Nonlinear Least Square Technique

New approximate secant equations are shown to result from the knowledge of (problem dependent) invariant subspace information, which in turn suggests improvements in quasi-Newton methods for unconstrained minimization. It is also shown that this type of information may often be extracted from the multigrid structure of discretized infinite dimensional problems. A new limited-memory BFGS using approximate secant equations is then derived and its encouraging behaviour illustrated on a small collection of multilevel optimization examples. The smoothing properties of this algorithm are considered next, and automatic generation of approximate eigenvalue information demonstrated. The use of this information for improving algorithmic performance is finally investigated on the same multilevel examples.

Artikel ini membahas tentang optimasi model deret geometri melalui studi relasi rekurensi homogen orde pertama. Latar belakang penelitian ini terletak pada kemunculan pola pertumbuhan dan peluruhan eksponensial yang sering terjadi dalam berbagai masalah ilmiah dan teknik, yang dapat dimodelkan secara efektif oleh relasi rekursif tersebut. Tujuan utamanya adalah untuk menyediakan pendekatan analitis yang komprehensif untuk memecahkan rekursif homogen orde pertama dan untuk mengeksplorasi aplikasi praktisnya dalam pemodelan deret geometri. Metodologi ini melibatkan penurunan solusi umum dari relasi rekursif a_{n+1}=d\bullet a_n, di mana d adalah rasio konstan, dan memvalidasi model dengan kondisi awal. Hasilnya menunjukkan bahwa solusinya berbentuk a_n=a_0\bullet d^n,\ \ yang mengonfirmasi kesetaraan dengan deret geometri. Lebih jauh, penelitian ini membahas strategi untuk mengoptimalkan parameter model guna meningkatkan akurasi dan penerapan dalam skenario dunia nyata. Ini termasuk teknik estimasi parameter dan proses validasi model. Implikasi dari penelitian ini meluas ke bidang-bidang seperti ilmu komputer, ekonomi, dan ilmu pengetahuan alam, di mana pemodelan diskrit dari proses pertumbuhan sangat penting. Artikel ini menyimpulkan bahwa menguasai relasi rekursif homogen orde pertama menyediakan alat yang ampuh untuk analisis teoritis dan pemecahan masalah praktis dalam matematika diskrit.

This study aims to find the time-dependent potential terms in the two inverse problems of the third-order pseudo-parabolic with initial and various boundary conditions supplemented by the overdetermination data. The nonlinear inverse problems have significant applications in physics and engineering fields. We proved the existence and uniqueness of the solution of the two problems are being proved, but they still need to be proposed (since tiny perturbations in input data cause considerable errors in the output potential term). Consequently, the regularized methods should be employed. A finite difference schema is used for solving direct problems. In contrast, the inverse problems were reformulated as nonlinear least-square minimization and solved efficiently by optimizing MATLAB routine lsqnonlin. Tikhonov's regularization method was applied to get stable results. The numerical results were explained by presenting a test example for each problem. In addition, the stability was discussed by utilizing the Von Neumann stability analysis. The results showed that the time-dependent potential terms were reconstructed successfully and were stable and accurate.

Using Taylor's series we propose a modified secant relation to get a more accurate approximation of the second curvature of the objective function. Then, based on this modified secant relation we present a new BFGS method for solving unconstrained optimization problems. The proposed method make use of both gradient and function values while the usual secant relation uses only gradient values. Under appropriate conditions, we show that the proposed method is globally convergent without needing convexity assumption on the objective function. Comparative results show computational efficiency of the proposed method in the sense of the Dolan-Moré performance profiles.

Using Taylor's series we propose a modified secant relation to get a more accurate approximation of the second curvature of the objective function. Then, based on this modified secant relation we present a new BFGS method for solving unconstrained optimization problems. The proposed method make use of both gradient and function values while the usual secant relation uses only gradient values. Under appropriate conditions, we show that the proposed method is globally convergent without needing convexity assumption on the objective function. Comparative results show computational efficiency of the proposed method in the sense of the Dolan-More performance profiles.

This paper presents a circuit of a high-precision, wide ranged, analog clock generator with on-chip programmability feature using Floating-gate transistors. The programmable oscillator can attain a continuous range of time-periods lying in the programming precision range of Floating Gates. The circuit consists of two sub circuits: Current Generator circuit and Wave Generator circuit. The current of current generator circuit is programmable and mirrored to the wave generator to generate the desired square wave. The topology is well suited to applications like clocking high performance ADCs and DACs as well as used as the internal clock in structured analog CMOS designs. A simulation model of the circuit was built in T-Spice, 0.35µm CMOS process. The circuit results in finely tuned clock with programmability precision of about 13bit [1]. Simulation results show high amount of temperature insensitivity (0.507ns/°C) for a large range of thermal conditions. The proposed circuit can compensate any change in temperature. The circuit design can be operated at low supply voltage i.e., 1v.

Motorcycles generate different sound patterns under dissimilar working conditions. The generated sound pattern gives a clue of the fault. Mainly the parts of the engine that lead to change in sound are cylinder kit, crank, timing chain, and valve. The parts of the exhaust system that change sound under fault are muffler and silencer. In this study, we analyze the sound signals produced by motorcycles to locate the faults in subsystems. The work proceeds in three stages, the first stage detects the fault, the second stage identifies the faulty subsystem and finally the third stage locates the fault. The overall classification accuracy of the first stage is 0.8019. The work finds interesting applications in troubleshooting of machinery, electronic gadgets, musical instruments and the like.

We proposed a matrix-free direction with an inexact line search technique to solve system of nonlinear equations by using double direction approach. In this article, we approximated the Jacobian matrix by appropriately constructed matrix-free method via acceleration parameter. The global convergence of our method is established under mild conditions. Numerical comparisons reported in this paper are based on a set of large-scale test problems and show that the proposed method is efficient for large-scale problems.

This paper presents an improved diagonal Secant-like method using two-step approach for solving large scale systems of nonlinear equations. In this scheme, instead of using direct updating matrix in every iteration to construct the interpolation curves, we chose to use an implicit updating approach to obtain an enhanced approximation of the Jacobian matrix which only requires a vector storage. The fact that the proposed method solves systems of nonlinear equations without the cost of storing the weighted matrix can be considered as a clear advantage of this method over some variants of Newton's methods.

A new method for approximating quantities that involve physical properties derivatives in equation‐oriented process design is presented. It is a hybrid algorithm that makes combined use of Newton's method and the Schubert update. In doing so, available analytical derivative information is used in an optimal way.This hybrid algorithm is surprisingly close, in terms of the number of iterations required for solution, to an implementation of Newton's method that uses finite difference approximations of any unavailable physical properties derivatives. However, the number of rigorous thermodynamics calculations is usually about 50% fewer for the hybrid method. This can result in a substantial savings for problems in which the physical properties calculations dominate the simulation time. Two examples are presented to support these claims.

This article studies the nonlocal inverse boundary value problem for a rectangular domain, a secondorder, elliptic equation and a two-dimensional equation. The main objective of the article is to find the unidentified coefficient and provide a solution to the problem. The two-dimensional second-order, convection equation is solved directly using the finite difference method (FDM). However, the inverse problem was successfully solved the MATLAB subroutine lsqnonlin from the optimization toolbox after reformulating it as a nonlinear regularized least-square optimization problem with a simple bound on the unknown quantity. Considering that the problem under study is often ill-posed and that even a small error in the input data can have a large impact on the outcome, Tikhonov's regularization technique is used to obtain stable and regularized results.

With an electrical grid shifting toward Distributed Generation (DG), the emerging use of renewable energy resources is continuously creating challenges to maintain an acceptable electrical power quality thought-out the grid; Therefore, in an energy market where loads are becoming more and more sensitive in a distributed generation filled with polluting nonlinear loads, power quality improvement devices such Active Power Filters (APFs) have to evolve to meet the new standards, since theirs conventional control strategies can't properly operate when multiple power quality problems happens at once, even the one using AI based control as it will be proven in this paper. In this paper a neural network based Active Power Filter will be tested in a DG environment where both current and voltage harmonics, along with fast frequency variation occurs, we will see how the PLL can downgrade its performances enormously under such hostile conditions, We propose to solve this problem by replaci...

(DoS) merupakan sebuah fenomena yang sedang menjadi topik hangat belakangan ini. Intensitas serangan DoS semakin meningkat setiap harinya dengan ditemukannya jenis serangan baru dengan tipe yang sama yaitu Distributed Denial of Service (DDoS). Kedua serangan tersebut menyerang korban dengan cara membanjiri kanal trafik korban dengan banyak kiriman paket pada satu waktu. Hal ini membuat aliran paket yang menuju komputer korban menjadi tersendat sehingga memungkinkan koran tidak mendapatkan paket yang diinginkan karena padatnya trafik pada jaringannya. Metode LRD dan Self-Similarity merupakan metode yang cocok dengan sifat trafik jaringan yaitu variability dan burstiness. Pada metode LRD dinyatakan bahwa trafik jaringan menunjukan sebagai memori jangka panjang dimana tingkah laku tersebut berkorelasi melalui waktu yang terpisah jauh. Hal ini menunjukan bahwa setiap paket yang dikirim dan diterima memiliki korelasi dan hubungan tertentu meskipun waktu antar kedatangan paket terpisah cukup jauh. Dalam DDoS kemungkinan korelasi dan hubungan tersebut tidak terjadi dalam waktu yang dekat sekalipun. Ini membuat pendeteksian menggunakan DDoS menggunakan LRD menjadi salah satu metode terbaik. Self-Similarity adalah sebuah skala dari invarian yang selalu memiliki kesamaan, jadi ketika selfsimilarity digunakan kedalam pemodelan trafik, maka terlihat plot dari trafik tersebut memiliki kesamaan, walaupun secara waktu memiliki perbedaan.

This paper reports a real-time localization algorithm system that has a main function to determine the location of devices accurately. The model can locate the smartphone position passively (which do not need a set on a smartphone) as long as the Wi-Fi is turned on. The algorithm uses Intersection Density, and the Nonlinear Least Square Algorithm (NLS) method that utilizes the Lavenberg-Marquart method. To minimize the localization error, Kalman Filter (KF) is used. The algorithm is computed under Matlab approach. The most obtained model will be implemented in this Wi-Fi tracker system using RSSI-based distance for indoor crowd monitoring. According to the experiment result, KF can improve Hit ratio of 81.15 %. Hit ratio is predicting results of a location that is less than 5 m from the actual area (location). It can be obtained from several RSSI scans, the calculation is as follows: the number of non-error results divided by the number of RSSI scans and multiplied by 100%.

This paper presents two modified Hager-Zhang (HZ) Conjugate Gradient methods for solving large-scale system of monotone nonlinear equations. The methods were developed by combining modified forms of the one-parameter method by Hager and Zhang (2006) and the hyperplane projection technique. Global convergence and numerical results of the methods are established. Preliminary numerical results show that the proposed methods are promising and more efficient compared to the methods presented by Mushtak and Keyvan (2018) and Sun et al. (2017).

A q-Levenberg-Marquardt method is an iterative procedure that blends a q-steepest descent and q-Gauss-Newton methods. When the current solution is far from the correct one the algorithm acts as the q-steepest descent method. Otherwise the algorithm acts as the q-Gauss-Newton method. A damping parameter is used to interpolate between these two methods. The q-parameter is used to escape from local minima and to speed up the search process near the optimal solution.

The study aimed at evaluating two models (Swartzendruber and Horton) using furrow infiltration data measured in Samaru, Zaria. These measurements were carried out on the three field plots A, B and C. Infiltration parameters were generated from the data measured from plot A and B and fitted into the models for predictions of water infiltrated depths at different time intervals. The predictions from the models were compared with the measured field data from plot C. Statistical indices such as coefficient of determination (R 2), root mean square error (RMSE) and T-test at 5% level of significance were used to determine the best performing model. The results show that, value of R 2 and RMSE recorded for Horton's model was 0.996 and 3.05, while the value of R 2 and RMSE recorded for Swartzendruber's model was 0.998 and 3.01, respectively. The T-test values obtained for Horton's model was 2.93, while 2.51 was recorded for Swartzendruber's model. The results of the evaluation indicated that both models were considered suitable because they presented high values of R 2 and low values of RMSE as suggested by Aminu (2019). Similarly, the results obtained from the T-test statistics indicated that there is a no significant difference between the models evaluated because the calculated values of 'T' 2.93 and 2.51were greater than the tabulated value 2.365 at 5% confidence interval. Therefore, this study recommends that both models should be used to predict infiltration rates of soils in the study area.

Infiltration study is very crucial in modelling water requirement of crops during their growth season. Infiltration rate measurements were carried out on dryland areas of Sokoto, Sudan savanna ecological zone of Nigeria; using the double ring infiltrometer. Disturbed and undisturbed soil samples were used to determine some physical characteristics (Texture, Saturated hydraulic Conductivity, particle density, bulk density, porosity and organic matter) of the soil. The results showed that the cumulative infiltration predicted by Horton infiltration model was very close to the field measurements for all the spots from the average values (3.35, 2.83 and 1.71 cm/min) and predicted rates (2.37,2.34 and 1.54 cm/min) with coefficient of determination (R2) close to unity (0.98, 0.97, 0.97) for the three spots. The study showed that the Horton infiltration model can be applied to estimate infiltration characteristics of some soils in Sudan Savanna of Nigeria.

In this paper, a matrix-free method for solving large-scale system of nonlinear equations is presented. The method is derived via quasi-Newton approach, where the approximation to the Broyden's update is done by constructing diagonal matrix using acceleration parameter. A fascinating feature of the method is that it is a matrix-free, so is suitable for solving large-scale problems. Furthermore, the convergence analysis of the new method is discussed based on some standard condition. Preliminary numerical results on some test problems show that the method is promising.

The electrocardiogram (ECG) has significant clinical importance for analyzing most cardiovascular diseases. ECGs beat morphologies, beat durations, and amplitudes vary from subject to subject and diseases to diseases. Therefore, ECG morphology-based modeling has long-standing research interests. This work aims to develop a simplified ECG model based on a minimum number of parameters that could correctly represent ECG morphology in different cardiac dysrhythmias. A simple mathematical model based on the sum of two Gaussian functions is proposed. However, fitting more than one Gaussian function in a deterministic way has accuracy and localization problems. To solve these fitting problems, two hybrid optimization methods have been developed to select the optimal ECG model parameters. The first method is the combination of an approximation and global search technique (ApproxiGlo), and the second method is the combination of an approximation and multi-start search technique (ApproxiMul)....

As the number of vehicles in roads increases, information of traffic density becomes crucial to municipalities for making better decisions about road management and to the environment for reduced carbon emission. Here, the problem of traffic density estimation is addressed when there is continuous influx of vehicle data. First the traffic density is modeled by the clusters of the speed groups that are centered after Kernel Density Estimation technique is implemented for the probability density function of the speed data. Then, empirical cumulative distribution function of data is found by Kolmogorov-Smirnov Test. A peak detection algorithm is used to estimate speed centers of the clusters. Since the estimation model has linear and non-linear components, the estimation of variance values and kernel weights are found by a nonlinear Least Square approach with separation of parameters property. Finally, the tracking of former and latter estimations of a road is calculated by using Scalar Kalman Filtering with scalar state-scalar observation generality level. For all example data sets, the minimum mean square error of kernel weights is found to be less than 0.002 while error of mean values is found to be less than 0.261.

With increasing population, the determination of traffic density becomes critical in managing urban city roads for safer driving and low carbon emissions. In this study, kernel density estimation is utilized in order to estimate traffic density more accurately when the speeds of vehicles are available for a given region. For the proposed approach, as a first step, the probability density function of the speed data is modeled by kernel density estimation. Then the speed centers from the density function are modeled as clusters. The cumulative distribution function of the speed data is then determined by Kolmogorov-Smirnov test, whose complexity is less when compared to the other techniques and whose robustness is high when outliers exist. Then the mean values of clusters are estimated from the smoothed density function of the distribution function, followed by a peak detection algorithm. The estimates of variance values and kernel weights, on the other hand, are found by a nonlinear least square approach. As the estimation problem has linear and nonlinear components, the nonlinear least square with separation of parameters approach is adopted, instead of dealing with a high complexity nonlinear equation. Simulations are carried out in order to assess the performance of the proposed approach. It is observed that the error between cumulative distribution functions is less than 1% , an indication that the traffic densities are estimated accurately. For an assumed traffic condition that bears five speed clusters, the minimum mean square error of kernel weights is found to be less than 0.00004. The proposed approach was also applied to real data from sample road traffic, and the speed center and the variance were accurately estimated. By using the proposed approach, accurate traffic density estimation is realized, providing extra information to the municipalities for better planning of their cities.

This paper aims to provide a methodology to construct parametrically the Efficient Frontier (EF) of Power Generation Portfolio (PGP). The methodology works as follows. First, we obtain two sets of the shares of the assets: one that guarantee the maximal expected return on the PGP; and another that guarantee the minimal risk of the PGP. The EF corresponds to the parametric equation of the risk-return profiles from the minimal risk to the maximal expected return of the PGP. We apply our methodology to replicate the results from three existing papers. The present methodology allows to and different and more coherent results than those obtained in the original papers. The analysis suggests that there are optimal investment alternatives that have been denied by previous analysis. This fact creates a bias in the design of investment policies in electricity generation. One limitation of the paper is that the analysis relies on the assumption that the covariances of the returns of the diffe...

In this paper, a matrix-free method for solving large-scale system of nonlinear equations is presented. The method is derived via quasi-Newton approach, where the approximation to the Broyden's update is done by constructing diagonal matrix using acceleration parameter. A fascinating feature of the method is that it is a matrix-free, so is suitable for solving large-scale problems. Furthermore, the convergence analysis of the new method is discussed based on some standard condition. Preliminary numerical results on some test problems show that the method is promising.

Dalam suatu sistem supervisory control diperlukan perangkat lunak antar muka yang menjadi penghubung antara manusia (operator) dengan mesin atau peralatan yang dikendalikan. Perangkat lunak tersebut umumnya disebut sebagai HMI (Human Machine Interface) yang berupa Graphical User Interface (GUI) berbasis komputer. Makalah ini menjelaskan tentang pengembangan perangkat lunak HMI untuk sistem kontrol dan monitoring jarak jauh pada  pilot plant biodiesel (metil ester) berbasis TCP/IP. Penggunaan format penyimpanan data standar (teknologi XML) dilakukan untuk kemudahan penggunaan koleksi data HMI oleh sub sistem lain. Metodologi pengembangan perangkat lunak mengikuti pendekatan Object Oriented Software Engineering (OOSE) dengan menggunakan notasi UML (Unified Modeling Language). Pengkodean dalam tahap implementasi dilakukan dengan menggunakan bahasa pemrograman Java untuk pengurangan aspek ketergantungan pada produk-produk berlisensi. Dari hasil kegiatan diperoleh sebuah perangkat lunak ...

A new hybrid quasi-Newton search direction ( HQNEI ) is proposed. It uses the update formula of Broyden–Fletcher–Goldfarb–Shanno (BFGS) with a certain conjugate gradient (CG) parameter by a nested direction. The global convergence analysis and superlinear rate, addtionaly with sufficient descent are proved using exact line search. Finally, the computation comparisons are made with original hybrid parents; BFGS and CG, through the efficiency in terms of iteration numbers and CPU-running time showing the superior of HQNEI. Therefore, the results marked preference of HQNEI from other two producer algorithms.

ABSTRAK PERKIRAAN DESAIN BASIS CURAH HUJAN DI SEMENANJUNG MURIA. Nilai desain basis diperlukan dalam memperhitungkan aspek kesalamatan PLTN. Perhitungan telah dilakukan terhadap parameter meteorologi curah hujan. Data curah hujan diperoleh dari stasiun pemantauan milik Departemen Permukiman dan Prasarana Wilayah dan PTPN IX untuk daerah Bangsri, Beji, Jatisari, keling dan Jepara. Data curah hujan maksimum dalam periode 24 jam memiliki unit mm/hari. Pada studi terdahulu, konsultan Newjec menggunakan data dari stasiun Ujung Watu dengan periode pemantauan selama kurang dari 2 tahun. Pada perhitungan ini digunakan paling sedikit data selama 19 tahun. Data diolah menggunakan dua pendekatan: statistik dan numerik, menggunakan persamaan Gumbel (Generalized Extreme Value distrbutions-GEV Type I). Dari data diperoleh rerata curah hujan sebesar 248 mm/hari (metode numerik) dan 258 mm/hari (metode statistik) perioda pengulangan (return period) 50 tahun. ABSTRACT PRECIPITATION DESIGN BASIS CALC...

Brain-computer interface (BCI) is an active domain which has attracted attention of the research community in recent years. It offers huge potential as a technology which can estimate the intention of a user by analysis of brain signals and establish a communication channel directly between a human brain and an external device. Electroencephalography (EEG) is the most popular signal acquisition technique due to its ease of use and simplicity. In EEG-based BCI systems, electrodes are placed on specific positions on the scalp of the subject to record electrical activity. The BCI system consists of sequential stages of signal acquisition, its preprocessing, feature extraction and feature classification. It is an active research area which has a focus on improving classification accuracy in motor imagery-based BCI systems. The first stage in a BCI system is to acquire EEG signals from different positions of the scalp of the human subject. The acquired brain signals are preprocessed to remove artifacts before these are fed to feature the extraction stage. In this paper, independent component analysis (ICA) technique is used to remove artifacts from acquired signals. Filter bank common spatial pattern (FBCSP) technique is then used for feature extraction and feature selection. A feature classification approach based on support vector machine (SVM) is proposed in this work and its performance is enhanced by optimizing its polynomial kernel parameters. Selection of kernel parameters is done by grid search method using the fivefold cross-validation procedure. The proposed approach is then executed on publicly available data set 2a of BCI Competition IV. Results show that the proposed approach offers higher classification accuracy and lower misclassification rate as compared to other methods executed on the same dataset, as reported in literature.

In digital images, impulse noise (such as salt and pepper noise) detection and removal is an important process as the images are corrupted by those noise because of transmission and acquisition. The main aim of the noise removal is to suppress the noise when preserving the edge information. The median filter and its derivatives are usually used for this purpose. These filtering techniques usually applied to the overall image and modify the pixel value. The modification in pixel values will be performed in unaffected pixels also. Hence the sufficient removal of impulse using this technique will leads to the reduction in quality of images. In this paper, Adaptive Fuzzy Switching Filter is proposed which is based on fuzzy logic for removing the impulse noise from the affected image. The proposed technique involves three phases. The first phase will detect the impulse noise by considering grayscale distribution among neighboring pixels. In the second phase, grayscale values for the pixels are determined based on the values of neighboring pixels. The final phase implements the fuzzy switching for further improvement in the image preservation. The fuzzy membership function used in the proposed technique is half open fuzzy membership function. The experimental result shows that the proposed adaptive fuzzy switching filter has the better capability of removing the impulse noise from the corrupted image.

The median M-type K-nearest neighbour (MM-KNN) filter to remove impulse noise from corrupted images is presented. This filter uses R and M estimators combined with different influence functions. Simulation results have shown that the restoration performance is better than that of other known filters.

Pada penelitian ini, dilakukan perancangan identifikasi sistem pada sebuah wahana terbang tanpa awak yaitu quadrotor. Sistem dinamik quadrotor memiliki 4 masukan berupa kecepatan 4 motor dan 3 keluaran berupa sudut yang dibentuk quadrotor. Pengambilan data masukan dan keluaran dilakukan secara serial dengan menggunakan RF YS-1020UA. Struktur model sistem yang digunakan adalah struktur auto regressif moving average with exogenous input(ARMAX). Metode estimasi parameter dilakukan dengan metode recursive extended least square (RELS). Validasi model terestimasi diperoleh dengan perhitungan root mean square error (RMSE) dan validasi berdasarkan grafik.

This paper aims to provide a methodology to construct parametrically the Efficient Frontier (EF) of Power Generation Portfolio (PGP). The methodology works as follows. First, we obtain two sets of the shares of the assets: one that guarantee the maximal expected return on the PGP; and another that guarantee the minimal risk of the PGP. The EF corresponds to the parametric equation of the risk-return profiles from the minimal risk to the maximal expected return of the PGP. We apply our methodology to replicate the results from three existing papers. The present methodology allows to and different and more coherent results than those obtained in the original papers. The analysis suggests that there are optimal investment alternatives that have been denied by previous analysis. This fact creates a bias in the design of investment policies in electricity generation. One limitation of the paper is that the analysis relies on the assumption that the covariances of the returns of the different assets is zero. This assumption leads to gains in tractability, clarity, and in the scope of the methodology formulated.

Kebutuhan air bersih di PDAM Kota Pontianak untuk golongan pelanggan sosial, nonniaga, niaga, dan industri telah dianalisis dengan menggunakan metode Levenberg-Marquardt. Hasil analisis menunjukan bahwa fungsi matematika terbaik untuk pelanggan sosial adalah polinomial orde 5 dengan nilai korelasi sebesar 0,905 dan RMSE sebesar 4,77×102. Kemudian untuk pelanggan nonniaga dan pelanggan niaga, fungsi matematika terbaik adalah polinomial orde 6 dengan nilai korelasi masing-masing pelanggan adalah 0,956 dan 0,878 serta RMSE sebesar 9,25×104 dan 2,91×104. Selanjutnya untuk pelanggan industri, fungsi matematika terbaik adalah polinomial orde 4 dengan nilai korelasi sebesar 0,719 dan RMSE sebesar 8,43×102 . Berdasarkan fungsi matematika yang telah diperoleh, hasil prediksi kebutuhan air bersih dari tahun 2017 hingga tahun 2019 menunjukkan bahwa golongan pelanggan niaga memiliki peningkatan yang paling tinggi dengan rata-rata laju pertumbuhan sebesar 80,4%.

Simple modifications of the limited-memory BFGS method (L-BFGS) for large scale unconstrained optimization are considered, which consist in corrections of the used difference vectors (derived from the idea of conjugate directions), utilizing information from the preceding iteration. For quadratic objective functions, the improvement of convergence is the best one in some sense and all stored difference vectors are conjugate for unit stepsizes. The algorithm is globally convergent for convex sufficiently smooth functions. Numerical experiments indicate that the new method often improves the L-BFGS method significantly.

A block version of the Broyden-Fletcher-Goldfarb-Shanno (BFGS) variable metric update formula and its modifications are investigated. In spite of the fact that this formula satisfies the quasi-Newton conditions with all used difference vectors and that the improvement of convergence is the best one in some sense for quadratic objective functions, for general functions, it does not guarantee that the corresponding direction vectors are descent directions. To overcome this difficulty, but at the same time utilize the advantageous properties of the block BFGS update, a block version of the limited-memory variable metric BNS method for large-scale unconstrained optimization is proposed. The global convergence of the algorithm is established for convex sufficiently smooth functions. Numerical experiments demonstrate the efficiency of the new method.

This contribution contains a description of efficient methods for large-scale unconstrained optimization. Many of them have been developed recently by the authors. It concerns limited memory methods for general smooth optimization, variable-metric bundle methods for partially separable nonsmooth optimization, hybrid methods for sparse least squares and methods for solving large-scale trust-region subproblems. These methods are compared with other methods by extensive computational experiments.

Binary decision variable, 1 if MD located at bus i communicates with protection room located at bus j and 0, otherwise. w i Binary decision variable, 1 if bus i is equipped with MD and 0, otherwise. t ij Delay of data transmission from bus i to bus j. x kj , y fj , z fj Auxiliary binary variables. ε Small positive number, say 0.01.

In this paper, a matrix-free method for solving large-scale system of nonlinear equations is presented. The method is derived via quasi-Newton approach, where the approximation to the Broyden's update is done by constructing diagonal matrix using acceleration parameter. A fascinating feature of the method is that it is a matrix-free, so is suitable for solving large-scale problems. Furthermore, the convergence analysis of the new method is discussed based on some standard condition. Preliminary numerical results on some test problems show that the method is promising.

Like the Polak-Ribière-Polyak (PRP) and Hestenes-Stiefel (HS) methods, the classical Liu-Storey (LS) conjugate gradient scheme is widely believed to perform well numerically. This is attributed to the in-built capability of the method to conduct a restart when a bad direction is encountered. However, the scheme's inability to generate descent search directions, which is vital for global convergence, represents its major shortfall. In this article, we present an LS-type scheme for solving system of monotone nonlinear equations with convex constraints. The scheme is based on the approach by Wang et al. (2020) and the projection scheme by Solodov and Svaiter (1998). The new scheme satisfies the important condition for global convergence and is suitable for non-smooth nonlinear problems. Furthermore, we demonstrate the method's application in restoring blurry images in compressed sensing. The scheme's global convergence is established under mild assumptions and preliminary numerical results show that the proposed method is promising and performs better than two recent methods in the literature.

Like the Polak-Ribière-Polyak (PRP) and Hestenes-Stiefel (HS) methods, the classical Liu-Storey (LS) conjugate gradient scheme is widely believed to perform well numerically. This is attributed to the in-built capability of the method to conduct a restart when a bad direction is encountered. However, the scheme's inability to generate descent search directions, which is vital for global convergence, represents its major shortfall. In this article, we present an LS-type scheme for solving system of monotone nonlinear equations with convex constraints. The scheme is based on the approach by Wang et al. (2020) and the projection scheme by Solodov and Svaiter (1998). The new scheme satisfies the important condition for global convergence and is suitable for non-smooth nonlinear problems. Furthermore, we demonstrate the method's application in restoring blurry images in compressed sensing. The scheme's global convergence is established under mild assumptions and preliminary numerical results show that the proposed method is promising and performs better than two recent methods in the literature.

In this paper, a matrix-free method for solving large-scale system of nonlinear equations is presented. The method is derived via quasi-Newton approach, where the approximation to the Broyden's update is done by constructing diagonal matrix using acceleration parameter. A fascinating feature of the method is that it is a matrix-free, so is suitable for solving large-scale problems. Furthermore, the convergence analysis of the new method is discussed based on some standard condition. Preliminary numerical results on some test problems show that the method is promising.

Two basic disadvantages of the symmetric rank one (SR1) update are that the SR1 update may not preserve positive definiteness when starting with a positive definite approximation and the SR1 update can be undefined. A simple remedy to these problems is to restart the update with the initial approximation, mostly the identity matrix, whenever these difficulties arise. However, numerical experience shows that restart with the identity matrix is not a good choice. Instead of using the identity matrix we used a positive multiple of the identity matrix. The used positive scaling factor is the optimal solution of the measure defined by the problem-maximize the determinant of the update subject to a bound of one on the largest eigenvalue. This measure is motivated by considering the volume of the symmetric difference of the two ellipsoids, which arise from the current and updated quadratic models in quasi-Newton methods. A replacement in the form of a positive multiple of the identity matrix is provided for the SR1 update when it is not positive definite or undefined. Our experiments indicate that with such simple initial scaling the possibility of an undefined update or the loss of positive definiteness for the SR1 method is avoided on all iterations. Keywords Symmetric rank one • Volume of ellipsoid • Unconstrained optimization 1 Introduction This paper is concerned with quasi-Newton methods for finding a local minimum of the unconstrained optimization problem min x∈R n f (x).

Process models play important role in computer aided process engineering, since most of advanced process monitoring, control, and optimization algorithms relay on a model of the process. In most of the cases, some parameters of the model should be estimated based on some experiments. One of the factors affecting the model prediction quality is the accuracy of these estimated parameters. Establishing optimal experiment design can maximise the confidence on the parameters, hereby increasing the confidence on the model prediction. The aim of this paper is to work out a modern experiment design tool to minimize the number of experiments while maximizing of their information content. This paper illustrates the applicability of ES for the design of feeding profile for a fed-batch biochemical reactor. The results illustrate that if the model structure is not accurate, the evolutionary strategy can result in more satisfactory parameter values than the classical sequential quadratic programm...

Symmetric rank-one update (SR1) is known to have good numerical performance among the quasi-Newton methods for solving unconstrained optimization problems as evident from the recent study of Farzin et al. (2011), However, it is well known that the SR1 update may not preserve positive definiteness even when updated from a positive definite approximation and can be undefined with zero denominator. In this paper, we propose some scaling strategies to overcome these well known shortcomings of the SR1 update. Numerical experiment showed that the proposed strategies are very competitive, encouraging and have exhibited a clear improvement in the numerical performance over SR1 algorithms with some existing strategies in avoiding zero denominator and preserving positive-definiteness.

Quasi-Newton (QN) methods are generally held to be the most efficient minimization methods for solving unconstrained optimization problems. Among the QN methods, symmetric rank-one (SR1) is one of the very competitive formulas. In the present paper, we propose a new SR1 method. The new technique attempts to improve the quality of the SR1 Hessian by employing the scaling of the identity in a certain sense. However, since at some iterations these updates might be singular, indefinite or undefined, this paper proposes an updates criterion based on the eigenvalues of the SR1 update to measure this quality. Hence, the new method is employed only to improve the approximation of the SR1 Hessian. It is shown that the numerical results support the theoretical considerations for the usefulness of this criterion and show that the proposed method improves the performance of the SR1 update substantially.

Two basic disadvantages of the symmetric rank one (SR1) update are that the SR1 update may not preserve positive definiteness when starting with a positive definite approximation and the SR1 update can be undefined. A simple remedy to these problems is to restart the update with the initial approximation, mostly the identity matrix, whenever these difficulties arise. However, numerical experience shows that restart with the identity matrix is not a good choice. Instead of using the identity matrix we used a positive multiple of the identity matrix. The used positive scaling factor is the optimal solution of the measure defined by the problem-maximize the determinant of the update subject to a bound of one on the largest eigenvalue. This measure is motivated by considering the volume of the symmetric difference of the two ellipsoids, which arise from the current and updated quadratic models in quasi-Newton methods. A replacement in the form of a positive multiple of the identity matrix is provided for the SR1 update when it is not positive definite or undefined. Our experiments indicate that with such simple initial scaling the possibility of an undefined update or the loss of positive definiteness for the SR1 method is avoided on all iterations. Keywords Symmetric rank one • Volume of ellipsoid • Unconstrained optimization 1 Introduction This paper is concerned with quasi-Newton methods for finding a local minimum of the unconstrained optimization problem min x∈R n f (x).