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This paper proposes a modified scaled spectral-conjugate-based algorithm for finding solutions to monotone operator equations. The algorithm is a modification of the work of Li and Zheng in the sense that the uniformly monotone assumption on the operator is relaxed to just monotone. Furthermore, unlike the work of Li and Zheng, the search directions of the proposed algorithm are shown to be descent and bounded independent of the monotonicity assumption. Moreover, the global convergence is established under some appropriate assumptions. Finally, numerical examples on some test problems are provided to show the efficiency of the proposed algorithm compared to that of Li and Zheng.

We desire in this work to propose an algorithm to solve the problem:

Solving problems of form (

Some notable methods for finding solution to (1) are: Newton’s method, quasi-Newton method, Gauss–Newton method, Levenberg–Marquardt method, and their variants [

In this work, motivated by the strong condition imposed on the operator by Li and Zheng [

Notations: unless or otherwise stated, the symbol

In this section, we will begin by recalling a three-term spectral-conjugate gradient method for solving (

Based on the three-term direction above, we will propose a modified scaled three-term derivative-free algorithms for solving (

STDF1:

STDF2:

To obtain a lower bound for the term

PSTDF1:

PSTDF2:

From (

Let

The constraint set

The operator

The operator

In the following algorithm, we generate approximate solutions to problem (

PSTDF.

where

In this section, we will establish the convergence analysis of the proposed algorithm. However, we require the following important lemmas. The following lemma shows that the proposed directions are descent.

The search directions defined by (

Multiplying both sides of (

Also, multiplying both sides of (

Hence, for all

The lemma below shows that the linesearch (

Suppose Assumptions

For all

Suppose Assumptions

From Lemma

Since

All are now set to establish the convergence of the proposed algorithm.

Suppose Assumptions

Furthermore, the sequence

Suppose that

By (

To complete the proof of the theorem, we need to show that the search direction

For

Now for

Similarly, from (

Letting

Multiplying (

This contradicts (

Because

This segment of the paper would demonstrate the computational efficiency of the PSTDF algorithm relative to STDF algorithm [

Table

List of test problems with references.

S/N | Problem and reference |
---|---|

1 | Modified exponential function 2 [ |

2 | Logarithmic function [ |

3 | Nonsmooth function [ |

4 | Strictly convex function I [ |

5 | Tridiagonal exponential function [ |

6 | Nonsmooth function [ |

7 | Problem 4 in [ |

8 | Problem 9 in [ |

The result of the experiments in Tabular form can be found in the link

Performance profiles for the number of iterations (NOI).

Performance profiles for the number of function evaluations (NFE).

Performance profiles for the CPU time (in seconds).

In this paper, a modified scaled algorithm based on the spectral-conjugate gradient method for solving nonlinear monotone operator equations was proposed. The algorithm replaces the stronger assumption of uniformly monotone on the operator in the work of Li and Zheng (2020) with just monotone, which is weaker. Interestingly, the search directions were shown to be descent independent of line search and also without monotonicity assumption (unlike in the work of Li and Zheng). Furthermore, the convergence results were established under monotonicity and Lipschitz continuity assumptions on the operator. Numerical experiments on some benchmark problems were conducted to illustrate the good performance of the proposed algorithm.

No data were used to support this study.

The authors declare that they have no conflicts of interest.

All authors contributed equally to the manuscript and read and approved the final manuscript.

The first, fifth, and the sixth authors acknowledge with thanks, the Department of Mathematics and Applied Mathematics at the Sefako Makgatho Health Sciences University. The second author was financially supported by the Rajamangala University of Technology Phra Nakhon (RMUTP) Research Scholarship.