An Accelerated Conjugate Gradient Algorithm for Solving Nonlinear Monotone Equations and Image Restoration Problems

Abstract

Combining the three-term conjugate gradient method of Yuan and Zhang and the acceleration step length of Andrei with the hyperplane projection method of Solodov and Svaiter, we propose an accelerated conjugate gradient algorithm for solving nonlinear monotone equations in this paper. The presented algorithm has the following properties: (i) All search directions generated by the algorithm satisfy the sufficient descent and trust region properties independent of the line search technique. (ii) A derivative-free search technique is proposed along the direction to obtain the step length αk. (iii) If ϕk=−αkhk−hwkTdk>0, then an acceleration scheme is used to modify the step length in a multiplicative manner and create a point. (iv) If the point satisfies the given condition, then it is the next point; otherwise, the hyperplane projection technique is used to obtain the next point. (v) The global convergence of the proposed algorithm is established under some suitable conditions. Numerical comparisons with other conjugate gradient algorithms show that the accelerated computing scheme is more competitive. In addition, the presented algorithm can also be applied to image restoration.

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

“…If φ k is outside the interval [0, 1], to maintain the convex combination in (11), it can be fxed by…”

“…Although iterative approaches are recognized for their simplicity and good convergence, each iteration of these methods requires a substantial amount of space to compute and store the Jacobian or comparable to the Jacobian, which is not conducive to solving large-scale nonlinear monotone equations. In order to efciently solve problem (1) and avoid solving a linear system of equations at each iteration, the emergence of the derivative-free method is necessary [9][10][11].…”