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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]

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]

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]

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]

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]

• The double influence function method including flattening deformation influence function and deflection deformation influence function are established and combined used to solve the deformations between the rolls. • A model,considering the dispersion distribution characteristics of backing roll,for calculating the contact pressure between the rolls of the Sendzimir 20-high mill based on the double influence function method is established. • The conjugate gradient method is used to solve the contact pressure between the rolls by combining the force balance equation, the moment balance equation and the deformation compatibility equation group of rolls system. • The solving accuracy of the model established is consistent with that obtained by finite element method and the computation time can be greatly reduced. This paper established a model for calculating the contact pressure between the rolls of the Sendzimir 20-high mill based on the influence function method with considering the dispersion distribution characteristics of backup rolls. Stone's and Hirschcock's formula are used to calculate the rolling force between the work rolls and the rolling stock. The force balance equations and the moment balance equations of the roll system are established to determine the relationship between the contact pressure of rolls and the rolling force. The deformation-compatibility equation is established by combining the flattening deformation, deflection deformation, and rigid displacement of the rolls and roll crown. The influence function method is used to solve the flattening deformation and deflection deformation between the rolls. The conjugate gradient method is used to solve the contact pressure between the rolls by combining the force balance equation group, the moment balance equation group, and the deformation compatibility equation group of the roll system. The contact pressure between rolls obtained by the model established is compared to that by the finite element method. The results show that the calculation accuracy of the model established in this paper is consistent with that of the finite element method. Significantly, the contact pressure between the backup roll and the second intermediate roll can be accurately calculated by considering the dispersion distribution characteristics of the backup roll, and the computation time can be greatly reduced. [ABSTRACT FROM AUTHOR]

The conjugate gradient (CG) method is a useful tool for obtaining the optimum point for unconstrained optimization problems since it does not require a second derivative or its approximations. Moreover, the conjugate gradient method can be applied in many fields such as machine learning, deep learning, neural network, and many others. This paper constructs a fourterm conjugate gradient method that satisfies the descent property and convergence properties to obtain the stationary point. The new modification was constructed based on Liu and Storey’s conjugate gradient method, two-term conjugate gradient method, and three-term conjugate gradient method. To analyze the efficiency and robustness, we used more than 150 optimization functions from the CUTEst library with different dimensions and shapes. The numerical results show that the new modification outperforms the recent conjugate gradient methods such as CG-Descent, Dai and Liao, and others in terms of number of functions evaluations, number of gradient evaluations, number of iterations, and CPU time. [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.

Here, we propose a practical method for solving nonsmooth convex problems by using conjugate gradient-type methods. The conjugate gradient method is one of the most remarkable methods to solve smooth and large-scale optimization problems. As a result of this fact, We present a modified HS conjugate gradient method. In the case that we have a nonsmooth convex problem, by the Moreau-Yosida regularization, we convert the nonsmooth objective function to a smooth function and then we use our method, by making use of a nonmonotone line search, for solving a nonsmooth convex optimization problem. We prove that our algorithm converges to an optimal solution under standard condition. Our algorithm inherits the performance of HS conjugate gradient method. [ABSTRACT FROM AUTHOR]

In this work, we focused on minimizing a strongly convex smooth function over the common fixed-point constraints. We proposed an extrapolated fixed-point optimization method, which is a modified version of the extrapolated sequential constraint method with conjugate gradient direction. We proved the convergence of the generated sequence to the unique solution to the considered problem without boundedness assumption. We also investigated some numerical experiments to underline the effectiveness and performance of the proposed method. [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.

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.

In this work, we consider Cauchy-type problems for Laplace's equation with a dynamical boundary condition on a part of the domain boundary. We construct a discrete-in-time, meshless method for solving two inverse problems for recovering the space–time-dependent source and boundary functions in dynamical and Dirichlet boundary conditions. The approach is based on Green's second identity and the forward-in-time discretization of the non-stationary problem. We derive a global connection that relates the source of the dynamical boundary condition and Dirichlet and Neumann boundary conditions in an integral equation. First, we perform time semi-discretization for the dynamical boundary condition into the integral equation. Then, on each time layer, we use Trefftz-type test functions to find the unknown source and Dirichlet boundary functions. The accuracy of the developed method for determining dynamical and Dirichlet boundary conditions for given over-determined data is first-order in time. We illustrate its efficiency for a high level of noise, namely, when the deviation of the input data is above 10% on some part of the over-specified boundary data. The proposed method achieves optimal accuracy for the identified boundary functions for a moderate number of iterations. [ABSTRACT FROM AUTHOR]

We established a new conjugate gradient method with an efficient restart direction for solving large scale unconstrained optimization problems. The modified method was proposed under the Polak-Ribière-Polyak conjugate gradient method. Under the strong Wolfe line search, the search direction of the new method was sufficiently descent and its global convergence property could be proved. Compared with other methods having good numerical performances, numerical experiments and image restorations showed that the modified method was more effective. [ABSTRACT FROM AUTHOR]

Scalarization approaches transform vector optimization problems (VOPs) into single-objective optimization but have trade-offs: information loss, subjective weight assignments, and limited representation of the Pareto front. To address these limitations, alternative strategies like conjugate gradient (CG) techniques are valuable for their simplicity and less memory usage. The paper introduces three CG techniques for VOPs, including two CG techniques that satisfy sufficient descent conditions (SDC) without a line search. These two CG techniques are combined with the third CG technique, a variant of the Polak–Ribiére–Polyak (PRP) technique, resulting in two hybrid CG techniques. Global convergence of these hybrids is achieved, without convexity assumptions, under standard assumptions and Wolfe line search. Numerical analysis and comparisons with nonnegative PRP and Liu–Storey (LS) CG techniques showcase the implementation and effectiveness of our hybrid CG techniques. The results demonstrate the promise of our hybrids techniques.

In this paper, a three-parameter subspace conjugate gradient method is proposed for solving large-scale unconstrained optimization problems. By minimizing the quadratic approximate model of the objective function on a new special three-dimensional subspace, the embedded parameters are determined and the corresponding algorithm is obtained. The global convergence result of a given method for general nonlinear functions is established under mild assumptions. In numerical experiments, the proposed algorithm is compared with SMCG_NLS and SMCG_Conic, which shows that the given algorithm is robust and efficient. [ABSTRACT FROM AUTHOR]

In this paper, a new subspace gradient method is proposed in which the search direction is determined by solving an approximate quadratic model in which a simple symmetric matrix is used to estimate the Hessian matrix in a three-dimensional subspace. The obtained algorithm has the ability to automatically adjust the search direction according to the feedback from experiments. Under some mild assumptions, we use the generalized line search with non-monotonicity to obtain remarkable results, which not only establishes the global convergence of the algorithm for general functions, but also R-linear convergence for uniformly convex functions is further proved. The numerical performance for both the traditional test functions and image restoration problems show that the proposed algorithm is efficient. [ABSTRACT FROM AUTHOR]

A mathematical model consisting of weakly coupled time fractional one parabolic PDE and one ODE equations describing dynamical processes in porous media is our physical motivation. As is often performed, by solving analytically the ODE equation, such a system is reduced to an integro-parabolic equation. We focus on the numerical reconstruction of a diffusion coefficient at finite number space-points measurements. The well-posedness of the direct problem is investigated and energy estimates of their solutions are derived. The second order in time and space finite difference approximation of the direct problem is analyzed. The approach of Lagrangian multiplier adjoint equations is utilized to compute the Fréchet derivative of the least-square cost functional. A numerical solution based on the conjugate gradient method (CGM) of the inverse problem is studied. A number of computational examples are discussed. [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.

In studying planar multi-bar mechanisms with multiple degrees of freedom, mathematical modeling is undoubtedly a way to get closer to the expected trajectory. Compared with the analytical method, the optimization method has higher accuracy in solving nonlinear equations, and it can be searched and iterated in an extensive range until it meets the real engineering solution. The research on the overlap ratio of the Slitting Shear essentially aims to study the kinematics of the two-DOF mechanism. In this study, a one-DOF mathematical model of mechanical motion was established for the Slitting Shear without considering the overlap ratio. The mathematical model was then verified through simulation. The overlap ratio was introduced based on the above mathematic model, and the three-DOF mathematic model of the shear mechanism was established. Finally, the setting of the overlap ratio was optimized using the conjugate gradient method and the global optimal NMinmize function. Under the condition of satisfying the overlap ratio, the recommended value for the overlap ratio setting of the two-DOF mechanism, which approximates the pure rolling cut, was determined. [ABSTRACT FROM AUTHOR]

Compressed imaging reconstruction technology can reconstruct high-resolution images with a small number of observations by applying the theory of block compressed sensing to traditional optical imaging systems, and the reconstruction algorithm mainly determines its reconstruction accuracy. In this work, we design a reconstruction algorithm based on block compressed sensing with a conjugate gradient smoothed l 0 norm termed BCS-CGSL0. The algorithm is divided into two parts. The first part, CGSL0, optimizes the SL0 algorithm by constructing a new inverse triangular fraction function to approximate the l 0 norm and uses the modified conjugate gradient method to solve the optimization problem. The second part combines the BCS-SPL method under the framework of block compressed sensing to remove the block effect. Research shows that the algorithm can reduce the block effect while improving the accuracy and efficiency of reconstruction. Simulation results also verify that the BCS-CGSL0 algorithm has significant advantages in reconstruction accuracy and efficiency. [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 three-term hybrid form of the Hestenes-Stiefel (HS) and the Polak-Ribiére-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.

Unconstrained optimization problems often arise from mining of big data and scientific computing. On the basis of a modified gradient-difference, this article aims to present a new threeterm conjugate gradient algorithm to efficiently solve unconstrained optimization problems. Compared with the existing nonlinear conjugate gradient algorithms, the search directions in this algorithm are always sufficiently descent independent of any line search, as well as having conjugacy property. Using the standard Wolfe line search, global and local convergence of the proposed algorithm is proved under mild assumptions. Implementing the developed algorithm to solve 750 benchmark test problems available in the literature, it is shown that the numerical performance of this algorithm is remarkable, especially in comparison with that of the other similar efficient algorithms. [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.

In the first order reliability method (FORM), the Hasofer–Lind and Rackwitz–Flessler (HL–RF) algorithm sometimes encounters numerical instability problems due to the highly nonlinear limit state function (LSF). In this paper, an improved HL–RF algorithm introducing the hybrid conjugate gradient method with adaptive Barzilai–Borwein step sizes is developed to enhance the robustness and efficiency of the original HL–RF method. The proposed algorithm is composed of two stages, the steepest descent method is performed in the first stage to move to the vicinity of the most probable failure point (MPFP) and provide a good initial location for the second stage. Along with the hybrid conjugate search direction defined by the descent direction of the non-differential merit function, the second stage quantizes the adaptive Barzilai–Borwein step sizes to accelerate the process of locating the final MPFP under the nonmonotone line search rule. Eight illustrative examples with nonlinear LSFs are analyzed in detail to validate the performance of the proposed algorithm compared with other first order reliability methods. The results indicate that the proposed algorithm is not only computationally efficient but also robust in terms of convergence, especially for those problems with super nonlinear LSFs and with nonlinear LSFs involving high-frequency noise terms. • Non-differential merit function is chosen to monitor the iterations. • Novel hybrid conjugate search direction of the most probable point is defined. • Global Barzilai–Borwein steps accelerate the convergence speed. • Optimal initial step size is set to be 1 in the nonmonotone line search stage. • Problems with high nonlinearities and high-frequency noises are solved efficiently. [ABSTRACT FROM AUTHOR]

Abstract The conjugate gradient method can be applied in many fields, such as neural networks, image restoration, machine learning, deep learning, and many others. Polak–Ribiere–Polyak and Hestenses–Stiefel conjugate gradient methods are considered as the most efficient methods to solve nonlinear optimization problems. However, both methods cannot satisfy the descent property or global convergence property for general nonlinear functions. In this paper, we present two new modifications of the PRP method with restart conditions. The proposed conjugate gradient methods satisfy the global convergence property and descent property for general nonlinear functions. The numerical results show that the new modifications are more efficient than recent CG methods in terms of number of iterations, number of function evaluations, number of gradient evaluations, and CPU time.