A Three‐Term Modified PRP Conjugate Gradient Method for Unconstrained Optimization: Application to Image Restoration

ABSTRACT The PRP method is widely recognized as one of the most classical conjugate gradient methods, yet its convergence is generally difficult to guarantee under conventional weak assumptions, making its improvement a long‐standing open research problem. To improve the convergence property of the PRP method, this paper introduces a convex combination term into the denominator of the PRP conjugate parameter and incorporates a fluctuation factor into its numerator. Building on these two modifications, we propose a modified three‐term PRP conjugate gradient method. Its search direction satisfies the sufficient descent condition independently of any line search. Under mild assumptions, we establish the global convergence of the proposed method. Moreover, numerical comparisons against several representative existing methods demonstrate that the proposed method achieves superior performance in solving both general unconstrained optimization problems and image restoration problems.