A Spectral RMIL+ Conjugate Gradient Method for Unconstrained Optimization With Applications in Portfolio Selection and Motion Control

Aliyu Muhammed Awwal,Mustafa Mamat,Maulana Malik,Ibrahim Mohammed Sulaiman,Kanokwan Sitthithakerngkiet,Poom Kumam

doi:10.1109/access.2021.3081570

Abstract

The Spectral conjugate gradient (SCG) methods are among the efficient variants of CG algorithms which are obtained by combining the spectral gradient parameter and CG parameter. The success of SCG methods relies on effective choices of the step-size $\alpha _{k}$ and the search direction $d_{k}$ . This paper presents an SCG method for unconstrained optimization models. The search directions generated by the new method possess sufficient descent property without the restart condition and independent of the line search procedure used. The global convergence of the new method is proved under the weak Wolfe line search. Preliminary numerical results are presented which show that the method is efficient and promising, particularly for large-scale problems. Also, the method was applied to solve the robotic motion control problem and portfolio selection problem.

Highlights

In this paper, we consider the optimization model: min f (x), x ∈ Rn. (1)The function f : Rn → R has continuous partial derivatives, whose gradient ∇f (x) = g(x) is available
Remark 1: From the analysis above, we have shown that the proposed sRMIL+ satisfies the descent condition (16) regardless of the line search
PRELIMINARY RESULTS This section presents the numerical results on 130 benchmark test problems to illustrate the efficiency of the proposed sRMIL+ method and compare the performance with the RMIL+ method [21], PRP method [6], [7], spectral FR (sFR) method [40], and sPRP method [33]

Summary

INTRODUCTION

We consider the optimization model: min f (x), x ∈ Rn. The function f : Rn → R has continuous partial derivatives, whose gradient ∇f (x) = g(x) is available. Where the authors replaced gk−1 2 with dk−1 2 in the denominator of classical PRP CG parameter and showed that the method satisfies (8) and further proved its global convergence under exact minimization condition. The convergence analysis was discussed under a modified Armijo line search and results obtained from numerical computations showed that sFR method is efficient and promising compare to PRP method. Liu et al [33] extended the work of Birgin and Martinez, [24] and Zhang et al, [32] to proposed a general spectral parameter that will reduce to the main CG algorithm if an exact line search is used.

NEW ALGORITHM

PRELIMINARY RESULTS

APPLICATION IN PORTFOLIO SELECTION

CONCLUSION