scholarly journals A limited memory BFGS subspace algorithm for bound constrained nonsmooth problems

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xiangrong Li
Keyword(s):  
Subspace Algorithm ◽  
Limited Memory ◽  
Nonsmooth Problems

scholarly journals Analysis of limited-memory BFGS on a class of nonsmooth convex functions

2020 ◽  
Author(s):  
Azam Asl ◽  
Michael L Overton
Keyword(s):  
Gradient Method ◽  
Large Scale ◽  
Line Search ◽  
Piecewise Linear ◽  
Fixed Number ◽  
Linear Functions ◽  
Limited Memory ◽  
Nonsmooth Problems ◽  
Wolfe Line Search ◽  
Large Scale Unconstrained Optimization

Abstract The limited-memory BFGS (Broyden-Fletcher-Goldfarb-Shanno) method is widely used for large-scale unconstrained optimization, but its behavior on nonsmooth problems has received little attention. L-BFGS (limited memory BFGS) can be used with or without ‘scaling’; the use of scaling is normally recommended. A simple special case, when just one BFGS update is stored and used at every iteration, is sometimes also known as memoryless BFGS. We analyze memoryless BFGS with scaling, using any Armijo–Wolfe line search, on the function $f(x) = a|x^{(1)}| + \sum _{i=2}^{n} x^{(i)}$, initiated at any point $x_0$ with $x_0^{(1)}\not = 0$. We show that if $a\ge 2\sqrt{n-1}$, the absolute value of the normalized search direction generated by this method converges to a constant vector, and if, in addition, $a$ is larger than a quantity that depends on the Armijo parameter, then the iterates converge to a nonoptimal point $\bar x$ with $\bar x^{(1)}=0$, although $f$ is unbounded below. As we showed in previous work, the gradient method with any Armijo–Wolfe line search also fails on the same function if $a\geq \sqrt{n-1}$ and $a$ is larger than another quantity depending on the Armijo parameter, but scaled memoryless BFGS fails under a weaker condition relating $a$ to the Armijo parameter than that implying failure of the gradient method. Furthermore, in sharp contrast to the gradient method, if a specific standard Armijo–Wolfe bracketing line search is used, scaled memoryless BFGS fails when $a\ge 2 \sqrt{n-1}$regardless of the Armijo parameter. Finally, numerical experiments indicate that the results may extend to scaled L-BFGS with any fixed number of updates $m$, and to more general piecewise linear functions.

scholarly journals Attitude and Heading Estimation for Indoor Positioning Based on the Adaptive Cubature Kalman Filter

Micromachines ◽  
2021 ◽  
Vol 12 (1) ◽  
pp. 79
Author(s):  
Jijun Geng ◽  
Linyuan Xia ◽  
Dongjin Wu
Keyword(s):  
Kalman Filter ◽  
Angular Rate ◽  
Indoor Positioning ◽  
Location Based Services ◽  
Statistical Characteristics ◽  
Fading Memory ◽  
Limited Memory ◽  
Cubature Kalman Filter ◽  
The Mean ◽  
Heading Estimation

The demands for indoor positioning in location-based services (LBS) and applications grow rapidly. It is beneficial for indoor positioning to combine attitude and heading information. Accurate attitude and heading estimation based on magnetic, angular rate, and gravity (MARG) sensors of micro-electro-mechanical systems (MEMS) has received increasing attention due to its high availability and independence. This paper proposes a quaternion-based adaptive cubature Kalman filter (ACKF) algorithm to estimate the attitude and heading based on smart phone-embedded MARG sensors. In this algorithm, the fading memory weighted method and the limited memory weighted method are used to adaptively correct the statistical characteristics of the nonlinear system and reduce the estimation bias of the filter. The latest step data is used as the memory window data of the limited memory weighted method. Moreover, for restraining the divergence, the filter innovation sequence is used to rectify the noise covariance measurements and system. Besides, an adaptive factor based on prediction residual construction is used to overcome the filter model error and the influence of abnormal disturbance. In the static test, compared with the Sage-Husa cubature Kalman filter (SHCKF), cubature Kalman filter (CKF), and extended Kalman filter (EKF), the mean absolute errors (MAE) of the heading pitch and roll calculated by the proposed algorithm decreased by 4–18%, 14–29%, and 61–77% respectively. In the dynamic test, compared with the above three filters, the MAE of the heading reduced by 1–8%, 2–18%, and 2–21%, and the mean of location errors decreased by 9–22%, 19–31%, and 32–54% respectively by using the proposed algorithm for three participants. Generally, the proposed algorithm can effectively improve the accuracy of heading. Moreover, it can also improve the accuracy of attitude under quasistatic conditions.

scholarly journals On-Shelf Utility Mining of Sequence Data

2021 ◽  
Vol 16 (2) ◽  
pp. 1-31
Author(s):  
Chunkai Zhang ◽  
Zilin Du ◽  
Yuting Yang ◽  
Wensheng Gan ◽  
Philip S. Yu
Keyword(s):  
High Efficiency ◽  
Sequence Data ◽  
Real Life ◽  
Search Space ◽  
Upper Bounds ◽  
Utility Mining ◽  
Limited Memory ◽  
Time Periods ◽  
High Utility ◽  
Synthetic Datasets

Utility mining has emerged as an important and interesting topic owing to its wide application and considerable popularity. However, conventional utility mining methods have a bias toward items that have longer on-shelf time as they have a greater chance to generate a high utility. To eliminate the bias, the problem of on-shelf utility mining (OSUM) is introduced. In this article, we focus on the task of OSUM of sequence data, where the sequential database is divided into several partitions according to time periods and items are associated with utilities and several on-shelf time periods. To address the problem, we propose two methods, OSUM of sequence data (OSUMS) and OSUMS + , to extract on-shelf high-utility sequential patterns. For further efficiency, we also design several strategies to reduce the search space and avoid redundant calculation with two upper bounds time prefix extension utility ( TPEU ) and time reduced sequence utility ( TRSU ). In addition, two novel data structures are developed for facilitating the calculation of upper bounds and utilities. Substantial experimental results on certain real and synthetic datasets show that the two methods outperform the state-of-the-art algorithm. In conclusion, OSUMS may consume a large amount of memory and is unsuitable for cases with limited memory, while OSUMS + has wider real-life applications owing to its high efficiency.

Theory of Boundary Eigensolutions in Engineering Mechanics

2000 ◽  
Vol 68 (1) ◽  
pp. 101-108 ◽  
Author(s):  
A. R. Hadjesfandiari ◽  
G. F. Dargush
Keyword(s):  
Weight Function ◽  
Mixed Boundary ◽  
Traditional Approach ◽  
Mixed Boundary Conditions ◽  
Boundary Element Methods ◽  
Positive Weight ◽  
Nonsmooth Problems ◽  
Integral Equation Methods ◽  
Potential Problems ◽  
Engineering Mechanics

A theory of boundary eigensolutions is presented for boundary value problems in engineering mechanics. While the theory is quite general, the presentation here is restricted to potential problems. Contrary to the traditional approach, the eigenproblem is formed by inserting the eigenparameter, along with a positive weight function, into the boundary condition. The resulting spectra are real and the eigenfunctions are mutually orthogonal on the boundary, thus providing a basis for solutions. The weight function permits effective treatment of nonsmooth problems associated with cracks, notches and mixed boundary conditions. Several ideas related to the convergence characteristics are also introduced. Furthermore, the connection is made to integral equation methods and variational methods. This paves the way toward the development of new computational formulations for finite element and boundary element methods. Two numerical examples are included to illustrate the applicability.

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