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Properties, Extensions and Application of Piecewise Linearization for Euclidean Norm Optimization in R²

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2022年第2期195卷 418-448页

作者： Duguet, Alois Artigues, Christian Houssin, Laurent Ngueveu, Sandra Ulrich Univ Toulouse INP CNRS LAAS CNRS Toulouse France Univ Toulouse ISAE SUPAERO Toulouse France

This work considers nonconvex mixed integer nonlinear programming where nonlinearity comes from the presence of the two-dimensional euclidean norm in the objective or the constraints. We build from the euclidean norm piecewise linearization proposed by Camino et al. (Comput. Optim. Appl. https://***/10.1007/s10589-019-00083-z, 2019) that allows to solve such nonconvex problems via mixed-integer linear programming with an arbitrary approximation guarantee. Theoretical results are established that prove that this linearization is able to satisfy any given approximation level with the minimum number of pieces. An extension of the piecewise linearization approach is proposed. It shares the same theoretical properties for elliptic constraints and/or objective. An application shows the practical appeal of the elliptic linearization on a nonconvex beam layout mixed optimization problem coming from an industrial application.

关键词： mixed integer linear programming mixed integer nonlinear programming Euclidean norm linearization Approximation guarantee Multibeam satellites

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UAV Placement and Resource Allocation for Intelligent Reflecting Surface Assisted UAV-Based Wireless Networks

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IEEE COMMUNICATIONS LETTERS 2022年第5期26卷 1106-1110页

作者： Nguyen, Minh Dat Le, Long Bao Girard, Andre Univ Quebec INRS EMT Montreal PQ H5A 1K6 Canada

We design an unmanned aerial vehicle (UAV) based wireless network with wireless access and backhaul links leveraging an intelligent reflecting surface (IRS). This design aims to maximize the sum rate achieved by ground users (GUs) through optimizing the UAV placement, IRS phase shifts, and sub-channel assignments considering the wireless backhaul capacity constraint. To tackle the underlying mixed integer non-linear optimization problem (MINLP), we first derive the closed-form IRS phase shift solution;we then optimize the sub-channel assignment and UAV placement by using the alternating optimization method. Specifically, we propose an iterative sub-channel assignment method to efficiently utilize the bandwidth and balance bandwidth allocation for wireless access and backhaul links while maintaining the backhaul capacity constraint. Moreover, we employ the successive convex approximation (SCA) method to solve the UAV placement optimization sub-problem. We show the effectiveness of our proposed design via extensive numerical studies.

关键词： Autonomous aerial vehicles Optimization Wireless networks Resource management Bandwidth Rayleigh channels Power control UAV placement IRS sub-channel assignment wireless backhaul mixed integer nonlinear programming

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On the exact separation of cover inequalities of maximum-depth

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OPTIMIZATION LETTERS 2022年第2期16卷 449-469页

作者： Catanzaro, Daniele Coniglio, Stefano Furini, Fabio Catholic Univ Louvain Ctr Operat Res & Econometr CORE Louvain La Neuve Belgium Univ Southampton Dept Math Sci Southampton Hants England Consiglio Nazl Ric IASI CNR Ist Anal Sistemi Informat Antonio Ruberti Rome Italy

We investigate the problem of separating cover inequalities of maximum-depth exactly. We propose a pseudopolynomial-time dynamic-programming algorithm for its solution, thanks to which we show that this problem is weakly NP-hard (similarly to the problem of separating cover inequalities of maximum violation). We carry out extensive computational experiments on instances of the knapsack and the multi-dimensional knapsack problems with and without conflict constraints. The results show that, with a cutting-plane generation method based on the maximum-depth criterion, we can optimize over the cover-inequality closure by generating a number of cuts smaller than when adopting the standard maximum-violation criterion. We also introduce the Point-to-Hyperplane Distance Knapsack Problem (PHD-KP), a problem closely related to the separation problem for maximum-depth cover inequalities, and show how the proposed dynamic programming algorithm can be adapted for effectively solving the PHD-KP as well.

关键词： Knapsack problem Cover inequalities Dynamic programming mixed integer nonlinear programming Cutting plane generation

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The nature of optimal policies for deterministic finite-horizon inventory models

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INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE-OPERATIONS & LOGISTICS 2022年第1期9卷 39-60页

作者： Benkherouf, Lakdere Gilding, Brian H. Kuwait Univ Coll Sci Dept Stat & Operat Res POB 5969 Safat 13060 Kuwait Kuwait Univ Coll Sci Dept Math Safat Kuwait

This paper concerns a general formulation for a longstanding problem. This is the problem of determining a replenishment schedule that minimises the total cost of stocking and holding an inventory in a deterministic finite-horizon inventory model. The formulation permits the treatment of seemingly unrelated models in a single framework. These include classical lot-size models, batching models, repair models, recovery models and others. Admissible control policies are restricted to a partition of some closed interval on the real line. The solution of a mixed integer nonlinear programming problem (MINLP) delivers the optimal partition. It is shown that the MINLP possesses an optimal solution under very mild conditions. The theory of submodular functions on a lattice is the key to handling the integer variable. This theory permits the recovery and generalisations of earlier results on the interleaving property of optimal partitions and a convexity property of the value of the objective function. Past intractable inventory models with demand driven by a general differential equation and in the absence and in the presence of shortages and inflation are solved. This generalises a number of existing results in the literature.

关键词： Inventory model finite time horizon optimal schedule mixed integer nonlinear programming submodular function

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A n - D ant colony optimization with fuzzy logic for air traffic flow management

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OPERATIONAL RESEARCH 2022年第5期22卷 5035-5053页

作者： Ntakolia, Charis Lyridis, Dimitrios, V Hellen Air Force Acad Dekelia Air Base Dept Aeronaut Studies Sect Mat Engn Machining Technol & Prod Management 1919 Acharnes Athens Greece Natl Tech Univ Athens Lab Maritime Transport Athens 15772 Greece

Recent studies show that the number of flights is expected to be increased significantly by 2030, leading to air traffic capacity and congestion issues in the air sectors. This challenging management of the anticipated volume of flights has emerged new derivatives and procedures from the European Union and EUROCONTROL. Aligned with the new vision of future Air Traffic Flow Management (ATFM), such as Trajectory Based Operations, this study proposes a mixed integer nonlinear formulation of ATFM based on 4D trajectories and free flight aspects. The model targets to minimize the total costs derived from airborne and ground holding delays, speed deviations, route alterations and cancellation policies. To solve the proposed nonlinear formulation, a novel n - D ant colony optimization algorithm integrated with fuzzy logic (n - DACOF) is presented. Each flight level is represented as graph and the n - D stands for the n number of permitted flight levels. n - DACOF can solve the ATFM problem by constructing a route moving among n graphs. Due to the multi-objective formulation, fuzzy logic permits the qualitative evaluation of the generated routes by the algorithm. The results showed that n - DACOF outperformed the baseline algorithm ACO, as well as, the CPLEX solver within computing time limits.

关键词： mixed integer nonlinear programming 4D trajectories Air traffic flow management Ant colony optimization Fuzzy logic Metaheuristic algorithm

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A Multi-Objective Optimization Model for Microgrid Optimal Operation with Cooperative Game Theory Approach 20

A Multi-Objective Optimization Model for Microgrid Optimal O...

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20th IEEE India Council International Conference, INDICON 2023

作者： Sudhakar, Munnangi Srinivas, R.S. Kamireddy, Ravivarma Lokeshgupta, Bhamidi Dr. Y.S.R Anu College of Engineering & Technology Department of Electrical and Electronics Engineering Guntur India Srm University Department of Electrical and Electronics Engineering AP Guntur India

ISBN: (纸本)9798350305593

The optimal operation of microgrid (MG) is an important problem to attain significant benefits, which mainly improves the cost reduction in energy operation and also lowers the emission of environmental pollutants. The integration of renewable energy sources is one of the key factors to achieve significant microgrid operational benefits. A multi-objective MG optimal operation problem is formulated in this paper with different equality and inequality constraints to obtain significant economic and environmental benefits. The primary objective of the proposed model is to simultaneously minimization of the daily MG operation cost and environmental pollution under islanded mode and grid-connected mode. The proposed multi-objective problem in this paper is converted into a single-objective problem by employing a cooperative game theory approach and a mixed integer nonlinear programming (MINLP) solver is used to solve the final single-optimization problem. The effectiveness of the proposed model is evaluated by investigating multiple scenarios and case studies. The comprehensive analysis of the simulation results indicates that the proposed multi-objective MG optimization model by using the cooperative game theory approach can be able to provide significant economic and environmental benefits simultaneously. © 2023 IEEE.

关键词： Cooperative game theory approach Distributed energy resources Microgrid mixed integer nonlinear programming Multi-objective optimization

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Matheuristics vs. metaheuristics for joint lot-sizing and dynamic pricing problem with nonlinear demands

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COMPUTERS & OPERATIONS RESEARCH 2024年 163卷

作者： Terzi, Mourad Ouazene, Yassine Yalaoui, Alice Yalaoui, Farouk Univ Technol Troyes Comp Sci & Digital Soc Lab 12 Rue Marie Curie F-10004 Troyes France Univ Technol Troyes Chaire Connected Innovat 12 Rue Marie CurieCS 42060 F-10004 Troyes CS France Univ Technol Troyes Comp Sci & Digital Soc Lab LIST3N 12 rue Marie Curie F- 10004 Troyes France

This paper considers a joint dynamic pricing and production planning decisions problem for a profit-maximizing firm that produces and sells multiple products. The objective is to develop a coordinated decision approach for multi-product pricing and lot sizing decisions for a manufacturer considering a limited production capacity. The demand for each product is assumed to be iso-elastic and integrates the complementarity and the substitution effects between the products. First, the problem is formulated as a non-convex mixed integer nonlinear programming model (MINLP) incorporating capacity constraints, setup costs, and nonlinear demand functions. Then, since the model is nonlinear and non-convex, a set of approximate approaches based on the Genetic algorithm, Late Acceptance Hill Climbing and Simulated Annealing methods are designed to solve this problem. Based on this study, the performances of two variants of approximate methods: matheuristics and metaheuristics are discussed and analyzed. The extensive experimental study, performed on real-world inspired instances, shows that matheuristic methods with setup-variables encoding scheme outperform the rest of the methods. The research outcomes show that coordinating a decision-making process by optimizing both prices and production plans simultaneously can result in significant profit for a company. However, one must consider the joint effect of the parameters of the demand function as well as the impact of the production capacity. This comprehensive understanding enables the company to avoid excessive investments in less lucrative products with lower sales potential, thereby ensuring resource allocation aligns with profitability and market demand.

关键词： Dynamic pricing Lot-sizing Metaheuristics Matheuristics nonlinear demand model mixed integer nonlinear programming

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Piecewise Linearization of Bivariate nonlinear Functions: Minimizing the Number of Pieces Under a Bounded Approximation Error 7th

Piecewise Linearization of Bivariate Nonlinear Functions: Mi...

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7th International Symposium on Combinatorial Optimization (ISCO)

作者： Duguet, Alois Ngueveu, Sandra Ulrich Univ Toulouse LAAS CNRS CNRS INP Toulouse France

ISBN: (纸本)9783031185298;9783031185304

This work focuses on the approximation of bivariate functions into piecewise linear ones with a minimal number of pieces and under a bounded approximation error. Applications include the approximation of mixed integer nonlinear optimization problems into mixed integer linear ones that are in general easier to solve. A framework to build dedicated linearization algorithms is introduced, and a comparison to the state of the art heuristics shows their efficiency.

关键词： Piecewise linear approximation Bivariate nonlinear functions mixed integer nonlinear programming Heuristics

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A multi-level vehicle-to-grid optimal scheduling approach with EV economic dispatching model

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ENERGY REPORTS 2021年 7卷 22-37页

作者： Guo, Song Qiu, Zejing Xiao, Chupeng Liao, Hui Huang, Yuping Lei, Ting Wu, Dan Jiang, Qian Nari Grp Corp State Grid Elect Power Res Inst Luoyu Rd 134 Wuhan 430074 Peoples R China Wuhan Efficiency Evaluat Co Ltd State Grid Elect Power Res Inst Wuhan 430074 Peoples R China Chinese Acad Sci Guangzhou Inst Energy Convers Nengyuan Rd 2 Guangzhou 510640 Peoples R China State Grid Shanghai Elect Power Co Yuanshen Rd 1122 Shanghai 201204 Peoples R China

As electric vehicles (EVs) are eco-friendly and have the feature of flexible power storage, the electric vehicle development is strongly supported by many governments. The market share of EVs has been rising steadily in recent years to promote the long-term growth of the available EV battery resources to participate in vehicle to grid (V2G). When a large number of electric vehicles can connect to grids randomly for charging or discharging, this inevitably brings large load fluctuation in regions and new challenges to the operation scheduling and control of power systems. Therefore, this paper proposed a new multi-level optimal V2G scheduling approach, ensuring the smooth operation and control from the V2G control center to the EV users. This paper also introduced a new EV economic dispatch optimization model to minimize the operating costs of regional V2G systems. A case analysis of the IEEE 33-bus system with 100 EVs verified the feasibility of the proposed model and indicated that the proper size control of total EV battery capacities with ramp adjustment can reduce additional load fluctuations caused by large-scale vehicles to grid. Moreover, the multi-stage TOU pricing mode is another effective measures that can trigger a next round of EV charging and discharging service. (C) 2021 The Authors. Published by Elsevier Ltd.

关键词： Electric vehicle V2G operation Multi-level scheduling approach EV economic dispatching mixed integer nonlinear programming

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Conflict Analysis for MINLP

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INFORMS JOURNAL ON COMPUTING 2021年第2期33卷 421-435页

作者： Berthold, Timo Witzig, Jakob Fair Isaac Germany GmbH D-64625 Bensheim Germany Zuse Inst Berlin D-14195 Berlin Germany

The generalization of mixed integer program (MIP) techniques to deal with nonlinear, potentially nonconvex, constraints has been a fruitful direction of research for computational mixed integer nonlinear programs (MINLPs) in the last decade. In this paper, we follow that path in order to extend another essential subroutine of modern MIP solvers toward the case of nonlinear optimization: the analysis of infeasible subproblems for learning additional valid constraints. To this end, we derive two different strategies, geared toward two different solution approaches. These are using local dual proofs of infeasibility for LP-based branch-and-bound and the creation of nonlinear dual proofs for NLP-based branch-and-bound, respectively. We discuss implementation details of both approaches and present an extensive computational study, showing that both techniques can significantly enhance performance when solving MINLPs to global optimality. Summary of Contribution: This original article concerns the advancement of exact general-purpose algorithms for solving one of the largest and most prominent problem classes in optimization, mixed integer nonlinear programs (MINLPs). It demonstrates how methods for conflict analysis that learn from infeasible subproblems can be transferred to nonlinear optimization. Further, it develops theory for how nonlinear dual infeasibility proofs can be derived from a nonlinear relaxation. This paper features a thoroughly computational study regarding the impact of conflict analysis techniques on the overall performance of a state-of-the-art MINLP solver when solving MINLPs to global optimality.

关键词： mixed integer nonlinear programming infeasibility analysis conflict analysis branch-and-bound

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