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On the stationarity for nonlinear optimization problems with polyhedral constraints

MATHEMATICAL programming 2024年第1-2期205卷 107-134页

作者： di Serafino, Daniela Hager, William W. W. Toraldo, Gerardo Viola, Marco Univ Naples Federico II Dept Math & Applicat Naples Italy Univ Florida Dept Math Gainesville FL USA Univ Campania L Vanvitelli Dept Math & Phys Caserta Italy Univ Coll Dublin Sch Math & Stat Dublin Ireland

For polyhedral constrained optimization problems and a feasible point x, it is shown that the projection of the negative gradient on the tangent cone, denoted del(Omega) f(x), has an orthogonal decomposition of the form beta(x) + phi(x). At a stationary point, del(Omega) f(x) = 0 so parallel to del(Omega) f(x)parallel to reflects the distance to a stationary point. Away from a stationary point, parallel to beta(x)parallel to and parallel to phi(x)parallel to measure different aspects of optimality since beta(x) only vanishes when the KKT multipliers at x have the correct sign, while phi(x) only vanishes when x is a stationary point in the active manifold. As an application of the theory, an active set algorithm is developed for convex quadratic programs which adapts the flow of the algorithm based on a comparison between parallel to beta(x)parallel to and parallel to phi(x)parallel to.

关键词： nonlinear programming Polyhedral constraints Stationarity Gradient projection Proportionality

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An approximation algorithm for mobile multi-agent monitoring and routing problem

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FLEXIBLE SERVICES AND MANUFACTURING JOURNAL 2025年 1-27页

作者： Kim, Gwang Jeong, Yoonjea Chosun Univ Div Business Adm Gwangju South Korea Pukyong Natl Univ Div Syst Management & Safety Engn Busan 48513 South Korea

In this study, we present a monitoring scheme with a group of agents, that is considered a practical challenge in operations management. In particular, mobile multi-agents, such as drones, can facilitate the implementation of monitoring tasks in more efficient and flexible manners. However, comparing to a monitoring system with stationary agents, a monitoring problem with mobile multi-agents must incorporate the routing plan of agents together. Accordingly, this study provides a monitoring (patrolling) and routing model coupled with mobile agents. The focal interest of the paper is to obtain the optimal routes of agents such that the total utilities from the monitoring process are maximized over a specific duration of the planning horizon. To reflect a real-world situation, we examine a three-dimensional space along with a stochastic process of event occurrence. The corresponding model is formulated based on an integer programming model with a nonlinear objective function, also known as an NP-hard problem. In addition, we show the mathematical formulation based on a submodular maximization problem and propose a heuristic algorithm in light of submodularity to guarantee sub-optimal solutions along with the efficiency of the algorithm via numerical experiments.

关键词： Mobile multi-agent systems Monitoring problem Route planning nonlinear programming Submodularity

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Cost efficiency in water supply systems: An applied review on optimization models for the pump scheduling problem

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2025年第1期323卷 1-19页

作者： Bras, Marlene Moura, Ana Andrade-Campos, Antonio Univ Aveiro Dept Econ Management Ind Engn & Tourism Res Unit Governance Competitiveness & Publ Pol GOV Aveiro Portugal Univ Aveiro Ctr Mech Technol & Automat DEM TEMA Dept Mech Engn Aveiro Portugal Intelligent Syst Associate Lab LASI Guimaraes Portugal

The need for efficient pump operation in water supply systems (WSS) has become increasingly important over time, driven by the growing energy consumption and the associated energy costs. Forecasts for 2050 anticipate a global increase in water demand by 55%, indicating an increasing surge in WSS energy consumption. Control of pumping stations, which consume 70% of the energy in WSS, is the most critical area for optimization. This optimization challenge is commonly referred as the pump scheduling problem (PSP), and can be addressed using a variety of mathematical formulations. While numerous formulations exist to solve this optimization problem, the large majority of the studies are focus on the optimization techniques, sidelining the problem formulation. Due to the unique physical characteristics of each WSS, individual mathematical formulations may exhibit different levels of performance. In addition to general pumps' operation optimization, the employment of variable speed pumps (VSP) can lead to significant energy savings compared to fixed speed pumps (FSP). However, despite their apparent benefits, many established optimization models for the PSP have not yet incorporated VSP decision variables into their formulations. Therefore, this work aims to review the main mathematical formulations for the pump scheduling problem for WSS with VSP and to present a quantitative comparative study of three mathematical formulations applied to a case study in the literature. The comparative analysis here presented revealed that the optimization model based on duty cycles is more cost-efficient when compared to alternative approaches discussed in the literature.

关键词： nonlinear programming Linear programming Water supply systems Pump scheduling problem Optimization models

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Recursive Polynomial Method for Fast Collision Avoidance Maneuver Design

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IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS 2025年第2期61卷 2914-2925页

作者： Pavanello, Zeno Pirovano, Laura Armellin, Roberto Univ Auckland Te Punaha Atea Space Inst Auckland 1010 New Zealand

A simple and reliable algorithm for collision avoidance maneuvers (CAMs), capable of computing impulsive, multi-impulsive, and low-thrust maneuvers, is proposed. The probability of collision (PoC) is approximated by a polynomial of arbitrary order as a function of the control, transforming the CAM design into a polynomial program, which also considers the change in the time of closest approach and the linear evolution of the covariances due to the maneuver. The solution procedure is initiated by computing the CAM via a first-order greedy optimization approach, wherein the control action is applied in the direction of the gradient of PoC to maximize its change. Successively, the polynomial is truncated at higher orders, and the solution of the previous order is used to linearize the constraint. This enables achieving accurate solutions even for highly nonlinear safety metrics and dynamics. Since the optimization process comprises only polynomial evaluations, the method is computationally efficient, with run times typically below 1 s. Moreover, no restrictions on the considered dynamics are necessary;therefore, results are shown for Keplerian, J2, and circular restricted three-body problem dynamics.

关键词： Polynomials Earth Aerodynamics Planetary orbits Computational modeling Aerospace and electronic systems Space vehicles Satellites Vehicle dynamics Tensors Fuel optimal control nonlinear programming optimization methods polynomial approximation space vehicle control

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Time-optimal global trajectory planning for autonomous valet parking: An improved hybrid A-star algorithm-based optimization control approach

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INTERNATIONAL JOURNAL OF ADVANCED ROBOTIC SYSTEMS 2025年第2期22卷

作者： Qiu, Duoyang Li, Xianyang Yang, Hao Zhu, Xiaomeng Hefei Univ Sch Adv Mfg Engn Hefei Peoples R China Hefei Univ Anhui Prov Engn Technol Res Ctr Intelligent Vehicl Hefei Peoples R China Univ Sci & Technol China Grad Sch Sci Isl Branch 99 Jinxiu Ave Hefei Anhui Peoples R China

Autonomous valet parking is an L4-level self-driving technology that solves the "last-mile freedom" of automobile users. Aiming at the motion planning problem involved in the decision-making layer of autonomous valet parking, this article proposes a trajectory planning method that obtains global time optimization by solving the optimization problem. The method is based on the path points obtained by the improved hybrid A-star algorithm as the reference waypoints. The global trajectory planning problem is segmented. In each section, the physical system constraints of the vehicle, the boundary condition constraints, and obstacle avoidance constraints in the parking process are comprehensively considered, and the parking trajectory planning task is described as an optimal control problem. The optimal problem is transformed into a nonlinear programming problem by the numerical optimization method. Finally, the global optimal trajectory is obtained on the basis of ensuring the continuity of state variables and control variables. Simulation experiments verify the effectiveness of the algorithm for horizontal and vertical parking. The real vehicle test shows that the vehicle can be safely, quickly, and accurately parked in the parking space.

关键词： autonomous parking motion planning trajectory planning optimal control nonlinear programming

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Trajectory optimization of unmanned aerial vehicles in the electromagnetic environment

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OPTIMIZATION AND ENGINEERING 2025年第1期26卷 159-198页

作者： Atayev, Anvarbek Fliege, Jorg Zemkoho, Alain Univ Warwick Math Inst Coventry CV4 7AL England Univ Southampt Math Sci Southampton SO17 1BJ England

We consider a type of routing problems common in defence and security, in which we control a fleet of unmanned aerial vehicles (UAVs) that have to reach one or more target locations without being detected by an adversary. Detection can be carried out by a variety of sensors (radio receivers, cameras, personnel, etc) placed by the adversary around the target sites. We model the act of detecting a UAV from first principles by noting that sensors work by monitoring frequencies in the electromagnetic spectrum for signals or noise emitted. By this, we are able to provide a flexible and versatile nonlinear optimisation framework in which the problem is modeled as a novel trajectory optimisation problem with paths of the UAVs as continuous arcs in an Euclidean space. The flexibility of our approach is exhibited by the fact that we can easily consider various relevant objectives, among them minimising the overall probability of detection and maximising the location error that the adversary experiences when trying to locate our UAVs. Our model is also versatile enough to consider the act of jamming, in which one or more of our UAVs intentionally send out signals to interfere with the operations of the adversary's sensors. Numerical results show the flexibility of our framework, and that we can solve realistic instances of this problem type.

关键词： nonlinear programming OR in defence Trajectory optimisation

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Multi-drone rescue search in a large network

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2025年第3期324卷 787-798页

作者： Jaillet, Patrick Gonzalez, Victor MIT Operat Res Ctr Cambridge MA 02139 USA MIT Operat Res Ctr Elect Engn & Comp Sci Cambridge MA USA

Natural disasters are recurring emergencies that can result innumerous deaths and injuries. When a natural disaster occurs, rescue teams can be sent to help affected survivors, but deploying them efficiently is a challenge. Rescuers not knowing where affected survivors are located poses a significant challenge in delivering aid. With the development of new technologies, there are new possibilities to reduce this uncertainty, alleviating this challenge. One can first send out automated drones to locate affected survivors and then send rescue teams to their locations. We develop a model for the search process and construct mathematical methods to construct efficient search routes. We utilize a divide and conquer technique to determine the routes that are most likely to yield an efficient search. We combine this with our mathematical methods to construct efficient search routes in real-time and a method to update these routes in real-time as drones gather information.

关键词： nonlinear programming Large-scale optimization Routing Networks

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Degree reduction techniques for polynomial optimization problems

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2025年第2期322卷 401-413页

作者： Gonzalez-Rodriguez, Brais Naoum-Sawaya, Joe Western Univ Ivey Business Sch 1255 Western Rd London ON N6G 4W1 Canada

This paper presents anew approach to quadrify a polynomial programming problem, i.e. reduce the polynomial program to a quadratic program, before solving it. The proposed approach, QUAD-RLT, exploits the Reformulation-Linearization Technique (RLT) structure to obtain smaller relaxations that can be solved faster and still provide high quality bounds. QUAD-RLT is compared to other quadrification techniques that have been previously discussed in the literature. The paper presents theoretical as well as computational results showing the advantage of QUAD-RLT compared to other quadrification techniques. Furthermore, rather than quadrifying a polynomial program, QUAD-RLT is generalized to reduce the degree of the polynomial to any degree. Computational results show that reducing the degree of the polynomial to a degree that is higher than two provides computational advantages in certain cases compared to fully quadrifying the problem. Finally, QUAD-RLT along with other quadrification/degree reduction schemes are implemented and made available in the freely available software RAPOSa.

关键词： nonlinear programming Quadrification Degree reduction Reformulation-linearization technique Polynomial programming

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An interior point method for nonlinear constrained derivative-free optimization

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OPTIMIZATION METHODS & SOFTWARE 2025年

作者： Brilli, A. Liuzzi, G. Lucidi, S. Sapienza Univ Rome Dept Comp Control & Management Engn A Ruberti Rome Italy

In this paper, we consider constrained optimization problems where both the objective and constraint functions are of the black-box type. Furthermore, we assume that the nonlinear inequality constraints are non-relaxable, i.e. the values of the objective function and constraints cannot be computed outside of the feasible region. This situation happens frequently in practice especially in the black-box setting where function values are typically computed by means of complex simulation programs which may fail to execute if the considered point is outside of the feasible region. For such problems, we propose a new derivative-free optimization method which is based on the use of a merit function that handles inequality constraints by means of a log-barrier approach and equality constraints by means of an exterior penalty approach. We prove the convergence of the proposed method to KKT stationary points of the problem under standard assumptions (we do not require any convexity assumption). Furthermore, we also carry out a preliminary numerical experience on standard test problems and comparison with state-of-the-art solvers showing the efficiency of the proposed method.

关键词： Derivative-free optimization nonlinear programming interior point methods

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Asteroid mining: ACT&Friends’ results for the GTOC12 problem

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Astrodynamics 2025年第1期9卷 19-40页

作者： Dario Izzo Marcus Märtens Laurent Beauregard Max Bannach Giacomo Acciarini Emmanuel Blazquez Alexander Hadjiivanov Jai Grover Gernot Heißel Yuri Shimane Chit Hong Yam European Space Technology and Research Center Noordwijk2201 AZthe Netherlands European Space Operations Centre Darmstadt64293Germany Surrey Space Center University of SurreyGuildfordGU27XHUK School of Aerospace Engineering Georgia Institute of TechnologyAtlantaGeorgia30332USA Mission Design and Operations Group ispace inc.Sumitomo Fudosan Hamacho Building 3F3-42-3Nihonbashi HamachoChuo-kuTokyo103-0007Japan

In 2023, the 12th edition of Global Trajectory Competition was organized around the problem referred to as “Sustainable Asteroid Mining”. This paper reports the developments that led to the solution proposed by ESA’s Advanced Concepts Team. Beyond the fact that the proposed approach failed to rank higher than fourth in the final competition leader-board, several innovative fundamental methodologies were developed which have a broader application. In particular, new methods based on machine learning as well as on manipulating the fundamental laws of astrodynamics were developed and able to fill with remarkable accuracy the gap between full low-thrust trajectories and their representation as impulsive Lambert transfers. A novel technique was devised to formulate the challenge of optimal subset selection from a repository of pre-existing optimal mining trajectories as an integer linear programming problem. Finally, the fundamental problem of searching for single optimal mining trajectories (mining and collecting all resources), albeit ignoring the possibility of having intra-ship collaboration and thus sub-optimal in the case of the GTOC12 problem, was efficiently solved by means of a novel search based on a look-ahead score and thus making sure to select asteroids that had chances to be re-visited later on.

关键词： GTOC low thrust asteroid mining machine learning mixed-integer linear programming nonlinear programming

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