PENYELARASAN ARAH VEKTOR GRADIEN UNTUK MENENTUKAN STEP SIZE METODE STEEPEST  DESCENT PADA FUNGSI NONLINEAR  KUADRATIK BANYAK VARIABEL

S. IDAMAN; B. P. SILALAHI; S. GURITMAN

doi:10.29244/jmap.17.1.47-60

PENYELARASAN ARAH VEKTOR GRADIEN UNTUK MENENTUKAN STEP SIZE METODE STEEPEST DESCENT PADA FUNGSI NONLINEAR KUADRATIK BANYAK VARIABEL

Journal of Mathematics and Its Applications ◽

10.29244/jmap.17.1.47-60 ◽

2018 ◽

Vol 17 (1) ◽

pp. 47

Author(s):

S. IDAMAN ◽

B. P. SILALAHI ◽

S. GURITMAN

Keyword(s):

Steepest Descent ◽

Step Size ◽

Running Time

Masalah optimisasi banyak variabel dapat diselesaikan dengan berbagai metode untuk mendapatkan solusi yang optimal. Salah satu metode yang paling sederhana yaitu metode steepest descent. Metode steepest descent menggunakan vektor gradien untuk menentukan arah pencarian disetiap iterasi kemudian ditentukan step size sebagai jarak perubahan solusi yang dipengaruhi oleh vektor gradien. Step size ( ) pada metode steepest descent sangat mempengaruhi kecepatan kekonvergenan metode ini. Sehingga diperlukan penentuan step size yang tepat untuk mempercepat kekonvergenan metode steepest descent. Penelitian ini akan memodifikasi step size pada metode steepest descent dengan menentukan step size yang dapat menghasilkan arah (vektor gradien) yang mendekati vektor eigen dari matriks Heisse suatu fungsi kuadratik definit positif banyak variabel. Hasil numerik menunjukkan bahwa step size yang diperoleh pada penelitian ini bisa mereduksi jumlah iterasi dan running time lebih baik dari pada metode steepest descent biasa terutama untuk kasus ill-conditioned yaitu kasus lamanya metode steepest descent mencapai kekonvergenan yang disebabkan oleh perbandingan (rasio) yang besar antara nilai eigen terbesar dan nilai eigen terkecil dari matriks Heisse.

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Self-Interference Cancellation in Full Duplex Communication using Steepest Descent Method

International Journal for Research in Applied Science and Engineering Technology ◽

10.22214/ijraset.2021.35281 ◽

2021 ◽

Vol 9 (VI) ◽

pp. 1433-1442

Author(s):

N. Alivelu Manga

Keyword(s):

Interference Cancellation ◽

Descent Method ◽

Steepest Descent ◽

Full Duplex ◽

Steepest Descent Method ◽

Interference Signal ◽

Least Mean Square ◽

Echo Canceller ◽

Step Size ◽

Signal Processing Technique

The present-day communication system uses Frequency Division Duplex (FDD) to emulate the benefits of Full Duplex Communication. But it requires more bandwidth as the cost of the spectrum is very high it becomes a major limitation. To overcome this problem implementation of Full Duplex Communication is the best solution. Implementation of full duplex communication is difficult because of a significant problem called self-interference. while transmitting and receiving signals on the same frequency band, receiving signal is interfered with the transmitted signal this phenomenon is called self-interference. The objective of this project is to minimize that self-interference signal from the received signal by using signal processing technique, LMS echo cancellation. Least Mean Square (LMS) echo canceller whose coefficients are updated iteratively is used to cancel the self-interference. An algorithm based on steepest descent method is used to obtain coefficients that change iteratively with varying step size to solve Weiner-Hopfs equation.

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A modified limited memory steepest descent method motivated by an inexact super-linear convergence rate analysis

IMA Journal of Numerical Analysis ◽

10.1093/imanum/drz059 ◽

2020 ◽

Author(s):

Ran Gu ◽

Qiang Du

Keyword(s):

Convergence Rate ◽

Linear Convergence ◽

Descent Method ◽

Steepest Descent ◽

Steepest Descent Method ◽

Step Size ◽

Linear Convergence Rate ◽

Limited Memory ◽

Rate Analysis ◽

Convergence Rate Analysis

Abstract How to choose the step size of gradient descent method has been a popular subject of research. In this paper we propose a modified limited memory steepest descent method (MLMSD). In each iteration we propose a selection rule to pick a unique step size from a candidate set, which is calculated by Fletcher’s limited memory steepest descent method (LMSD), instead of going through all the step sizes in a sweep, as in Fletcher’s original LMSD algorithm. MLMSD is motivated by an inexact super-linear convergence rate analysis. The R-linear convergence of MLMSD is proved for a strictly convex quadratic minimization problem. Numerical tests are presented to show that our algorithm is efficient and robust.

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Adaptive hybrid steepest descent algorithms involving an inertial extrapolation term for split monotone variational inclusion problems

10.22541/au.162222867.71201407/v1 ◽

2021 ◽

Author(s):

Zheng Zhou ◽

Bing Tan ◽

Songxiao Li

Keyword(s):

Descent Method ◽

Steepest Descent ◽

Steepest Descent Method ◽

Variational Inclusion ◽

Iterative Sequence ◽

Inclusion Problem ◽

Step Size ◽

Hybrid Steepest Descent Method ◽

Variational Inclusion Problem ◽

Size Criterion

In this paper, we discuss the split monotone variational inclusion problem and propose two new inertial algorithms in infinite-dimensional Hilbert spaces. As well as, the iterative sequence by the proposed algorithms converges strongly to the solution of a certain variational inequality with the help of the hybrid steepest descent method. Furthermore, an adaptive step size criterion is considered in suggested algorithms to avoid the difficulty of calculating the operator norm. Finally, some numerical experiments show that our algorithms are realistic and summarize the known results.

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A Comparative Study of First-Order Reliability Method-Based Steepest Descent Search Directions for Reliability Analysis of Steel Structures

Advances in Civil Engineering ◽

10.1155/2017/8643801 ◽

2017 ◽

Vol 2017 ◽

pp. 1-10 ◽

Cited By ~ 2

Author(s):

Hamed Makhduomi ◽

Behrooz Keshtegar ◽

Mehdi Shahraki

Keyword(s):

Reliability Index ◽

Steel Structures ◽

Steepest Descent ◽

Building Code ◽

Step Size ◽

First Order Reliability Method ◽

First Order ◽

Confidence Levels ◽

Reliability Method ◽

Bending Capacity

Three algorithms of first-order reliability method (FORM) using steepest descent search direction are compared to evaluate the reliability index of structural steel problems which are designed by the Iranian National Building code. The FORM formula is modified based on a dynamic step size which is computed based on the merit functions named modified Hasofer-Lind and Rackwitz-Fiessler (MHL-RF) method. The efficiency of the gradient, HL-RF, and MHL-RF method was compared for a bar structure under tensile capacity, a multispan beam under bending capacity, a connection under tension load, and a column under axial force. The results illustrated that the MHL-RF method is more efficient than the HL-RF and gradient method. The designed steel components by the Iranian National Building code showed good confidence levels with the reliability index in the range from 2.5 to 3.0.

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A steepest descent algorithm for the optimal control of a cascaded hydropower system

International Journal of Electrical and Computer Engineering (IJECE) ◽

10.11591/ijece.v10i4.pp4136-4144 ◽

2020 ◽

Vol 10 (4) ◽

pp. 4136

Author(s):

Olalekan Ogunbiyi ◽

Cornelius T. Thomas ◽

Oludare Y. Ogundepo ◽

Isaac O. A. Omeiza ◽

Jimoh Akanni ◽

...

Keyword(s):

Power Generation ◽

Descent Method ◽

Steepest Descent ◽

Steepest Descent Method ◽

Hydroelectric Power ◽

Step Size ◽

Infinite Dimensional ◽

Optimal Power ◽

Hydropower System ◽

And Control

Optimal power generation along the cascaded Kainji-Jebba hydroelectric power system had been very difficult to achieve. The reservoirs operating heads are being affected by possible variation in impoundments upstream, stochastic factors that are weather-related, availability of the turbo-alternators and power generated at any time. Proposed in this paper, is an algorithm for solving the optimal release of water on the cascaded hydropower system based on steepest descent method. The uniqueness of this work is the conversion of the infinite dimensional control problem to a finite one, the introduction of clever techniques for choosing the steepest descent step size in each iteration and the nonlinear penalty embedded in the procedure. The control algorithm was implemented in an Excel VBA® environment to solve the ormulated Lagrange problem within an accuracy of 0.03%. It is recommended for use in system studies and control design for the optimal power generation in the cascaded hydropower system.

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METODE STEEPEST DESCENT DENGAN UKURAN LANGKAH BARU UNTUK PENGOPTIMUMAN NIRKENDALA

Journal of Mathematics and Its Applications ◽

10.29244/jmap.14.1.1-10 ◽

2015 ◽

Vol 14 (1) ◽

pp. 1

Author(s):

D. WUNGGULI ◽

B. P. SILALAHI ◽

S. GURITMAN

Keyword(s):

Steepest Descent ◽

Running Time

Metode steepest descent adalah metode gradien sederhana untuk pengoptimuman. Metode ini memiliki kekonvergenan yang lambat dalam menuju ke solusi optimum, hal ini terjadi karena langkahnya yang berbentuk zig-zag. Barzilai dan Borwein berusaha menyempurnakan metode ini dengan memodifikasi algoritme sehingga hasilnya berjalan cukup baik untuk masalah dengan dimensi yang besar. Hasil metode Barzilai dan Borwein ini telah memicu banyak penelitian pada metode steepest descent, diantaranya terdapat metode Alternatif Minimisasi dan metode Yuan. Dalam tulisan ini telah dimodifikasi metode steepest descent dengan ukuran langkah baru. Hasil modifikasi ini kemudian dibandingkan dengan metode Barzilai dan Borwein, Alternatif Minimisasi dan metode Yuan dengan kasus fungsi kuadratik ditinjau dari iterasi dan running time. Rata-rata hasil perbandingan menunjukkan bahwa modifikasi dengan ukuran langkah baru ini memberikan hasil yang baik untuk dimensi yang kecil dan mampu menyaingi hasil metode Barzilai-Borwein dan metode Alternatif Minimisasi untuk dimensi yang besar. Ukuran langkah baru ini memiliki kekonvergenan yang lebih cepat dibandingkan dengan m

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The Added Value of Graphical Input and Display for Electron Lens Design

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100179701 ◽

1990 ◽

Vol 48 (1) ◽

pp. 190-191

Author(s):

B. Lencova ◽

G. Wisselink

Keyword(s):

Flux Density ◽

Numerical Data ◽

Added Value ◽

Lens Design ◽

Step Size ◽

Magnetic Lens ◽

Flux Lines ◽

Fine Mesh ◽

Electron Lens ◽

User Friendly

Recent progress in computer technology enables the calculation of lens fields and focal properties on commonly available computers such as IBM ATs. If we add to this the use of graphics, we greatly increase the applicability of design programs for electron lenses. Most programs for field computation are based on the finite element method (FEM). They are written in Fortran 77, so that they are easily transferred from PCs to larger machines.The design process has recently been made significantly more user friendly by adding input programs written in Turbo Pascal, which allows a flexible implementation of computer graphics. The input programs have not only menu driven input and modification of numerical data, but also graphics editing of the data. The input programs create files which are subsequently read by the Fortran programs. From the main menu of our magnetic lens design program, further options are chosen by using function keys or numbers. Some options (lens initialization and setting, fine mesh, current densities, etc.) open other menus where computation parameters can be set or numerical data can be entered with the help of a simple line editor. The "draw lens" option enables graphical editing of the mesh - see fig. I. The geometry of the electron lens is specified in terms of coordinates and indices of a coarse quadrilateral mesh. In this mesh, the fine mesh with smoothly changing step size is calculated by an automeshing procedure. The options shown in fig. 1 allow modification of the number of coarse mesh lines, change of coordinates of mesh points or lines, and specification of lens parts. Interactive and graphical modification of the fine mesh can be called from the fine mesh menu. Finally, the lens computation can be called. Our FEM program allows up to 8000 mesh points on an AT computer. Another menu allows the display of computed results stored in output files and graphical display of axial flux density, flux density in magnetic parts, and the flux lines in magnetic lenses - see fig. 2. A series of several lens excitations with user specified or default magnetization curves can be calculated and displayed in one session.

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