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The order of appearance of the product of the first and second Lucas numbers

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arXiv 2025年

作者： Xiao, Huiming Li, Hongjian Yuan, Pingzhi School of Mathematical Sciences South China Normal University Guangdong Guangzhou510631 China School of Mathematics and Statistics Guangdong University of Foreign Studies Guangdong Guangzhou510006 China

Let (Un)n≥0 and (Vn)n≥0 be the first and second Lucas sequences, respectively. Let m be a positive integer. Then the order of appearance of m in the first Lucas sequence is defined as the smallest positive integer k such that m divides Uk and denoted by τ(m). In this paper, we give explicit formulae for the terms τ(UmVn), τ(UmUn), τ(VmVn) and τ(UnUn+pUn+2p), where p ≥ 3 is a prime number. Copyright © 2025, The Authors. All rights reserved.

关键词： mixed-integer linear programming

C∞ RECTIFIABILITY OF STATIONARY VARIFOLDS

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arXiv 2025年

作者： Brena, Camillo de Lellis, Camillo Franceschini, Federico School of Mathematics Institute for Advanced Study 1 Einstein Dr. PrincetonNJ08540 United States

In this paper we prove that, for every integers m ≥ 2 and n ≥ 1, the support of any stationary m-dimensional integer rectifiable varifold V in an open set U ⊂ m+n is C∞ rectifiable, namely it can be covered, up to ... 详细信息

关键词： mixed-integer linear programming

Data-Driven Distributionally Robust mixed-integer Control through Lifted Control Policy

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arXiv 2025年

作者： Ma, Xutao Ning, Chao Du, Wenli Shi, Yang Department of Automation Shanghai Jiao Tong University Shanghai200240 China Key Laboratory of System Control and Information Processing Ministry of Education of China Shanghai200240 China the Key Laboratory of Smart Manufacturing in Energy Chemical Process Ministry of Education East China University of Science and Technology Shanghai200237 China the Department of Mechanical Engineering University of Victoria VictoriaBCV8W 3P6 Canada

This paper investigates the finite-horizon distributionally robust mixed-integer control (DRMIC) of uncertain linear systems. However, deriving an optimal causal feedback control policy to this DRMIC problem is computationally formidable for most ambiguity sets. To address the computational challenge, we propose a novel distributionally robust lifted control policy (DR-LCP) method to derive a high-quality approximate solution to this DRMIC problem for a rich class of Wasserstein metric-based ambiguity sets, including the Wasserstein ambiguity set and its variants. In theory, we analyze the asymptotic performance and establish a tight non-asymptotic bound of the proposed method. In numerical experiments, the proposed DR-LCP method empirically demonstrates superior performance compared with existing methods in the literature. © 2025, CC BY.

关键词： mixed-integer linear programming

An analysis of optimization problems involving ReLU neural networks

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arXiv 2025年

作者： Plate, Christoph Hahn, Mirko Klimek, Alexander Ganzer, Caroline Sundmacher, Kai Sager, Sebastian Otto von Guericke University Magdeburg Magdeburg Germany Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg Germany

Solving mixed-integer optimization problems with embedded neural networks with ReLU activation functions is challenging. Big-M coefficients that arise in relaxing binary decisions related to these functions grow exponentially with the number of layers. We survey and propose different approaches to analyze and improve the run time behavior of mixed-integer programming solvers in this context. Among them are clipped variants and regularization techniques applied during training as well as optimization-based bound tightening and a novel scaling for given ReLU networks. We numerically compare these approaches for three benchmark problems from the literature. We use the number of linear regions, the percentage of stable neurons, and overall computational effort as indicators. As a major takeaway we observe and quantify a trade-off between the often desired redundancy of neural network models versus the computational costs for solving related optimization problems. © 2025, CC BY.

关键词： mixed-integer linear programming

Learning Maritime Inventory Routing Optimization

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arXiv 2025年

作者： Chen, Rui Liu, Defeng Jiang, Nan Gupta, Rishabh Kilinc, Mustafa Lodi, Andrea The Chinese University of Hong Kong Shenzhen China Polytechnique Montréal Canada Cornell Tech United States ExxonMobil Technology and Engineering Company United States

We propose a machine learning-based local search approach for finding feasible solutions of large-scale maritime inventory routing optimization problems. Given the combinatorial complexity of the problems, we integrate a graph neural network-based neighborhood selection method to enhance local search efficiency. Our approach enables a structured exploration of vessel neighborhoods, improving solution quality while maintaining computational efficiency. Through extensive computational experiments on realistic instances, we demonstrate that our method outperforms direct mixed-integer programming in solution time. © 2025, CC BY.

关键词： mixed-integer linear programming

Demand Response Optimization MILP Framework for Microgrids with DERs

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arXiv 2025年

作者： Babu, K. Victor Sam Moses Chakraborty, Pratyush Pal, Mayukha ABB Ability Innovation Center Hyderabad500084 India Department of Electrical and Electronics Engineering BITS Pilani Hyderabad Campus Hyderabad500078 India

The integration of renewable energy sources in microgrids introduces significant operational challenges due to their intermittent nature and the mismatch between generation and demand patterns. Effective demand response (DR) strategies are crucial for maintaining system stability and economic efficiency, particularly in microgrids with high renewable penetration. This paper presents a comprehensive mixed-integer linear programming (MILP) framework for optimizing DR operations in a microgrid with solar generation and battery storage systems. The framework incorporates load classification, dynamic price thresholding, and multi-period coordination for optimal DR event scheduling. Analysis across seven distinct operational scenarios demonstrates consistent peak load reduction of 10% while achieving energy cost savings ranging from 13.1% to 38.0%. The highest performance was observed in scenarios with high solar generation, where the framework achieved 38.0% energy cost reduction through optimal coordination of renewable resources and DR actions. The results validate the framework’s effectiveness in managing diverse operational challenges while maintaining system stability and economic efficiency. © 2025, CC BY-NC-ND.

关键词： mixed-integer linear programming

The minimum edge-pancyclic graph of a given order

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arXiv 2025年

作者： Zhao, Xiamiao Yang, Yuxuan Lu, Mei Department of Mathematical Sciences Tsinghua University Beijing100084 China School of Science Beijing University of Posts and Telecommunications Beijing100876 China

A graph G of order n is called edge-pancyclic if, for every integer k with 3 ≤ k ≤ n, every edge of G lies in a cycle of length k. Determining the minimum size f(n) of a simple edge-pancyclic graph with n vertices seems difficult. Recently, Li, Liu and Zhan [5] gave both a lower bound and an upper bound of f(n). In this paper, we improve their lower bound by considering a new class of graphs and improve the upper bound by constructing a family of edge-pancyclic graphs. © 2025, CC BY.

关键词： mixed-integer linear programming

Experiments with Optimal Model Trees

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arXiv 2025年

作者： Roselli, Sabino Francesco Frank, Eibe Department of Electrical Engineering Chalmers University of Technology Gothenburg Sweden Department of Computer Science University of Waikato Hamilton New Zealand

Model trees provide an appealing way to perform interpretable machine learning for both classification and regression problems. In contrast to "classic" decision trees with constant values in their leaves, model trees can use linear combinations of predictor variables in their leaf nodes to form predictions, which can help achieve higher accuracy and smaller trees. Typical algorithms for learning model trees from training data work in a greedy fashion, growing the tree in a top-down manner by recursively splitting the data into smaller and smaller subsets. Crucially, the selected splits are only locally optimal, potentially rendering the tree overly complex and less accurate than a tree whose structure is globally optimal for the training data. In this paper, we empirically investigate the effect of constructing globally optimal model trees for classification and regression with linear support vector machines at the leaf nodes. To this end, we present mixed-integer linear programming formulations to learn optimal trees, compute such trees for a large collection of benchmark data sets, and compare their performance against greedily grown model trees in terms of interpretability and accuracy. We also compare to classic optimal and greedily grown decision trees, random forests, and support vector machines. Our results show that optimal model trees can achieve competitive accuracy with very small trees. We also investigate the effect on the accuracy of replacing axis-parallel splits with multivariate ones, foregoing interpretability while potentially obtaining greater accuracy. Copyright © 2025, The Authors. All rights reserved.

关键词： mixed-integer linear programming

Valid Cuts for the Design of Potential-based Flow Networks

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arXiv 2025年

作者： Börner, Pascal Klimm, Max Lutz, Annette Pfetsch, Marc E. Skutella, Martin Strubberg, Lea Department of Mathematics TU Darmstadt Germany Institute of Mathematics TU Berlin Germany

The construction of a cost minimal network for flows obeying physical laws is an important problem for the design of electricity, water, hydrogen, and natural gas infrastructures. We formulate this problem as a mixed-integer non-linear program with potential-based flows. The non-convexity of the constraints stemming from the potential-based flow model together with the binary variables indicating the decision to build a connection make these programs challenging to solve. We develop a novel class of valid inequalities on the fractional relaxations of the binary variables. Further, we show that this class of inequalities can be separated in polynomial time for solutions to a fractional relaxation. This makes it possible to incorporate these inequalities into a branch-and-cut framework. The advantage of these inequalities is lastly demonstrated in a computational study on the design of real-world gas transport networks. Copyright © 2025, The Authors. All rights reserved.

关键词： mixed-integer linear programming