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On the directional asymptotic approach in optimization theory 论优化理论中的定向渐近方法
IF 2.7 2区 数学 Q2 COMPUTER SCIENCE, SOFTWARE ENGINEERING Pub Date : 2024-07-05 DOI: 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.

作为我们研究的起点,我们证明了对于一个固定的阶(gamma),欧几里得空间中一个相当一般的非光滑优化问题的每个局部最小值要么是经典意义上的M静止(对应于阶1的静止)、满足阶 (gamma ) 的编码构造的静止条件,或者相对于临界方向以及阶 (gamma ) 在一定意义上是渐近静止的。通过用不强于方向度量次规则性的约束条件排除后一种情况,我们最终得到了新的必要最优性条件,包括阶1和阶(gamma )的极限变分工具的混合物。这些抽象发现是针对广泛的几何约束和 (gamma :=2) 类而提出的,并通过来自互补约束和非线性半有限优化的例子加以形象化。作为 (gamma :=1) 这一特殊设置的副产品,我们的一般方法产生了新的所谓方向渐近正则性条件,作为保证局部最小化 M-stationarity 的约束条件。我们将这些新的正则性条件与非光滑优化的标准约束条件进行了比较。此外,我们还将伪正则和准正则的方向性概念扩展到了任意的集值映射。结果表明,这些性质为方向渐近正则性的有效性提供了充分条件。最后,利用一种新颖的类似于 coderivative 的变分工具来构建方向渐近正则性存在的充分条件。对于几何约束来说,所有出现的对象都可以通过初始问题数据计算出来。
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引用次数: 0
Matrix discrepancy and the log-rank conjecture 矩阵差异和对数秩猜想
IF 2.7 2区 数学 Q2 COMPUTER SCIENCE, SOFTWARE ENGINEERING Pub Date : 2024-07-05 DOI: 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}))。
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引用次数: 0
On solving a rank regularized minimization problem via equivalent factorized column-sparse regularized models 通过等效因子化列稀疏正则化模型求解秩正则化最小化问题
IF 2.7 2区 数学 Q2 COMPUTER SCIENCE, SOFTWARE ENGINEERING Pub Date : 2024-07-03 DOI: 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。
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引用次数: 0
On the tightness of an SDP relaxation for homogeneous QCQP with three real or four complex homogeneous constraints 论具有三个实数或四个复数同质约束条件的同质 QCQP 的 SDP 松弛的紧密性
IF 2.7 2区 数学 Q2 COMPUTER SCIENCE, SOFTWARE ENGINEERING Pub Date : 2024-06-21 DOI: 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.

在本文中,我们考虑的问题是在三个实数或四个复数同质二次函数不等式或相等约束条件下,最小化一个一般同质二次函数。对于这个问题,我们提出了一个充分且必要的检验条件,以检测其标准半有限编程(SDP)松弛是否紧密。该检验条件仅基于 SDP 松弛及其对偶的最优解对。当紧密性得到确认时,就能同时在多项式时间内找到原始问题的全局最优解。当严密性不成立时,则证明 SDP 松弛及其对偶具有唯一最优解。此外,还为非均质情况指定了这种测试条件的拉格朗日版本。基于拉格朗日版本,证明了在两个约束的情况下,我们的检验条件包含了检验 SDP 紧缩性的几个最新充分条件。第三,作为检验条件的应用,在一定的精确条件下,S-lemma 和 Yuan's lemma 首先被推广到三实四复二次型,从而改进了文献中的一些经典结果。最后,提出了一个反例,说明检验条件不能简单地扩展到四实数或五复数同质二次约束。
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引用次数: 0
A slope generalization of Attouch theorem 阿图什定理的斜率一般化
IF 2.7 2区 数学 Q2 COMPUTER SCIENCE, SOFTWARE ENGINEERING Pub Date : 2024-06-20 DOI: 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。
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引用次数: 0
Generalized scaling for the constrained maximum-entropy sampling problem 受限最大熵抽样问题的广义缩放
IF 2.7 2区 数学 Q2 COMPUTER SCIENCE, SOFTWARE ENGINEERING Pub Date : 2024-06-20 DOI: 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.

受限最大熵采样问题是实验设计中出现的离散优化问题,精确求解该问题实例的最佳实用技术是通过分支与边界框架,利用目标函数的各种凹连续松弛来实现的。在这种情况下,一种标准的、在计算上非常重要的边界增强技术是通过单个正参数进行(普通)缩放。缩放可以调整连续松弛的形状,从而缩小上限与最优值之间的差距。我们将这一技术扩展到广义缩放,即采用一个正向参数向量,这样就有了更大的灵活性,从而有可能进一步缩小差距。我们给出的数学结果旨在支持计算最优广义缩放的算法方法,我们给出的计算结果证明了广义缩放在基准问题实例上的性能。
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引用次数: 0
A PTAS for the horizontal rectangle stabbing problem 水平矩形刺入问题的 PTAS
IF 2.7 2区 数学 Q2 COMPUTER SCIENCE, SOFTWARE ENGINEERING Pub Date : 2024-06-13 DOI: 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.

我们研究的是矩形切割问题,在这个问题中,我们要切割的是平面上 n 个轴线对齐的矩形,也就是说,我们要选择线段,使得每个矩形都有一条线段与它的两条相对边相交。在水平矩形刺入问题(刺入)中,我们的目标是找到一组总长度最小的水平线段,从而刺入所有矩形。在水平-垂直刺入问题(HV-Stabbing)中,目标是找到一组总长度最小的直线(即垂直或水平)线段,使所有矩形都被刺入。这两个变体都是 NP 难。Chan 等人(ISAAC,2018)通过提供恒定近似算法,开始了对这些问题的研究。最近,Eisenbrand 等人(A QPTAS for stabbing rectangles, 2021)提出了针对 Stabbing 问题的 QPTAS 和多项式时间 8 近似算法,但该问题是否存在 PTAS 尚无定论。在本文中,我们得到了 Stabbing 的 PTAS,从而解决了这个问题。对于HV-Stabbing,我们得到了一个((2+varepsilon ))近似值。我们还得到了 HV-Stabbing 特殊情况下的 PTAS:(i) 所有矩形都是正方形,(ii) 每个矩形的宽度最多等于它的高度,(iii) 所有矩形都是(Δ )大的,也就是说,至少有一条边的长度至少是(Δ ),而所有边的长度最多是 1。 我们的结果还意味着对其他问题的近似值的改进,比如广义最小曼哈顿网络。
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引用次数: 0
An asynchronous proximal bundle method 异步近端捆绑法
IF 2.7 2区 数学 Q2 COMPUTER SCIENCE, SOFTWARE ENGINEERING Pub Date : 2024-06-04 DOI: 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.

我们开发了一种用于解决非光滑凸优化问题的完全异步近似束方法。该算法可以直接替代传统的捆绑方法,即函数必须由计算函数值和子梯度的一阶神谕给出。该算法允许任意数量的主问题进程计算新的候选点,并允许神谕进程在这些候选点上评估函数。这些进程通过与单个监督进程通信共享信息,该监督进程类似于经典捆绑方法的主循环。所有进程并行运行,无需明确的同步步骤。相反,甲骨文计算的异步和可能过时的结果可以看作是一个不精确的函数甲骨文。因此,我们展示了我们的方法在弱假设条件下的收敛性,这与不精确方法和增量捆绑方法非常相似。特别是,我们展示了算法如何学习函数的重要结构特性,自动控制异步性引起的不准确性,从而保证整体收敛性。
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引用次数: 0
A unified framework for symmetry handling 处理对称性的统一框架
IF 2.7 2区 数学 Q2 COMPUTER SCIENCE, SOFTWARE ENGINEERING Pub Date : 2024-06-04 DOI: 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.

处理优化问题中的对称性对于设计高效的求解方法至关重要。在本文中,我们提出了一个通用框架,其中包含了许多现有的对称性处理方法。虽然这些方法大多是相互独立讨论的,但我们的框架允许同时应用不同的方法,从而超越它们各自的效果。此外,大多数现有的对称性处理方法只适用于二进制变量。我们的框架可以轻松地将这些方法推广到一般变量类型。数值实验证实,在大量实例上,我们的新框架优于在求解器 SCIP 中实施的最先进的对称性处理方法。
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引用次数: 0
Universal heavy-ball method for nonconvex optimization under Hölder continuous Hessians 赫尔德连续赫西亚条件下非凸优化的通用重球法
IF 2.7 2区 数学 Q2 COMPUTER SCIENCE, SOFTWARE ENGINEERING Pub Date : 2024-06-04 DOI: 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。
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引用次数: 0
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Mathematical Programming
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