A modified trust region algorithm by diagonally approximating the Hessian and hybridizing the trust region ratios in a fuzzy framework with application in image processing

Purpose Nowadays, fuzzy modeling has been technically considered as a meaningful way to address uncertainty in a wide range of applications. Hence, it can be used to target a classic uncertainty related to the best choice of the trust region (TR) ratio, recognized as an influential factor that highly affects the computational performance of the TR algorithms. Design/methodology/approach Determined according to the quality of the approximate local model of the given unconstrained minimization problem, researchers have justified the pros and cons of their choices for the TR ratio. By the way, an optimal choice for the TR ratio is still a matter of further study. Here, we plan to simultaneously benefit from the advantages of several effective choices of the TR ratio by first employing them to form a fuzzy number and then generating several quantifiable results as hybrid formulas for the TR ratio by applying the fuzzy ranking functions. Moreover, we suggest a local diagonal approximation for the Hessian to be used in the TR subproblem and make it possible to handle high-dimensional optimization models. Findings We analytically assess the effect of our modifications on the global convergence of an adaptive nonmonotone TR algorithm. We also conduct some computational experiments to provide further justification for our theoretical improvements. The results emphasize that the fuzzy concepts can helpfully make progress in the performance of the TR algorithms. Originality/value We propose a two-part method to enhance TR algorithms. First, we combine multiple update strategies for the TR radius using fuzzy logic to create a robust hybrid rule. Second, to ensure scalability, we approximate the Hessian matrix with a simple diagonal form, making the approach efficient for high-dimensional problems.