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All Journal International Journal of Industrial Optimization (IJIO)
Odekunle, Mathew Remilekun
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Decision Sciences, Operations Research & Management Engineering Industrial & Manufacturing Engineering

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A Dai-Liao Hybrid Hestenes-Stiefel and Fletcher-Revees Methods for Unconstrained Optimization Salihu, Nasiru; Odekunle, Mathew Remilekun; Saleh, Also Mohammed; Salihu, Suraj
International Journal of Industrial Optimization Vol 2, No 1 (2021)
Publisher : Universitas Ahmad Dahlan

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.12928/ijio.v2i1.3054

Abstract

Some problems have no analytical solution or too difficult to solve by scientists, engineers, and mathematicians, so the development of numerical methods to obtain approximate solutions became necessary. Gradient methods are more efficient when the function to be minimized continuously in its first derivative. Therefore, this article presents a new hybrid Conjugate Gradient (CG) method to solve unconstrained optimization problems. The method requires the first-order derivatives but overcomes the steepest descent method’s shortcoming of slow convergence and needs not to save or compute the second-order derivatives needed by the Newton method. The CG update parameter is suggested from the Dai-Liao conjugacy condition as a convex combination of Hestenes-Stiefel and Fletcher-Revees algorithms by employing an optimal modulating choice parameterto avoid matrix storage. Numerical computation adopts an inexact line search to obtain the step-size that generates a decent property, showing that the algorithm is robust and efficient. The scheme converges globally under Wolfe line search, and it’s like is suitable in compressive sensing problems and M-tensor systems.
A dai-liao hybrid conjugate gradient method for unconstrained optimization Salihu, Nasiru; Odekunle, Mathew Remilekun; Waziri, Mohammed Yusuf; Halilu, Abubakar Sani; Salihu, Suraj
International Journal of Industrial Optimization Vol 2, No 2 (2021)
Publisher : Universitas Ahmad Dahlan

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.12928/ijio.v2i2.4100

Abstract

One of todays’ best-performing CG methods is Dai-Liao (DL) method which depends on non-negative parameter  and conjugacy conditions for its computation. Although numerous optimal selections for the parameter were suggested, the best choice of  remains a subject of consideration. The pure conjugacy condition adopts an exact line search for numerical experiments and convergence analysis. Though, a practical mathematical experiment implies using an inexact line search to find the step size. To avoid such drawbacks, Dai and Liao substituted the earlier conjugacy condition with an extended conjugacy condition. Therefore, this paper suggests a new hybrid CG that combines the strength of Liu and Storey and Conjugate Descent CG methods by retaining a choice of Dai-Liao parameterthat is optimal. The theoretical analysis indicated that the search direction of the new CG scheme is descent and satisfies sufficient descent condition when the iterates jam under strong Wolfe line search. The algorithm is shown to converge globally using standard assumptions. The numerical experimentation of the scheme demonstrated that the proposed method is robust and promising than some known methods applying the performance profile Dolan and Mor´e on 250 unrestricted problems.  Numerical assessment of the tested CG algorithms with sparse signal reconstruction and image restoration in compressive sensing problems, file restoration, image video coding and other applications. The result shows that these CG schemes are comparable and can be applied in different fields such as temperature, fire, seismic sensors, and humidity detectors in forests, using wireless sensor network techniques.
Co-Authors Halilu, Abubakar Sani Saleh, Also Mohammed Salihu, Nasiru Salihu, Suraj Waziri, Mohammed Yusuf