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‎In this paper‎, a nonmonotone line search strategy is presented for minimization of the locally Lipschitz continuous function‎. ‎First‎, the Armijo condition is generalized along a descent direction at ‎the ‎current point. Then, a step length is selected along a descent direction satisfying the generalized Armijo condition. We show that there exists at least one step length satisfying the generalized Armijo condition. Next, the nonmonotone line search algorithm is proposed and its global convergence is proved. ‎Finally‎, ‎the proposed algorithm is implemented in ‎the‎ MATLAB environment and compared with some methods in‎ the subject literature. ‎It can be seen that the proposed method not only computes the global optimum also reduces the number of function evaluations than the monotone line search ‎method.‎
 

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The conjugate gradient method (CGM) stands out as one of the most rapidly growing and effective approaches for addressing unconstrained optimization problems. In recent years, significant efforts have been dedicated to adapting the CGM for tackling nonlinear optimization challenges. This research article introduces a new modification of the Fletcher–Reeves (FR) conjugate gradient projection method. The proposed method is characterized by its sufficient descent property, and its global convergence has been established under specific assumptions. Numerical experiments conducted on a range of functions from the CUTEr collection demonstrate the potential and effectiveness of the proposed methods.