scholarly journals A Lagrangian Multiplier Method for TDOA and FDOA Positioning of Multiple Disjoint Sources with Distance and Velocity Correlation Constraints

2020 ◽  
Vol 2020 ◽  
pp. 1-23 ◽  
Author(s):  
Bin Yang ◽  
Zeyu Yang ◽  
Ding Wang
Keyword(s):  
Source Localization ◽  
Optimization Problem ◽  
Inequality Constraints ◽  
Lagrangian Multiplier ◽  
Optimal Result ◽  
Constrained Optimization Problem ◽  
Exponential Functions ◽  
Velocity Correlation ◽  
Lagrangian Multipliers ◽  
Lagrangian Multiplier Method

This paper considers the source localization problem using time differences of arrival (TDOA) and frequency differences of arrival (FDOA) for multiple disjoint sources moving together with constraints on their distances and velocity correlation. To make full use of the synergistic improvement of multiple source localization, the constraints on all sources are combined together to obtain the optimal result. Unlike the existing methods that can achieve the normal Cramér-Rao lower bound (CRLB), our object is to further improve the accuracy of the estimation with constraints. On the basis of maximum likelihood criteria, a Lagrangian estimator is developed to solve the constrained optimization problem by iterative algorithm. Specifically, by transforming the inequality constraints into exponential functions, Lagrangian multipliers can be used to determine the source locations via Newton’s method. In addition, the constrained CRLB for source localization with distance and velocity correlation constraints is also derived. The estimated accuracy of the source positions and velocities is shown to achieve the constrained CRLB. Simulations are included to confirm the advantages of the proposed method over the existing methods.

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.

scholarly journals FNNC: Achieving Fairness through Neural Networks

2020 ◽  
Author(s):  
Manisha Padala ◽  
Sujit Gujar
Keyword(s):  
Gradient Descent ◽  
Optimization Problem ◽  
State Of The Art ◽  
High Accuracy ◽  
Stochastic Gradient Descent ◽  
Classification Models ◽  
Constrained Optimization Problem ◽  
Lagrangian Multipliers ◽  
Fairness Constraints ◽  
Generalization Errors

In classification models, fairness can be ensured by solving a constrained optimization problem. We focus on fairness constraints like Disparate Impact, Demographic Parity, and Equalized Odds, which are non-decomposable and non-convex. Researchers define convex surrogates of the constraints and then apply convex optimization frameworks to obtain fair classifiers. Surrogates serve as an upper bound to the actual constraints, and convexifying fairness constraints is challenging. We propose a neural network-based framework, \emph{FNNC}, to achieve fairness while maintaining high accuracy in classification. The above fairness constraints are included in the loss using Lagrangian multipliers. We prove bounds on generalization errors for the constrained losses which asymptotically go to zero. The network is optimized using two-step mini-batch stochastic gradient descent. Our experiments show that FNNC performs as good as the state of the art, if not better. The experimental evidence supplements our theoretical guarantees. In summary, we have an automated solution to achieve fairness in classification, which is easily extendable to many fairness constraints.

Simultaneous Optimal Robot Base Placement and Motion Planning Using Expanded Lagrangian Homotopy

2016 ◽  
Author(s):  
Audelia Gumarus Dharmawan ◽  
Shaohui Foong ◽  
Gim Song Soh
Keyword(s):  
Motion Planning ◽  
Constrained Optimization ◽  
Optimization Problem ◽  
Planning Process ◽  
Inequality Constraints ◽  
Future Research ◽  
Lagrangian System ◽  
Constrained Optimization Problem ◽  
Continuation Problem ◽  
Joint Limits

This paper presents a new approach to simultaneously determine the optimal robot base placement and motion plan for a prescribed set of tasks using expanded Lagrangian homotopy. First, the optimal base placement is formulated as a constrained optimization problem based on manipulability and kinematics of the robot. Then, the constrained optimization problem is expressed into the expanded Lagrangian system and subsequently converted into a homotopy map. Finally, the Newton-Raphson method is used to solve the constrained optimization problem as a continuation problem. The complete formulation for the case of a 6-DOF manipulator is presented and a planar optimal mobile platform motion planning approach is proposed. Numerical simulations confirm that the proposed approach is able to achieve the desired results. The approach also shows the potential for incorporating factors such as joint limits or collision avoidance into the motion planning process as inequality constraints and will be part of future research.

The Improvement of Newton Method and the Validation of Optimization Idea of Blind Walking Repeatedly

2011 ◽  
Vol 250-253 ◽  
pp. 4061-4064
Author(s):  
Chun Ling Zhang
Keyword(s):  
Saddle Point ◽  
Newton Method ◽  
Optimization Problem ◽  
Initial Point ◽  
Optimization Method ◽  
Maximum Point ◽  
Optimal Result ◽  
Feasible Domain ◽  
Parallel Arithmetic ◽  
Newton Direction

The existence of maximum point, oddity point and saddle point often leads to computation failure. The optimization idea is based on the reality that the optimum towards the local minimum related the initial point. After getting several optimal results with different initial point, the best result is taken as the final optimal result. The arithmetic improvement of multi-dimension Newton method is improved. The improvement is important for the optimization method with grads convergence rule or searching direction constructed by grads. A computational example with a saddle point, maximum point and oddity point is studied by multi-dimension Newton method, damped Newton method and Newton direction method. The importance of the idea of blind walking repeatedly is testified. Owing to the parallel arithmetic of modernistic optimization method, it does not need to study optimization problem with seriate feasible domain by modernistic optimization method.

scholarly journals Nonconvex constrained optimization by a filtering branch and bound

2020 ◽  
Author(s):  
Gabriele Eichfelder ◽  
Kathrin Klamroth ◽  
Julia Niebling
Keyword(s):  
Constrained Optimization ◽  
Optimization Problem ◽  
Feasible Solution ◽  
Constrained Optimization Problem ◽  
Numerical Tests ◽  
Nonconvex Constrained Optimization ◽  
Objective Value ◽  
Feasible Solutions ◽  
Quality Guarantee ◽  
Single Objective

AbstractA major difficulty in optimization with nonconvex constraints is to find feasible solutions. As simple examples show, the $$\alpha $$ α BB-algorithm for single-objective optimization may fail to compute feasible solutions even though this algorithm is a popular method in global optimization. In this work, we introduce a filtering approach motivated by a multiobjective reformulation of the constrained optimization problem. Moreover, the multiobjective reformulation enables to identify the trade-off between constraint satisfaction and objective value which is also reflected in the quality guarantee. Numerical tests validate that we indeed can find feasible and often optimal solutions where the classical single-objective $$\alpha $$ α BB method fails, i.e., it terminates without ever finding a feasible solution.

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