Conjugate Gradient Projection Method Research Articles

An improved Dai‐Liao‐style hybrid conjugate gradient‐based method for solving unconstrained nonconvex optimization and extension to constrained nonlinear monotone equations

In this work, for unconstrained optimization, we introduce an improved Dai‐Liao‐style hybrid conjugate gradient method based on the hybridization‐based self‐adaptive technique, and the search direction generated fulfills the sufficient descent and trust region properties regardless of any line search. The global convergence is established under standard Wolfe line search and common assumptions. Then, combining the hyperplane projection technique and a new self‐adaptive line search, we extend the proposed conjugate gradient method and obtain an improved Dai‐Liao‐style hybrid conjugate gradient projection method to solve constrained nonlinear monotone equations. Under mild conditions, we obtain its global convergence without Lipschitz continuity. In addition, the convergence rates for the two proposed methods are analyzed, respectively. Finally, numerical experiments are conducted to demonstrate the effectiveness of the proposed methods.

An improved spectral conjugate gradient projection method for monotone nonlinear equations with application

An improved spectral conjugate gradient projection method for monotone nonlinear equations with application

A conjugate gradient projection method with restart procedure for solving constraint equations and image restorations

A conjugate gradient projection method with restart procedure for solving constraint equations and image restorations

An inertial spectral conjugate gradient projection method for constrained nonlinear pseudo-monotone equations

An inertial spectral conjugate gradient projection method for constrained nonlinear pseudo-monotone equations

A modified inertial three-term conjugate gradient projection method for constrained nonlinear equations with applications in compressed sensing

A modified inertial three-term conjugate gradient projection method for constrained nonlinear equations with applications in compressed sensing

A three-term CGPM-based algorithm without Lipschitz continuity for constrained nonlinear monotone equations with applications

In this work, a new three-term conjugate gradient projection method (CGPM) with a new adaptive line search strategy is proposed for solving the large-scale nonlinear monotone equations with convex constraints. The proposed three-term CGPM-based algorithm has the following characteristics: (i) its search direction satisfies the sufficient descent and trust region properties which is independent of the line search; (ii) the nonlinear equations only satisfy continuous and monotone properties; (iii) its global convergence is analyzed and obtained without the Lipschitz continuity. Some preliminary numerical experiment results are reported and their corresponding performance profiles are illustrated, which show that the proposed algorithm is effective. Finally, the proposed algorithm is extended to solve sparse signal restoration problems.

A New Hybrid Conjugate Gradient Projection Method for Solving Nonlinear Monotone Equations

In this study, we propose a new modified hybrid conjugate gradient projection method with a new scale parameter φk for solving large‐scale nonlinear monotone equations. The proposed method includes two major features: projection techniques and sufficient descent property independent of line search technique. Global convergence of the proposed method is proved under some suitable assumptions. Finally, numerical results illustrating the robustness of the suggested strategy and its comparisons are shown.

First and second order unconditionally energy stable schemes for topology optimization based on phase field method

In this paper, we use the phase field method to deal with the compliance minimization problem in topology optimization. A modified Allen-Cahn type equation with two penalty terms is proposed. The equation couples the diffusive interface dynamics and the linear elasticity mechanics. We propose the first- and second-order unconditionally energy stable schemes for the evolution of phase field modeling. The linearly stabilized splitting scheme is applied to improve the stability. The Crank-Nicolson scheme is applied to achieve second-order accuracy in time. We prove the unconditional stabilities of our schemes in analysis. The finite element method and the projected conjugate gradient method combining with fast fourier transform are used to solve the compliance minimization problem. Several experimental results are presented to verify the efficiency and accuracy of the proposed schemes.

A derivative-free RMIL conjugate gradient projection method for convex constrained nonlinear monotone equations with applications in compressive sensing

In this paper, a derivative-free RMIL conjugate gradient projection method for solving large-scale nonlinear monotone equations with convex constraints is proposed. The proposed method is a modification of an RMIL conjugate gradient method combined with the projection techniques. We establish its global convergence under appropriate conditions. The method is then compared with other existing methods in the literature and the numerical results indicate that the method is efficient. Furthermore, the proposed method is used to recover a sparse signal from an incomplete and contaminated sampling measurements and the results are promising.

A generalized hybrid CGPM-based algorithm for solving large-scale convex constrained equations with applications to image restoration

In this paper, by improving a line search criterion to yield steplength, and designing a hybrid conjugate parameter to construct sufficient descent direction, we propose a generalized hybrid conjugate gradient projection method for solving large-scale monotone nonlinear equations with convex constraints. For the proposed method, we prove its global convergence under some mild conditions, and study its convergence rate. Numerical comparisons with two existing methods show that our method is promising for solving large-scale nonlinear constrained equations. Furthermore, the experiment results of dealing with image restoration problems also verify that the proposed method is effective.

A hybrid three-term conjugate gradient projection method for constrained nonlinear monotone equations with applications

In this paper, based on the three-term conjugate gradient method and the hybrid technique, we propose a hybrid three-term conjugate gradient projection method by incorporating the adaptive line search for solving large-scale nonlinear monotone equations with convex constraints. The search direction generated by the proposed method is close to the one yielded by the memoryless BFGS method, and has the sufficient descent property and the trust region property independent of line search technique. Under some mild conditions, we establish the global convergence of the proposed method. Our numerical experiments show the effectiveness and robustness of the proposed method in comparison with two existing algorithms in the literature. Moreover, we show applicability and encouraging efficiency of the proposed method by extending it to solve sparse signal restoration and image de-blurring problems.

A New Conjugate Gradient Projection Method for Convex Constrained Nonlinear Equations

The conjugate gradient projection method is one of the most effective methods for solving large-scale monotone nonlinear equations with convex constraints. In this paper, a new conjugate parameter is designed to generate the search direction, and an adaptive line search strategy is improved to yield the step size, and then, a new conjugate gradient projection method is proposed for large-scale monotone nonlinear equations with convex constraints. Under mild conditions, the proposed method is proved to be globally convergent. A large number of numerical experiments for the presented method and its comparisons are executed, which indicates that the presented method is very promising. Finally, the proposed method is applied to deal with the recovery of sparse signals.

Least-Square-Based Three-Term Conjugate Gradient Projection Method for ℓ1-Norm Problems with Application to Compressed Sensing

In this paper, we propose, analyze, and test an alternative method for solving the ℓ 1 -norm regularization problem for recovering sparse signals and blurred images in compressive sensing. The method is motivated by the recent proposed nonlinear conjugate gradient method of Tang, Li and Cui [Journal of Inequalities and Applications, 2020(1), 27] designed based on the least-squares technique. The proposed method aims to minimize a non-smooth minimization problem consisting of a least-squares data fitting term and an ℓ 1 -norm regularization term. The search directions generated by the proposed method are descent directions. In addition, under the monotonicity and Lipschitz continuity assumption, we establish the global convergence of the method. Preliminary numerical results are reported to show the efficiency of the proposed method in practical computation.

A conjugate gradient projection method for solving equations with convex constraints

In this paper, we propose a derivative-free iterative method for a class of equations with convex constraints appearing in a variety of the practical problems such as compressing sensing, fluid mechanics, plasma physics, nonlinear optics and solid state. In the iteration, our search direction can be viewed as an extension of a modified three-term CG method. By an appropriate line search and the projection step, our method is convergent to the solution. Our method inherits the advantages of CG method and projection method, and thus is suitable for solving large-scale non-smooth problem. Under the assumption that F is Lipschitz continuous and satisfies a weaker condition of monotonicity, our method is globally convergent. Numerical results show the efficiency of the proposed method.

An Efficient Projected Gradient Method for Convex Constrained Monotone Equations with Applications in Compressive Sensing

In this paper, a modified Polak-Ribière-Polyak conjugate gradient projection method is proposed for solving large scale nonlinear convex constrained monotone equations based on the projection method of Solodov and Svaiter. The obtained method has low-complexity property and converges globally. Furthermore, this method has also been extended to solve the sparse signal reconstruction in compressive sensing. Numerical experiments illustrate the efficiency of the given method and show that such non-monotone method is suitable for some large scale problems.

A derivative-free conjugate gradient projection method based on the memoryless BFGS update

A derivative-free conjugate gradient projection method based on the memoryless BFGS update

An efficient hybrid conjugate gradient-based projection method for convex constrained nonlinear monotone equations

Conjugate gradient projection methods are widely used for solving large-scale nonlinear systems of monotone equations because of their low memory requirements and research continues to be done to improve these methods. In this paper, we introduce a new hybrid conjugate gradient projection method for solving large-scale nonlinear monotone equations. This proposed method is shown to satisfy the descent property and its global convergence is also established. Numerical results show that the new method is very much promising when compared with other conjugate gradient projection methods.

A Modified Fletcher–Reeves Conjugate Gradient Method for Monotone Nonlinear Equations with Some Applications

One of the fastest growing and efficient methods for solving the unconstrained minimization problem is the conjugate gradient method (CG). Recently, considerable efforts have been made to extend the CG method for solving monotone nonlinear equations. In this research article, we present a modification of the Fletcher–Reeves (FR) conjugate gradient projection method for constrained monotone nonlinear equations. The method possesses sufficient descent property and its global convergence was proved using some appropriate assumptions. Two sets of numerical experiments were carried out to show the good performance of the proposed method compared with some existing ones. The first experiment was for solving monotone constrained nonlinear equations using some benchmark test problem while the second experiment was applying the method in signal and image recovery problems arising from compressive sensing.

A Modified Spectral PRP Conjugate Gradient Projection Method for Solving Large‐Scale Monotone Equations and Its Application in Compressed Sensing

In this paper, we develop an algorithm to solve nonlinear system of monotone equations, which is a combination of a modified spectral PRP (Polak‐Ribière‐Polyak) conjugate gradient method and a projection method. The search direction in this algorithm is proved to be sufficiently descent for any line search rule. A line search strategy in the literature is modified such that a better step length is more easily obtained without the difficulty of choosing an appropriate weight in the original one. Global convergence of the algorithm is proved under mild assumptions. Numerical tests and preliminary application in recovering sparse signals indicate that the developed algorithm outperforms the state‐of‐the‐art similar algorithms available in the literature, especially for solving large‐scale problems and singular ones.

Monitoring Earth's ionosphere by means of hardware-software complex using the GPS/GLONASS satellite systems

Near real-time one-dimensional vertical electron density profiles are determined from GPS-derived total electron content (TEC) data by means of the iterative conjugate gradient projection method (CGP). Electron density profiles are determined in near realtime (within minutes of the time of measurement) from short time series of slant TEC (STEC) approximately 5 minutes. Measured STEC values are obtained from dual frequency data from a single GPS satellite at a single dual frequency receiver station. Both code-based TEC derived from the P-observable (Ptec) and phase-based TEC derived from the carrier phase observable (Ltec) are used in the solution. The CGP method addresses the ill-posed inverse problem of determining the electron density profiles from TEC measurements through the application of a side constraint to the acceptable solution. This is an iterative method which approximates the solution of a least squares problem through a converging sequence of solutions. The accuracy of the results is verified by comparison to electron density determined from the ionograms measured with Digisondes (Pushkov Institute of Terrestrial Magnetizm, Ionosphere and Radio Wave Propagation, Russian Academy of Science) located at Troizk, Moscow region (55.5N, 37.3E). The results of a hardware-software complex intended for monitoring the Earth's ionosphere according to navigation satellite systems are presented. The anomalous behavior of the critical frequency of the F2-layer ionosphere at latitudes 57-59 degrees observed in December 2014 is detected.