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In this paper the 2-D steady-state heat transfer geometric inverse problem is solved using the finite element method, conjugate gradient method, and firefly algorithm. Based on the finite element method for solving the forward heat transfer model, and based on continuous iterative optimisation of the conjugate gradient method, the accuracy of the error function of the measured and estimated values is kept within a certain range, so that the geometry of the object under test can be calculated in an inverse way. In the study of the forward problem, the temperature field distribution is solved using circular pipes as the research objects, and the feasibility of applying the finite element method to heat transfer problems is verified. The inverse problem takes the circular tube as the object and considers two different corrosion defects on the inner wall of the circular tube. Simultaneously, the firefly algorithm is introduced based on the conjugate gradient method to stochastically optimize the temperature data and suppress the fluctuation of the inversion result. Numerical experiment results indicate that the method in this paper can perform more accurate geometric shape recognition when there is a certain temperature measurement error or the temperature measurement information is relatively complete. [ABSTRACT FROM AUTHOR]

This paper proposes an improved three-term conjugate gradient algorithm designed to solve nonlinear equations with convex constraints. The key features of the proposed algorithm are as follows: (i) It only requires that nonlinear equations have continuous and monotone properties; (ii) The designed search direction inherently ensures sufficient descent and trust-region properties, eliminating the need for line search formulas; (iii) Global convergence is established without the necessity of the Lipschitz continuity condition. Benchmark problem numerical results illustrate the proposed algorithm's effectiveness and competitiveness relative to other three-term algorithms. Additionally, the algorithm is extended to effectively address the image denoising problem. [ABSTRACT FROM AUTHOR]

Unconstrained optimization problems have a long history in computational mathematics and have been identified as being among the crucial problems in the fields of applied sciences, engineering, and management sciences. In this paper, a new variant of the conjugate descent method for solving unconstrained optimization problems is introduced. The proposed algorithm can be seen as a modification of the popular conjugate descent (CD) algorithm of Fletcher. The algorithm of the proposed method is well-defined, and the sequence of the directions of search is shown to be sufficiently descending. The convergence result of the proposed method is discussed under the common standard conditions. The proposed algorithm together with some existing ones in the literature is implemented to solve a collection of benchmark test problems. Numerical experiments conducted show the performance of the proposed method is very encouraging. Furthermore, an additional efficiency evaluation is carried out on problems arising from signal processing and it works well. [ABSTRACT FROM AUTHOR]

In linear system identification problems, the Wiener filter represents a popular tool and stands as an important benchmark. Nevertheless, it faces significant challenges when identifying long-length impulse responses. In order to address the related shortcomings, the solution presented in this paper is based on a third-order tensor decomposition technique, while the resulting sets of Wiener–Hopf equations are solved with the conjugate gradient (CG) method. Due to the decomposition-based approach, the number of coefficients (i.e., the parameter space of the filter) is greatly reduced, which results in operating with smaller data structures within the algorithm. As a result, improved robustness and accuracy can be achieved, especially in harsh scenarios (e.g., limited/incomplete sets of data and/or noisy conditions). Besides, the CG-based solution avoids matrix inversion operations, together with the related numerical and complexity issues. The simulation results are obtained in a network echo cancellation scenario and support the performance gain. In this context, the proposed iterative Wiener filter outperforms the conventional benchmark and also some previously developed counterparts that use matrix inversion or second-order tensor decompositions. [ABSTRACT FROM AUTHOR]

Quantum computing is an emerging field that has had a significant impact on optimization. Among the diverse quantum algorithms, quantum gradient descent has become a prominent technique for solving unconstrained optimization (UO) problems. In this paper, we propose a quantum spectral Polak–Ribiére–Polyak (PRP) conjugate gradient (CG) approach. The technique is considered as a generalization of the spectral PRP method which employs a q-gradient that approximates the classical gradient with quadratically better dependence on the quantum variable q. Additionally, the proposed method reduces to the classical variant as the quantum variable q approaches closer to 1. The quantum search direction always satisfies the sufficient descent condition and does not depend on any line search (LS). This approach is globally convergent with the standard Wolfe conditions without any convexity assumption. Numerical experiments are conducted and compared with the existing approach to demonstrate the improvement of the proposed strategy. [ABSTRACT FROM AUTHOR]

This paper discusses a method for taking into account rounding errors when constructing a stopping criterion for the iterative process in gradient minimization methods. The main aim of this work was to develop methods for improving the quality of the solutions for real applied minimization problems, which require significant amounts of calculations and, as a result, can be sensitive to the accumulation of rounding errors. However, this paper demonstrates that the developed approach can also be useful in solving computationally small problems. The main ideas of this work are demonstrated using one of the possible implementations of the conjugate gradient method for solving an overdetermined system of linear algebraic equations with a dense matrix. [ABSTRACT FROM AUTHOR]

This paper focuses on modifying the existing Conjugate Gradient (CG) method of Rivaie, Mustafa, Ismail and Leong (RMIL). The RMIL technique has been the subject of previous studies to enhance its effectiveness. In this study, a new CG search direction, IRMIL, has been presented. This new variation combines the scaled negative gradient, which acts as an initial direction, and a third-term parameter. This paper proves that the IRMIL satisfies the sufficient descent criteria. The method also exhibits global convergence characteristics for exact and strong Wolfe line searches. The method's efficacy is assessed using two distinct methodologies. The first methodology involved conducting numerical tests on conventional Unconstrained Optimisation (UO) problems. The test shows that, while the IRMIL method performs very similarly to other existing CG methods during exact line search, it excels during strong Wolfe line search and converges more quickly. For the second methodology, the NEWMRIL method is applied to solve issues regarding image restoration. Overall, IRMIL method exhibits excellent theoretical and numerical efficiency potential. [ABSTRACT FROM AUTHOR]

Kernel adaptive filter armed with information theoretic learning has gained popularity in the domain of time series online prediction. In particular, the generalized correntropy criterion (GCC), as a nonlinear similarity measure, is robust to non-Gaussian noise or outliers in time series. However, due to the nonconvex nature of GCC, optimal parameter estimation may be difficult. Therefore, this paper deliberately combines it with half-quadratic (HQ) optimization to generate the generalized HQ correntropy (GHC) criterion, which provides reliable calculations for convex optimization. After that, a novel adaptive algorithm called kernel generalized half-quadratic correntropy conjugate gradient (KGHCG) algorithm is designed by integrating GHC and the conjugate gradient method. The proposed approach effectively enhances the robustness of non-Gaussian noise and greatly improves the convergence speed and filtering accuracy, and its sparse version KGHCG-VP limits the dimension of the kernel matrix through vector projection, which successfully handles the bottleneck of high computational complexity. In addition, we also discuss the convergence properties, computational complexity and memory requirements in terms of theoretical analysis. Finally, online prediction simulation results with the benchmark Mackey–Glass chaotic time series and real-world datasets show that KGHCG and KGHCG-VP have better convergence and prediction performance. [ABSTRACT FROM AUTHOR]

The paper proposes a parallel algorithm for solving large overdetermined systems of linear algebraic equations with a dense matrix. This algorithm is based on the use of a modification of the conjugate gradient method, which is able to take into account rounding errors accumulated during calculations when making a decision to terminate the iterative process. The parallel algorithm is constructed in such a way that it takes into account the capabilities of the message passing interface (MPI) parallel programming technology, which is used for the software implementation of the proposed algorithm. The programming examples are shown using the Python programming language and the mpi4py package, but all programs are built in such a way that they can be easily rewritten using the C/C++/Fortran programming languages. The advantage of using the modern MPI-4.0 standard is demonstrated. [ABSTRACT FROM AUTHOR]

In this paper, we investigate an optimal control problem (OCP) for three-dimensional (3D) unsteady partial differential equations with convection term (UPDECT), extensively applied to the solidification process of billets. Firstly, we introduce the mathematical model of OCP and adopt a life cycle model method (LCMM) to numerically solve the 3D UPDECT. Secondly, the Lipschitz continuity of the cost function gradient is proved using the adjoint approach as a numerical method for designing optimal control systems. The cost function gradient of OCP for 3D UPDECT is analyzed based on a Hamiltonian costate system. Thirdly, we propose a modified trinomial conjugate gradient method (MTCGM) to solve the OCP for 3D UPDECT and prove the global convergence of MTCGM. Finally, the MTCGM is applied to the experimental simulation of continuous casting. According to experiment results, the optimizer based on MTCGM can significantly accelerate the convergence speed and reduce the iterative numbers, and the MTCGM provides a more stable temperature control effect. • An optimal control problem (OCP) for three-dimensional unsteady partial differential equation is studied. • The Lipschitz continuity of cost function gradient in the OCP is proved, and the Lipschitz constant is obtained. • The expression of gradient calculation of the OCP is analyzed based on Hamiltonian costate system. • A modified trinomial conjugate gradient method (MTCGM), and the global convergence of the MTCGM is proved. • The effectiveness of MTCGM is verified by experimental simulations. [ABSTRACT FROM AUTHOR]

The conjugate gradient (CG) method is an optimization method, which, in its application, has a fast convergence. Until now, many CG methods have been developed to improve computational performance and have been applied to real-world problems. In this paper, a new hybrid three-term CG method is proposed for solving unconstrained optimization problems. The search direction is a threeterm hybrid form of the Hestenes-Stiefel (HS) and the Polak-Ribi'ere-Polyak (PRP) CG coefficients, and it satisfies the sufficient descent condition. In addition, the global convergence properties of the proposed method will also be proved under the weak Wolfe line search. By using several test functions, numerical results show that the proposed method is most efficient compared to some of the existing methods. In addition, the proposed method is used in practical application problems for image restoration and portfolio selection. [ABSTRACT FROM AUTHOR]

This paper presents an approach for the solution of a zero-sum differential game associated with a nonlinear state-feedback H ∞ control problem. Instead of using the approximation methods for solving the corresponding Hamilton–Jacobi–Isaacs (HJI) partial differential equation, we propose an algorithm that calculates the explicit inputs to the dynamic system by directly performing minimization with simultaneous maximization of the same objective function. In order to achieve numerical robustness and stability, the proposed algorithm uses: quasi-Newton method, conjugate gradient method, line search method with Wolfe conditions, Adams approximation method for time discretization and complex-step calculation of derivatives. The algorithm is evaluated in computer simulations on examples of first- and second-order nonlinear systems with analytical solutions of H ∞ control problem. [ABSTRACT FROM AUTHOR]

Reliability is always considered as one of the most important factors in every system and it is commonly accepted to investigate reliability by employing reliability analysis problems. In this case, the need of existence of powerful reliability analysis methods has resulted in introducing different methods to solve first-order and higher-order reliability analysis problems. These methods have also been employed to ensure failure probability of a system is below an acceptable level. However, there are still disadvantages such as instability and inefficiency in some of these methods. The Hasofer - Lind and Rackwitz - Fiessler (HL-RF) method is still the most popular method to solve the first-order reliability analysis problems, but it will be shown in this paper that this method is not stable enough to solve highly non-linear problems. Also, it is found that the HL-RF method may show some inefficiency behaviour even when it converges to the optimum point. In this paper, a new reliability analysis method is introduced that solves first-order reliability analysis problems stably and efficiently. This method is based on the conjugate gradient direction that is used as a line search algorithm. The new reliability analysis method is called Conjugate Gradient Direction-Based (CGDB) method whose performance is compared with the performance of the existing HL-RF method by solving several numerical experiments at the end of this paper. [ABSTRACT FROM AUTHOR]

This paper proposes a novel ant colony hyperheuristic approach for reordering the rows and columns of symmetric positive definite matrices. This ant colony hyperheuristic approach evolves heuristics for bandwidth reduction applied to instances arising from specific application areas with the objective of generating low-cost reordering algorithms. This paper evaluates the resulting reordering algorithm in each application area against state-of-the-art reordering algorithms with the purpose of reducing the running times of the zero-fill incomplete Cholesky-preconditioned conjugate gradient method. The results obtained on a wide-ranging set of standard benchmark matrices show that the proposed approach compares favorably with state-of-the-art reordering algorithms when applied to instances arising from computational fluid dynamics, structural, and thermal problems. [ABSTRACT FROM AUTHOR]

The conjugate gradient methods are one of the most important techniques used to address problems involving minimization or maximization, especially nonlinear optimization problems with no constraints at all. That is because of their simplicity and low memory needed. They can be applied in many areas, such as economics, engineering, neural networks, image restoration, machine learning, and deep learning. The convergence of Fletcher-Reeves (FR) conjugate gradient method has been established under both exact and strong Wolfe line searches. However, it is performance in practice is poor. In this paper, to get good numerical performance from the FR method, a little modification is done. The global convergence of the modified version has been established for general nonlinear functions. Preliminary numerical results show that the modified method is very efficient in terms of number of iterations and CPU time. [ABSTRACT FROM AUTHOR]

Conjugate gradient method is an important method for solving large-scale unconstrained nonlinear optimization problems. In this paper, we take a little modification to CD conjugate gradient method and a modified CD conjugate gradient method is proposed. The presented method has sufficient descent properties. Under mild conditions, we prove that the method with strong Wolfe line search is globally convergent. At the end of this paper, we also present numerical experiments to show the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Thermal has a significant effect on high speed machine tools. When temperature grows up, it often causes processing errors and further reduces product quality. This paper aims to investigate magnitude of heat sources and temperature distribution in a horizontal lathe based on inverse method. In the present thermal model, there is not only consideration of spindle, but also analyzing relative structures. Simulated and experimental temperatures are used as input data to predict heat sources and temperature field. Effects of speed, number measured points, measurement errors, and measured distance on predicted results are analyzed. Results indicate that this inverse method can accurately estimate the heat sources based on two measured temperatures at front and rear outer rings. The trend of estimated heat sources is then compared to measured load rate. Results herein are useful information for designing horizontal lathe spindle and reducing thermal errors. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present a new conjugate gradient method for solving unconstrained nonlinear optimization problems, based on Perry's idea. An accelerated adaptive algorithm is proposed, where our search direction satisfies the sufficient descent condition. The global convergence is analyzed using the spectral analysis. The numerical results are described for a set of standard test problems, and it is shown that the performance of the proposed method is better than that of the CG-DESCENT, the mBFGS and the SPDOC. [ABSTRACT FROM AUTHOR]

In this paper, we study Cayley-transform-based gradient and conjugate gradient algorithms for optimization on Grassmann manifolds. We revisit the Cayley transform on Grassmann manifolds as a retraction in the framework of quotient manifolds constructed by Lie group actions and obtain an efficient formula for this retraction in low-rank cases. We also prove that this retraction is the restriction of the Cayley transform on Stiefel manifolds to horizontal spaces. To develop vector transports on Grassmann manifolds, we introduce a concept called induced vector transports on quotient manifolds. Based on this concept, three vector transports associated with the Cayley transform are obtained. The first vector transport is the traditional orthogonal projection onto horizontal spaces, whereas the other two vector transports are newly proposed herein. We show that one of the new vector transports satisfies the Ring–Wirth non-expansion condition and that the other is isometric. We also simplify the formulae of the new vector transports in low-rank cases. Riemannian gradient and conjugate gradient algorithms are established via the Cayley transform and the three abovementioned vector transports. Numerical experiments on two mean-of-subspaces problems demonstrate the effectiveness of the proposed algorithms. [ABSTRACT FROM AUTHOR]

A Polak–Ribière–Polyak (PRP) algorithm is one of the oldest and popular conjugate gradient algorithms for solving nonlinear unconstrained optimization problems. In this paper, we present a q-variant of the PRP (q-PRP) method for which both the sufficient and conjugacy conditions are satisfied at every iteration. The proposed method is convergent globally with standard Wolfe conditions and strong Wolfe conditions. The numerical results show that the proposed method is promising for a set of given test problems with different starting points. Moreover, the method reduces to the classical PRP method as the parameter q approaches 1. [ABSTRACT FROM AUTHOR]

Force identification of stochastic structures is very important in science and engineering, which also leads to the challenges in the field of computational mechanics. Monte-Carlo simulation (MCS) method is a robust and effective random simulation technique for the dynamic load identification problem of the stochastic structure. However, the MCS method needs large computational cost and is also inefficient for practical engineering applications because of the requirement of a large quantity of samples. In this paper, in order to improve computational efficiency of MCS, a novel algorithm is proposed based on the modified conjugate gradient method and matrix perturbation method. First, the new developed algorithm exploits matrix perturbation method to transform dynamic load identification problems for stochastic structures into equivalent deterministic dynamic load identification problems. Then the dynamic load identification can be realized using modified conjugate gradient method. Finally, the statistical characteristics of identified force are analyzed. The accuracy and efficiency of the newly developed computational method are demonstrated by several numerical examples. It has been found that the newly proposed algorithm can significantly improve the computational efficiency of MCS and it is believed to be a powerful tool for solving the dynamic load identification for stochastic structures. [ABSTRACT FROM AUTHOR]

In this paper, we present a family of conjugate gradient projection methods for solving large-scale nonlinear equations. At each iteration, it needs low storage and the subproblem can be easily solved. Compared with the existing solution methods for solving the problem, its global convergence is established without the restriction of the Lipschitz continuity on the underlying mapping. Preliminary numerical results are reported to show the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

The Conjugate Gradient Method is a numerical optimization technique that finds the optimal solution by focusing on the coefficient conjugate. This paper presents a new coefficients conjugate gradient method for removing impulse noise from images, which is based on a quadratic function and is proven to be globally convergent. Results show that it is an effective method for image restoration. [ABSTRACT FROM AUTHOR]

In this paper, based on the efficient Conjugate Descent (CD) method, two generalized CD algorithms are proposed to solve the unconstrained optimization problems. These methods are three-term conjugate gradient methods which the generated directions by using the conjugate gradient parameters and independent of the line search satisfy in the sufficient descent condition. Furthermore, under the strong Wolfe line search, the global convergence of the proposed methods are proved. Also, the preliminary numerical results on the CUTEst collection are presented to show effectiveness of our methods. [ABSTRACT FROM AUTHOR]

We consider a retrospective inverse heat conduction problem with nonstationary inhomogeneous Dirichlet boundary conditions. It is approximated by a Crank–Nicolson scheme that has the second order of approximation both in the spatial variable and in time. It is proposed to use the iterative method of conjugate gradients to determine the solution of the resulting system of linear algebraic equations. Examples are given of reconstructing smooth, nonsmooth, and discontinuous initial conditions, including the introduction of a "noise" characteristic, typical of additional conditions of inverse problems, and its smoothing using the Savitzky–Golay filter. [ABSTRACT FROM AUTHOR]

In this paper, the conjugate gradient method coupled with adjoint problem is used in order to solve the inverse heat conduction problem and estimation of the timedependent heat flux using the temperature distribution at a point in a three layer system with none homogeneous boundary conditions. Also, the effect of noisy data on final solution is studied. For solving this problem the general coordinate method is used. The inverse heat conduction problem of estimating the transient heat flux, applied on part of the boundary of an irregular region is solved in this paper. The present formulation is general and can be applied to the solution of boundary inverse heat conduction problems over any region that can be mapped into a rectangle. The obtained results for few selected examples show the good accuracy of the presented method. Also the solutions have good stability even if the input data includes noise. Applications of this model are in the thermal protect systems (t.p.s.) and heat shield systems. [ABSTRACT FROM AUTHOR]

In this paper, by using the smoothing Fischer-Burmeister function, we present a new smoothing conjugate gradient method for solving the nonlinear nonsmooth complementarity problems. The line search which we used guarantees the descent of the method. Under suitable conditions, the new smoothing conjugate gradient method is proved globally convergent. Finally, preliminary numerical experiments show that the new method is efficient. [ABSTRACT FROM AUTHOR]

In this paper, a boundary data completion problem for a diffusionreaction partial differential equations (PDEs) was considered in a 2D domain. In a first step, the classical KMF (Kozlov, Maz'ya, Fomin) algorithm was used with spectral element method to find an approximate solution to that problem. In a second step, an alternative way was proposed by using conjugate gradient method where a symmetric positive definite operator was designed. Obtained results are illustrated by some numerical tests by using Matlab software. [ABSTRACT FROM AUTHOR]

Highlights • A novel type of RIBEM combined with modified conjugate gradient method to solve the inverse problem of non-homogeneous is proposed for the first time. • The complex variable derivative method is introduced and then the sensitivity coefficient matrix can be accurately calculated. • The conjugate gradient method, which includes restart factor, solves the problem of slow convergence in traditional conjugate gradient methods. Abstract In order to analyze the heat conduction problem of composite materials in aerospace engineering, a modified conjugate gradient method is proposed to identify the physical parameters of transient heat conduction problems in this paper. In the positive problem, the boundary element method based on radial integration method is used to obtain the measured point temperature of the transient heat conduction problem with varying thermal conductivity. In the inverse problem, based on the advantages of high accuracy and fast convergence rate of the conjugate gradient method, a modified conjugate gradient method is used. On the one hand, the complex variable derivative method is introduced into the traditional conjugate gradient method, and then the sensitivity coefficient matrix can be accurately calculated. On the other hand, based on the linear convergence property of the conjugate gradient method, a restart factor is introduced in this paper. For an inverse problem containing n unknown parameters, the algorithm restarts from the steepest descent method after every n + 1 iterations to identify physical properties of transient non-homogeneous thermal conduction. The complex variable derivation method overcomes the problem that the traditional conjugate gradient method needs differential processing and low derivative accuracy in solving the inverse transient heat conduction problem. When solving the complicated inverse problems, conjugate gradient method, which includes restart factor, solves the problem of slow convergence in traditional conjugate gradient methods. Numerical examples are presented to demonstrate the accuracy and efficiency of the present method in identifying the parameters of transient non-homogeneous thermal conductivity. [ABSTRACT FROM AUTHOR]

In this paper, three modified Polak-Ribière-Polyak (PRP) conjugate gradient methods for unconstrained optimization are proposed. They are based on the two-term PRP method proposed by Cheng (Numer. Funct. Anal. Optim. 28:1217-1230, 2007), the three-term PRP method proposed by Zhang et al. (IMA J. Numer. Anal. 26:629-640, 2006), and the descent PRP method proposed by Yu et al. (Optim. Methods Softw. 23:275-293, 2008). These modified methods possess the sufficient descent property without any line searches. Moreover, if the exact line search is used, they reduce to the classical PRP method. Under standard assumptions, we show that these three methods converge globally with a Wolfe line search. We also report some numerical results to show the efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present accelerations of the Mann and CQ algorithms. We first apply the Picard algorithm to the smooth convex minimization problem and point out that the Picard algorithm is the steepest descent method for solving the minimization problem. Next, we provide the accelerated Picard algorithm by using the ideas of conjugate gradient methods that accelerate the steepest descent method. Then, based on the accelerated Picard algorithm, we present accelerations of the Mann and CQ algorithms. Under certain assumptions, we show that the new algorithms converge to a fixed point of a nonexpansive mapping. Finally, we show the efficiency of the accelerated Mann algorithm by numerically comparing with the Mann algorithm. A numerical example is provided to illustrate that the acceleration of the CQ algorithm is ineffective. [ABSTRACT FROM AUTHOR]

Analysis and design of linear periodic control systems are closely related to the discrete-time periodic matrix equations. In this paper, we propose an iterative algorithm based on the conjugate gradient method on the normal equations (CGNE) for finding the solution group of the general coupled periodic matrix equations By proving some properties of the algorithm, we show that the solution group of the periodic matrix equations can be obtained within a finite number of iterations in the absence of roundoff errors. Numerical examples are given to illustrate the efficiency and accuracy of the proposed algorithm. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present a very efficient and accurate numerical algorithm to identify a variable (space- and temperature-dependent) heat transfer coefficient in two-dimensional inverse steady-state heat conduction problems involving irregular heat-conducting body shapes in the presence of Dirichlet, Neumann, and Robin boundary conditions. In this numerical method, a boundary-fitted grid generation technique (elliptic) is used to discretize the physical domain (heat-conducting body) and solve for the steady-state heat conduction equation by approximating the derivatives of the field variable (temperature) by algebraic ones. This paper describes a very accurate and efficient sensitivity analysis scheme to compute the sensitivity of the temperatures to variation of the variable heat transfer coefficient. The main advantage of the sensitivity analysis scheme is that it does not require the solution of adjoint equation. The conjugate gradient method (CGM) is used to reduce the mismatch between the computed temperature on part of the boundary and the simulated measured temperature distribution. The obtained results confirm that the proposed algorithm is very accurate, efficient, and robust. [ABSTRACT FROM AUTHOR]

The conjugate gradient (CG) method is an efficient iterative method for solving large-scale strongly convex quadratic programming (QP). In this paper, we propose some generalized CG (GCG) methods for solving the l1-regularized (possibly not strongly) convex QP that terminate at an optimal solution in a finite number of iterations. At each iteration, our methods first identify a face of an orthant and then either perform an exact line search along the direction of the negative projected minimum-norm subgradient of the objective function or execute a CG subroutine that conducts a sequence of CG iterations until a CG iterate crosses the boundary of this face or an approximate minimizer of over this face or a subface is found. We determine which type of step should be taken by comparing the magnitude of some components of the minimum-norm subgradient of the objective function to that of its rest components. Our analysis on finite convergence of these methods makes use of an error bound result and some key properties of the aforementioned exact line search and the CG subroutine. We also show that the proposed methods are capable of finding an approximate solution of the problem by allowing some inexactness on the execution of the CG subroutine. The overall arithmetic operation cost of our GCG methods for finding an ∊-optimal solution depends on ∊ in O(log(1/∊)), which is superior to the accelerated proximal gradient method (Beck and Teboulle [Beck A, Teboulle M (2009) A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1):183-202], Nesterov [Nesterov Yu (2013) Gradient methods for minimizing composite functions. Math. Program. 140(1):125-161]) that depends on∊ in O(1/∊)). In addition, our GCG methods can be extended straightforwardly to solve box-constrained convex QP with finite convergence. Numerical results demonstrate that our methods are very favorable for solving ill-conditioned problems. [ABSTRACT FROM AUTHOR]

Complex-valued neural networks (CVNNs) have become a powerful modelling tool for complex-valued data processing. Because most of the critical points of CVNNs are saddle points, the gradient-based learning algorithms for CVNNs enjoy more chances to reach the global minima while suffering from slow convergence. To this end, we propose a hybrid complex spectral conjugate gradient learning algorithm for fast training CVNNs in this paper. The proposed algorithm combines the scaled negative gradient with a Barzilai–Borwein stepsize and an optimized conjugate term to define a new training direction, thus providing an accurate approximation of the second-order curvature of the objective function. The complex Wolfe conditions are employed to adaptively determine the optimal training stepsize. Under mild conditions, the descent property of the training direction and the convergence of the proposed algorithm are theoretically established. Simulation results on a number of benchmark complex-valued data processing problems demonstrate the efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

This paper considers the minimax design of two-dimensional (2D) finite impulse response (FIR) half-band filters. First, the design problem is formulated in a matrix form, where the half-band constraints are expressed as a pair of matrix equations. By matrix transformations, the constrained minimax problem is transformed into an unconstrained one. Then, we propose an efficient iterative reweighted least squares (IRLS) algorithm to solve this problem. The weighted least squares (WLS) subproblems arising from the IRLS algorithm are solved using a generalized conjugate gradient (GCG) algorithm. Moreover, the GCG algorithm is guaranteed to converge in a finite number of iterations. In the proposed algorithm, the design coefficients of filters are solved in their matrix form, leading to a great saving in computations and memory space. Design examples and comparisons with existing methods are provided to demonstrate the effectiveness and efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

The stationary point of optimization problems can be obtained via conjugate gradient (CG) methods without the second derivative. Many researchers have used this method to solve applications in various fields, such as neural networks and image restoration. In this study, we construct a three-term CG method that fulfills convergence analysis and a descent property. Next, in the second term, we employ a Hestenses-Stiefel CG formula with some restrictions to be positive. The third term includes a negative gradient used as a search direction multiplied by an accelerating expression. We also provide some numerical results collected using a strong Wolfe line search with different sigma values over 166 optimization functions from the CUTEr library. The result shows the proposed approach is far more efficient than alternative prevalent CG methods regarding central processing unit (CPU) time, number of iterations, number of function evaluations, and gradient evaluations. Moreover, we present some applications for the proposed three-term search direction in image restoration, and we compare the results with well-known CG methods with respect to the number of iterations, CPU time, as well as root-mean-square error (RMSE). Finally, we present three applications in regression analysis, image restoration, and electrical engineering. [ABSTRACT FROM AUTHOR]

In this paper, a new conjugate gradient method is suggested to estimate the multi-source dynamic loads acting on a surface of 3D plate. The algorithm reconstructs the distribution functions of the dynamic loads by the displacement responses at two receiving points. The dynamic Green's function and response function of dynamic forces excitation are obtained by finite element method. Under linear and time-invariant supposition, the response at a receiving point on the structure can be expressed as a convolution integral of the desired unknown force and the corresponding Green's function, where the continuous convolution functions are spatially and temporally discretized by trapezium formula. The preset method is suggested to identify the dynamic loads by minimizing the error functions. Finally, the multi-source dynamic loads are successfully identified by the proposed method, and the performance of numerical simulations is superior to Landweber iteration method, which demonstrates the efficiency and robustness of the proposed method. [ABSTRACT FROM AUTHOR]

An inverse optimization algorithm based on Quantum-Behaved Particle Swarm Optimization (QPSO) is examined and applied to estimate the unknown transient heat flux applied to certain boundaries in transient heat conduction problems. Results demonstrate the accuracy, stability and validity of the QPSO method in inverse estimation of the heat flux without prior knowledge of the functional form of the unknown quantities. This paper also addresses the high computational costs of QPSO and proposes a hybrid method to reduce the computational costs by combining the advantages of a gradient method and a stochastic method. Finally comparison of the proposed hybrid method and Conjugate Gradient Method (CGM) is also included. [ABSTRACT FROM AUTHOR]

Identification is considered to be among the main applications of inverse theory and its objective for a given physical system is to use data which is easily observable, to infer some of the geometric parameters which are not directly observable. In this paper, a parameter identification method using inverse problem methodology is proposed. The minimisation of the objective function with respect to the desired vector of design parameters is the most important procedure in solving the inverse problem. The conjugate gradient method is used to determine the unknown parameters, and Tikhonov's regularization method is then used to replace the original ill-posed problem with a well-posed problem. The simulation and experimental results are presented and compared. [ABSTRACT FROM AUTHOR]

This paper gives a modified PRP method which possesses the global convergence of nonconvex function and the R-linear convergence rate of uniformly convex function. Furthermore, the presented method has sufficiently descent property and characteristic of automatically being in a trust region without carrying out any line search technique. Numerical results indicate that the new method is interesting for the given test problems. [ABSTRACT FROM AUTHOR]

A new solution methodology for the constant matrix, decoupled power flow problem is presented in this paper. The proposed method uses the conjugate gradient method instead of the traditional direct solution of Ax = b for updating the power flow variables. The conjugate gradient method is accelerated with an approximate inverse matrix preconditioner obtained from a linear combination of matrix-valued Chebyshev polynomials. The new method has been tested on several systems of different sizes. In terms of speed, the method is comparable to the fast decoupled load flow [1] in serial environments but it is more amenable to parallel and vector processing since it contains only matrix-vector multiplications. [ABSTRACT FROM AUTHOR]