To guarantee the convergence of the conjugate gradient method, as discussed in many existing studies, the parameter \(\beta _k\) needs to meet the following requirement: \(| {{\beta _k}}| \le {{g_k^T{d_k}} \big {/} {g_{k - 1}^T{d_{k - 1}}}},\forall k\ge 0\) . During the iterative process of the algorithm, if \( {{g_k^T{d_k}} \big {/} {g_{k - 1}^T{d_{k - 1}}}}\rightarrow 0\) , then \(\beta _k\rightarrow 0\) and the search direction will follow the negative gradient direction, which affects the computational efficiency of the algorithm. In fact, for some classical methods, such as, PRP, HS and LS methods, the following condition may automatically hold when \(k\rightarrow \infty \) : \(| {{\beta _k}}| \le {{g_k^T{d_k}} \big {/} {g_{k - 1}^T{d_{k - 1}}}}\) . As a result, we designed an efficient algorithm and proved its convergence when \(\beta _k\) satisfies the condition \(| {{\beta _k}}| \le {{g_k^T{d_k}} \big {/} {g_{k - 1}^T{d_{k - 1}}}}, k\rightarrow \infty \) . Based on the Gilbert-Nocedal(GN) method, we propose a spectral conjugate gradient method for solving unconstrained optimization problems. This method possesses sufficient descent property and restart feature. Under some assumptions, the global convergence of this method is established. Numerical results show the effectiveness of the proposed method by comparing with some existing methods.
