Representation of the Lagrangian multipliers for coast periods of optimum trajectories.

AIAA Journal ◽  
1966 ◽  
Vol 4 (4) ◽  
pp. 729-730 ◽  
Author(s):  
PETER R. HEMPEL
Keyword(s):  
Lagrangian Multipliers ◽  
Optimum Trajectories

The boundary element method with Lagrangian multipliers

2008 ◽  
Vol 25 (6) ◽  
pp. 1303-1319 ◽  
Author(s):  
Gabriel N. Gatica ◽  
Martin Healey ◽  
Norbert Heuer
Keyword(s):  
Boundary Element Method ◽  
Boundary Element ◽  
Lagrangian Multipliers ◽  
Element Method

scholarly journals Joint 3-D Positioning and Power Allocation for UAV Relay Aided by Geographic Information

2021 ◽  
Author(s):  
Pengfei Yi ◽  
Liang Zhu ◽  
Lipeng Zhu ◽  
Zhenyu Xiao
Keyword(s):  
Power Allocation ◽  
Upper Bound ◽  
Three Dimensional ◽  
Geographic Information ◽  
Base Station ◽  
Lagrangian Multipliers ◽  
Loop Optimization ◽  
Convex Problem ◽  
Communication Links ◽  
Minimum Capacity

<div>In this paper, we study to employ geographic information to address the blockage problem of air-to-ground links between UAV and terrestrial nodes. In particular, a UAV relay is deployed to establish communication links from a ground base station to multiple ground users. To improve communication capacity, we fifirst model the blockage effect caused by buildings according to the three-dimensional (3-D) geographic information. Then, an optimization problem is formulated to maximize the minimum capacity among users by jointly optimizing the 3-D position and power allocation of the UAV relay, under the constraints of link capacity, maximum transmit power, and blockage. To solve this complex non-convex problem, a two-loop optimization framework is developed based on Lagrangian relaxation. The outer-loop aims to obtain proper Lagrangian multipliers to ensure the solution of the Lagrangian problem converge to the tightest upper bound on the original problem. The inner-loop solves the Lagrangian problem by applying the block coordinate descent (BCD) and successive convex approximation (SCA) techniques, where UAV 3-D positioning and power allocation are alternately optimized in each iteration. Simulation results confifirm that the proposed solution signifificantly outperforms two benchmark schemes and achieves a performance close to the upper bound on the UAV relay system.</div>

A New Constrained Shape Deformation Operator With Applications in Footwear Design

2008 ◽  
Author(s):  
Ying Xu ◽  
Ajay Joneja
Keyword(s):  
Numerical Methods ◽  
Optimization Problem ◽  
Shape Change ◽  
Geometric Approach ◽  
Shape Deformation ◽  
Uzawa Method ◽  
Constrained Optimization Problem ◽  
Integral Property ◽  
Lagrangian Multipliers ◽  
Footwear Design

In this paper, we address a problem that arises in several engineering applications: the deformation of a curve with constraints on its length. Since length is an integral property, typically computed by numerical methods, therefore implementing such shape change operations is non trivial. Recently some researchers have attempted to solve such problems for multi-resolution representations of curves. However, we take a differential geometric approach. The modification problem is formulated as constrained optimization problem, which is subsequently converted to an unconstrained min-max problem using Lagrangian multipliers. This problem is solved using the Uzawa method. The approach is implemented in MATLAB™, and some examples are presented in the paper.

Dynamic Cooperative Speed Optimization at Signalized Arterials with Various Platoons

2019 ◽  
Vol 2673 (5) ◽  
pp. 528-537 ◽  
Author(s):  
Xia Wu ◽  
Xiangmo Zhao ◽  
Qi Xin ◽  
Qiaoli Yang ◽  
Shaowei Yu ◽  
...  
Keyword(s):  
Traffic Signals ◽  
Programming Algorithm ◽  
Connected Vehicles ◽  
Driving Behaviors ◽  
Optimum Trajectories ◽  
Speed Optimization ◽  
The Status ◽  
Driving Model ◽  
Optimum Approach ◽  
Multiple Vehicles

Aggressive and inappropriate driving behaviors will lead to excessive fuel consumption. Both the Signal Phase and Timing (SPaT) and the status of preceding vehicles have significant impacts on driving behaviors. Drivers can obtain accurate SPaT information and the status of preceding vehicles via V2X communications. Many speed advisory strategies have been presented based on the consideration of this information. However, existing studies do not consider the cooperative optimization of multiple intersections and various platoons. Once connected vehicles travel through intersections with their own fuel-optimum trajectories, the following vehicles could be adversely affected by the preceding vehicles, leading to the following vehicles being stopped at the intersection. To address these problems, this paper presents an improved cooperative eco-driving model for when a vehicle passes two successive traffic signals during the green phase; a dynamic nonlinear programming algorithm is used to generate the optimal speed profile for various platoons considering the SPaT and the preceding vehicles’ status. Numerous simulations on VISSIM for uninformed and connected vehicles haves been conducted to make comparison analysis. It is apparent that the proposed eco-driving model produces a significant fuel saving. In addition, cooperative optimization for the various platoons and separate optimization of multiple vehicles were performed to seek the most effective solution. The results indicated that systematic optimization (cooperative optimization of the all vehicles) is identified as the fuel-optimum approach in comparison to the separate optimization.

Unilateral Impact and Contact of Elastic Structures Using Lagrangian Multipliers

2011 ◽  
Author(s):  
Sebastian Tatzko
Keyword(s):  
Equations Of Motion ◽  
Structural Equations ◽  
Contact Forces ◽  
Integration Scheme ◽  
Lagrangian Multiplier ◽  
Elastic Structures ◽  
Lagrangian Multipliers ◽  
Linear Elastic ◽  
Time Stepping ◽  
Numerical Effort

This paper deals with linear elastic structures exposed to impact and contact phenomena. Within a time stepping integration scheme contact forces are computed with a Lagrangian multiplier approach. The main focus is turned on a simplified solving method of the linear complementarity problem for the frictionless contact. Numerical effort is reduced by applying a Craig-Bampton transformation to the structural equations of motion.

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