu }} varepsilon ^{- frac{4 + 3 nu }{2 + 2 nu }}))</span> function and gradient evaluations, where <span>(nu in [0, 1])</span> and <span>(H_{nu })</span> are the Hölder exponent and constant, respectively. This complexity result covers the classical bound of <span>(O(varepsilon ^{-2}))</span> for <span>(nu = 0)</span> and the state-of-the-art bound of <span>(O(varepsilon ^{-7/4}))</span> for <span>(nu = 1)</span>. Our algorithm is <span>(nu )</span>-independent and thus universal; it automatically achieves the above complexity bound with the optimal <span>(nu in [0, 1])</span> without knowledge of <span>(H_{nu })</span>. In addition, the algorithm does not require other problem-dependent parameters as input, including the gradient’s Lipschitz constant or the target accuracy <span>(varepsilon )</span>. Numerical results illustrate that the proposed method is promising.</p>","PeriodicalId":18297,"journal":{"name":"Mathematical Programming","volume":"51 1","pages":""},"PeriodicalIF":2.7,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141251840","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}