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Smoothing penalty approach for solving second-order cone complementarity problems 解决二阶锥体互补问题的平滑惩罚法
IF 1.8 3区 数学 Q1 Mathematics Pub Date : 2024-09-18 DOI: 10.1007/s10898-024-01427-8
Chieu Thanh Nguyen, Jan Harold Alcantara, Zijun Hao, Jein-Shan Chen

In this paper, we propose a smoothing penalty approach for solving the second-order cone complementarity problem (SOCCP). The SOCCP is approximated by a smooth nonlinear equation with penalization parameter. We show that any solution sequence of the approximating equations converges to the solution of the SOCCP under the assumption that the associated function of the SOCCP satisfies a uniform Cartesian-type property. We present a corresponding algorithm for solving the SOCCP based on this smoothing penalty approach, and we demonstrate the efficiency of our method for solving linear, nonlinear and tensor complementarity problems in the second-order cone setting.

本文提出了一种解决二阶锥体互补问题(SOCCP)的平滑惩罚方法。SOCCP 由一个带有惩罚参数的平滑非线性方程逼近。我们证明,在 SOCCP 的相关函数满足均匀笛卡尔类型属性的假设下,近似方程的任何解序列都会收敛到 SOCCP 的解。我们基于这种平滑惩罚方法提出了相应的 SOCCP 求解算法,并证明了我们的方法在二阶圆锥环境下求解线性、非线性和张量互补问题的效率。
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引用次数: 0
Aircraft conflict resolution with trajectory recovery using mixed-integer programming 利用混合整数程序设计恢复轨迹的飞机冲突解决方法
IF 1.8 3区 数学 Q1 Mathematics Pub Date : 2024-09-17 DOI: 10.1007/s10898-024-01393-1
Fernando Dias, David Rey

To guarantee the safety of flight operations, decision-support systems for air traffic control must be able to improve the usage of airspace capacity and handle increasing demand. This study addresses the aircraft conflict avoidance and trajectory recovery problem. The problem of finding the least deviation conflict-free aircraft trajectories that guarantee the return to a target waypoint is highly complex due to the nature of the nonlinear trajectories that are sought. We present a two-stage iterative algorithm that first solves initial conflicts by manipulating their speed and heading control and then identifying each aircraft’s optimal time to recover its trajectory towards their nominal one. We extend existing mixed-integer programming formulations by modelling speed and heading control as continuous variables while recovery time is treated as a discrete variable. We develop a novel iterative approach which shows that the trajectory recovery costs can be anticipated by inducing avoidance trajectories with higher deviation, therefore obtaining earlier recovery time within a few iterations. Numerical results on benchmark conflict resolution problems show that this approach can solve instances with up to 30 aircraft within 10 min.

为保证飞行安全,空中交通管制决策支持系统必须能够提高空域容量的使用率,并应对日益增长的需求。本研究探讨了飞机冲突规避和轨迹恢复问题。由于要寻找的是非线性轨迹,因此寻找保证返回目标航点的最小偏差无冲突飞机轨迹问题非常复杂。我们提出了一种两阶段迭代算法,首先通过操纵速度和航向控制来解决初始冲突,然后确定每架飞机向其标称轨迹恢复的最佳时间。我们将速度和航向控制作为连续变量建模,而将恢复时间作为离散变量处理,从而扩展了现有的混合整数编程公式。我们开发了一种新颖的迭代方法,表明可以通过诱导偏差较大的回避轨迹来预计轨迹恢复成本,从而在几次迭代中提前获得恢复时间。解决基准冲突问题的数值结果表明,这种方法可以在 10 分钟内解决多达 30 架飞机的实例。
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引用次数: 0
Improved approximation algorithms for the k-path partition problem K 路径分割问题的改进近似计算法
IF 1.8 3区 数学 Q1 Mathematics Pub Date : 2024-09-13 DOI: 10.1007/s10898-024-01428-7
Shiming Li, Wei Yu, Zhaohui Liu

The k-path partition problem (kPP), defined on a graph (G=(V,E)), is a well-known NP-hard problem when (kge 3). The goal of the kPP is to find a minimum collection of vertex-disjoint paths to cover all the vertices in G such that the number of vertices on each path is no more than k. In this paper, we give two approximation algorithms for the kPP. The first one, called Algorithm 1, uses an algorithm for the (0,1)-weighted maximum traveling salesman problem as a subroutine. When G is undirected, the approximation ratio of Algorithm 1 is (frac{k+12}{7} -frac{6}{7k} ), which improves on the previous best-known approximation algorithm for every (kge 7). When G is directed, Algorithm 1 is a (left( frac{k+6}{4} -frac{3}{4k}right) )-approximation algorithm, which improves the existing best available approximation algorithm for every (kge 10). Our second algorithm, i.e. Algorithm 2, is a local search algorithm tailored for the kPP in undirected graphs with small k. Algorithm 2 improves on the approximation ratios of the best available algorithm for every (k=4,5,6). Combined with Algorithms 1 and 2, we have improved the approximation ratio for the kPP in undirected graphs for each (kge 4) as well as the approximation ratio for the kPP in directed graphs for each (kge 10). As for the negative side, we show that for any (epsilon >0) it is NP-hard to approximate the kPP (with k being part of the input) within the ratio (O(k^{1-epsilon })), which implies that Algorithm 1 is asymptotically optimal.

k 路径分割问题(kPP)定义在图(G=(V,E))上,是一个众所周知的 NP 难问题(当 (kge 3) 时)。kPP 的目标是找到覆盖 G 中所有顶点的顶点不相交路径的最小集合,使得每条路径上的顶点数不超过 k。第一种算法称为算法 1,它使用 (0,1)-weighted maximum traveling salesman 问题的算法作为子程序。当G是无向的,算法1的近似率是(frac{k+12}{7} -frac{6}{7k} ),这改进了之前已知的每(kge 7)的近似算法。当G是有向的,算法1是一个((left( frac{k+6}{4} -frac{3}{4k}right) )近似算法,它改进了现有的每一个(kge 10)的最佳近似算法。我们的第二种算法,即算法 2,是一种局部搜索算法,专为 k 较小的无向图中的 kPP 量身定制。算法 2 提高了现有最佳算法对每(k=4,5,6)个图的近似率。结合算法1和算法2,我们改进了无向图中每一个(k=4,5,6)的kPP近似率,以及有向图中每一个(k=10)的kPP近似率。至于反面,我们证明了对于任意(epsilon >0)来说,在比率(O(k^{1-epsilon }))内逼近kPP(k是输入的一部分)是NP-hard的,这意味着算法1是渐进最优的。
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引用次数: 0
On convergence of a q-random coordinate constrained algorithm for non-convex problems 论非凸问题 q 随机坐标约束算法的收敛性
IF 1.8 3区 数学 Q1 Mathematics Pub Date : 2024-09-12 DOI: 10.1007/s10898-024-01429-6
A. Ghaffari-Hadigheh, L. Sinjorgo, R. Sotirov

We propose a random coordinate descent algorithm for optimizing a non-convex objective function subject to one linear constraint and simple bounds on the variables. Although it is common use to update only two random coordinates simultaneously in each iteration of a coordinate descent algorithm, our algorithm allows updating arbitrary number of coordinates. We provide a proof of convergence of the algorithm. The convergence rate of the algorithm improves when we update more coordinates per iteration. Numerical experiments on large scale instances of different optimization problems show the benefit of updating many coordinates simultaneously.

我们提出了一种随机坐标下降算法,用于优化一个非凸目标函数,该函数受到一个线性约束和变量的简单约束。虽然通常在坐标下降算法的每次迭代中只同时更新两个随机坐标,但我们的算法允许更新任意数量的坐标。我们提供了算法的收敛性证明。当我们每次迭代更新更多坐标时,算法的收敛速度就会提高。在不同优化问题的大规模实例上进行的数值实验表明了同时更新多个坐标的好处。
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引用次数: 0
A QoS and sustainability-driven two-stage service composition method in cloud manufacturing: combining clustering and bi-objective optimization 云制造中以服务质量和可持续性为导向的两阶段服务组合方法:聚类与双目标优化相结合
IF 1.8 3区 数学 Q1 Mathematics Pub Date : 2024-09-12 DOI: 10.1007/s10898-024-01430-z
Chunhua Tang, Shuangyao Zhao, Han Su, Binbin Chen

Manufacturing service composition (MSC) is a core technology in cloud manufacturing (CMfg), which has been intensively studied to find an optimal composite service with the best quality of service (QoS). With the continuous expansion of CMfg platforms, the difficulty of MSC is gradually increasing. Large-scale platforms have put forward higher requirements for combination efficiency, and its open and dynamic environment makes service QoS exhibit strong uncertainty, leading to reliability issues of MSC. Meanwhile, the increased number of services and users makes it necessary for the platform to consider the sustainability issue, including economic, environmental, and social aspects, based on an operations management perspective. However, current studies only consider part of efficiency, reliability, and sustainability as optimization objectives in MSC allocation models, and do not take them into account simultaneously in an integrated manner. Therefore, this study proposes a two-stage method integrating clustering and multi-objective optimization for reliable and sustainable MSC allocation. Specifically, in the first stage, the K-means clustering technique and the QoS stability-based service pruning mechanism are integrated into the service clustering process to improve the reliability of candidate services and reduce the search space of combinations. In the second stage, a multi-objective optimization model with maximizing QoS and sustainability is proposed to find the optimal MSC, and the fast non-dominated sorting genetic algorithm is adopted to solve the model. Finally, a case study of the actual production of a customized automated guided vehicle verifies the effectiveness of the proposed two-stage method.

制造服务组合(MSC)是云制造(CMfg)中的一项核心技术,人们一直在深入研究如何找到具有最佳服务质量(QoS)的最优组合服务。随着 CMfg 平台的不断扩大,MSC 的难度也在逐渐增加。大型平台对组合效率提出了更高的要求,其开放、动态的环境使得服务质量表现出很强的不确定性,从而导致 MSC 的可靠性问题。同时,服务和用户数量的增加使得平台必须基于运营管理的视角考虑可持续发展问题,包括经济、环境和社会等方面。然而,目前的研究仅将效率、可靠性和可持续性作为地中海航运中心分配模型的部分优化目标,并没有将它们同时综合考虑。因此,本研究提出了一种集群和多目标优化于一体的两阶段方法,以实现可靠和可持续的 MSC 分配。具体来说,在第一阶段,将 K-means 聚类技术和基于 QoS 稳定性的服务剪枝机制整合到服务聚类过程中,以提高候选服务的可靠性并减少组合的搜索空间。第二阶段,提出了一个兼顾 QoS 最大化和可持续性的多目标优化模型来寻找最优 MSC,并采用快速非支配排序遗传算法来求解该模型。最后,通过对定制自动导引车实际生产的案例研究,验证了所提出的两阶段方法的有效性。
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引用次数: 0
A reformulation-enumeration MINLP algorithm for gas network design 用于燃气管网设计的重构-枚举 MINLP 算法
IF 1.8 3区 数学 Q1 Mathematics Pub Date : 2024-09-03 DOI: 10.1007/s10898-024-01424-x
Yijiang Li, Santanu S. Dey, Nikolaos V. Sahinidis

Gas networks are used to transport natural gas, which is an important resource for both residential and industrial customers throughout the world. The gas network design problem is generally modelled as a nonconvex mixed-integer nonlinear integer programming problem (MINLP). The challenges of solving the resulting MINLP arise due to the nonlinearity and nonconvexity. In this paper, we propose a framework to study the “design variant” of the problem in which the variables are the diameter choices of the pipes, the flows, the potentials, and the states of various network components. We utilize a nested loop that includes a two-stage procedure that involves a convex reformulation of the original problem in the inner loop and an efficient enumeration scheme in the outer loop. We conduct experiments on benchmark networks to validate and analyze the performance of our framework.

天然气是全世界居民和工业用户的重要资源,天然气网络用于运输天然气。天然气网络设计问题通常被模拟为一个非凸混合整数非线性整数编程问题(MINLP)。由于非线性和非凸性,解决由此产生的 MINLP 问题面临挑战。在本文中,我们提出了一个研究该问题 "设计变量 "的框架,其中的变量包括管道直径选择、流量、电势以及各种网络组件的状态。我们利用了一个嵌套循环,其中包括一个两阶段程序,在内环中对原始问题进行凸重构,在外环中采用高效的枚举方案。我们在基准网络上进行了实验,以验证和分析我们框架的性能。
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引用次数: 0
Budget-constrained profit maximization without non-negative objective assumption in social networks 社交网络中无非负目标假设的预算受限利润最大化
IF 1.8 3区 数学 Q1 Mathematics Pub Date : 2024-08-14 DOI: 10.1007/s10898-024-01406-z
Suning Gong, Qingqin Nong, Yue Wang, Dingzhu Du

In this paper, we study the budget-constrained profit maximization problem with expensive seed endorsement, a derivation of the well-studied influence maximization and profit maximization in social networks. While existing research requires the non-negativity of the objective profit function, this paper considers real-world scenarios where costs may surpass revenue. Specifically, our problem can be regarded as maximizing the difference between a non-negative submodular function and a non-negative modular function under a knapsack constraint, allowing for negative differences. To tackle this challenge, we propose two algorithms. Firstly, we employ a twin greedy and enumeration technique to design a polynomial-time algorithm with a quarter weak approximation ratio, providing a balance between computational efficiency and solution quality. Then, we incorporate a threshold decreasing technique to enhance the time complexity of the first algorithm, yielding an improved computational efficiency while maintaining a reasonable level of solution accuracy. To our knowledge, this is the first paper to study the profit maximization beyond non-negativity and to propose polynomial-time algorithms with a constant bicriteria approximation ratio.

在本文中,我们研究了具有昂贵种子背书的预算受限利润最大化问题,这是社交网络中影响最大化和利润最大化问题的衍生。现有研究要求目标利润函数为非负,而本文则考虑了成本可能超过收入的实际情况。具体来说,我们的问题可以看作是在knapsack约束条件下,最大化非负次模态函数和非负模态函数之间的差值,允许负差值。为了应对这一挑战,我们提出了两种算法。首先,我们采用孪生贪婪和枚举技术,设计了一种具有四分之一弱逼近率的多项式时间算法,在计算效率和求解质量之间取得了平衡。然后,我们采用阈值递减技术来提高第一种算法的时间复杂度,从而在提高计算效率的同时保持合理的求解精度。据我们所知,这是第一篇研究非负性之外的利润最大化并提出具有恒定双标准近似率的多项式时间算法的论文。
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引用次数: 0
A two-phase sequential algorithm for global optimization of the standard quadratic programming problem 标准二次编程问题全局优化的两阶段顺序算法
IF 1.8 3区 数学 Q1 Mathematics Pub Date : 2024-08-12 DOI: 10.1007/s10898-024-01423-y
Joaquim Júdice, Valentina Sessa, Masao Fukushima

We introduce a new sequential algorithm for the Standard Quadratic Programming Problem (StQP), which exploits a formulation of StQP as a Linear Program with Linear Complementarity Constraints (LPLCC). The algorithm is finite and guarantees at least in theory a (delta )-approximate global minimum for an arbitrary small (delta ), which is a global minimum in practice. The sequential algorithm has two phases. In Phase 1, Stationary Points (SP) with strictly decreasing objective function values are computed. Phase 2 is designed for giving a certificate of global optimality for the last SP computed in Phase 1. Two different Nonlinear Programming Formulations for LPLCC are proposed for each one of these phases, which are solved by efficient enumerative algorithms. New procedures for computing a lower bound for StQP are also proposed, which are easy to implement and give tight bounds in general. Computational experiments with a number of test problems from known sources indicate that the two-phase sequential algorithm is, in general, efficient in practice. Furthermore, the algorithm seems to be an efficient way to study the copositivity of a matrix by exploiting an StQP with this matrix.

我们为标准二次编程问题(StQP)引入了一种新的顺序算法,它利用了将StQP表述为具有线性互补约束的线性规划(LPLCC)的方法。该算法是有限的,至少在理论上保证了任意小的(delta )的近似全局最小值,这也是实际中的全局最小值。顺序算法分为两个阶段。在第一阶段,计算目标函数值严格递减的静止点(SP)。第 2 阶段的目的是为第 1 阶段计算出的最后一个 SP 提供全局最优证明。针对 LPLCC 的每个阶段提出了两种不同的非线性编程公式,并通过高效的枚举算法加以解决。此外,还提出了计算 StQP 下限的新程序,这些程序易于实现,并能在一般情况下给出严格的下限。利用已知来源的大量测试问题进行的计算实验表明,两阶段顺序算法在实践中总体上是高效的。此外,通过利用矩阵的 StQP,该算法似乎是研究矩阵共存性的有效方法。
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引用次数: 0
The limitation of neural nets for approximation and optimization 神经网络在近似和优化方面的局限性
IF 1.8 3区 数学 Q1 Mathematics Pub Date : 2024-08-09 DOI: 10.1007/s10898-024-01426-9
T. Giovannelli, O. Sohab, L. N. Vicente

We are interested in assessing the use of neural networks as surrogate models to approximate and minimize objective functions in optimization problems. While neural networks are widely used for machine learning tasks such as classification and regression, their application in solving optimization problems has been limited. Our study begins by determining the best activation function for approximating the objective functions of popular nonlinear optimization test problems, and the evidence provided shows that ReLU and SiLU exhibit the best performance on both training and testing data. We then analyze the accuracy of function value, gradient, and Hessian approximations for such objective functions obtained through interpolation/regression models and neural networks. When compared to interpolation/regression models, neural networks can deliver competitive zero- and first-order approximations (at a high training cost) but underperform on second-order approximation. However, it is shown that combining a neural net activation function with the natural basis for quadratic interpolation/regression can waive the necessity of including cross terms in the natural basis, leading to models with fewer parameters to determine. Lastly, we provide evidence that the performance of a state-of-the-art derivative-free optimization algorithm can hardly be improved when the gradient of an objective function is approximated using any of the surrogate models considered, including neural networks.

我们有兴趣评估在优化问题中使用神经网络作为近似和最小化目标函数的代理模型。虽然神经网络被广泛用于分类和回归等机器学习任务,但其在解决优化问题方面的应用却很有限。我们的研究首先确定了近似常用非线性优化测试问题目标函数的最佳激活函数,所提供的证据表明,ReLU 和 SiLU 在训练和测试数据上都表现出最佳性能。然后,我们分析了通过插值/回归模型和神经网络获得的此类目标函数的函数值、梯度和赫塞斯近似值的准确性。与插值/回归模型相比,神经网络可以提供有竞争力的零阶和一阶近似(训练成本较高),但在二阶近似方面表现不佳。不过,研究表明,将神经网络激活函数与二次插值/回归的自然基相结合,可以免除在自然基中加入交叉项的必要性,从而减少模型需要确定的参数。最后,我们提供的证据表明,当使用包括神经网络在内的任何代用模型对目标函数梯度进行逼近时,最先进的无导数优化算法的性能很难得到改善。
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引用次数: 0
Relaxed projection methods for solving variational inequality problems 解决变分不等式问题的松弛投影法
IF 1.8 3区 数学 Q1 Mathematics Pub Date : 2024-08-08 DOI: 10.1007/s10898-024-01398-w
Pham Ngoc Anh
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引用次数: 0
期刊
Journal of Global Optimization
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