Some Hybrid Conjugate Gradient Methods Based on Barzilai-Borwein Approach for Solving Two-Dimensional Unconstrained Optimization Problems

Rahpeymaii, Farzad; Rostami, Majid

doi:10.30473/coam.2025.74997.1317

Articles in Press

Document Type : Research Article

Authors

Department of Basic Sciences‎, ‎Technical and Vocational University (TVU)‎, ‎Tehran‎, ‎Iran‎.

10.30473/coam.2025.74997.1317

Abstract

The conjugate gradient ({CG}) method is one of the simplest and most widely used approaches for unconstrained optimization, and our focus is on two-dimensional problems with numerous practical applications. We devise three hybrid {CG} methods in which the hybrid parameter is constructed from the Barzilai–Borwein process, and in these hybrids, the weaknesses of each constituent method are mitigated by the strengths of the others. The conjugate gradient parameter is formed as a linear combination of two well-known CG parameters, blended by a scalar, enabling our new methods to solve the targeted problems efficiently. Under mild assumptions, we establish the descent property of the generated directions and prove the global convergence of the hybrid schemes. Numerical experiments on ten practical examples indicate that the proposed hybrid {CG} methods outperform standard {CG} methods for two-dimensional unconstrained optimization.

Highlights

Three hybrid conjugate gradient (CG) methods for two-dimensional unconstrained optimization.

Each hybrid method constructs the CG parameter as a linear combination of two classical CG parameters, blended via a scalar, with the Barzilai–Borwein (BB) process guiding the hybridization.
The hybrids are designed to overcome the limitations of individual CG methods by pairing their complementary advantages.
Established descent property for the generated search directions and proven global convergence under mild assumptions.
Numerical experiments on ten practical 2D problems demonstrate that the hybrid CG methods outperform standard CG in terms of iterations, function evaluations, and CPU time.
Demonstrates that combining CG strategies in pairwise configurations can consistently enhance iterative performance for unconstrained optimization.

Keywords

Main Subjects

Optimization

References

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Some Hybrid Conjugate Gradient Methods Based on Barzilai-Borwein Approach for Solving Two-Dimensional Unconstrained Optimization Problems

References

References

Articles in Press, Accepted Manuscript Available Online from 16 October 2025

Articles in Press, Accepted Manuscript
Available Online from 16 October 2025