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 fo...
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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.
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 implement...
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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.
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 ...
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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.
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...
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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.
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 autonom...
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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.
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 advers...
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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.
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 chal...
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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.
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...
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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.
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-relaxa...
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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.
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...
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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.
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