<p>Recent years have witnessed a growing adoption of the inertial-relaxed mechanism to accelerate the convergence of iterative schemes. Despite these advances, relatively few studies have investigated the Liu-Storey (LS) and Conjugate-Descent (CD) conjugate gradient methods when combined with the inertial-relaxed mechanism and convex combination technique. To address this gap, we propose an inertial-relaxed hybrid LS-CD-type conjugate gradient projection (CGP) algorithm, where an inertial extrapolation step is incorporated into the algorithmic structure. The resulting search direction preserves sufficient descent and trust-region characteristics, while the conjugate parameter is determined as a convex combination of the LS and CD methods with the convex coefficient chosen to satisfy the conjugacy condition. We establish the global convergence of the proposed algorithm under mild hypotheses, without requiring Lipschitz continuity condition. Numerical experiments on constrained nonlinear equations and sparse signal recovery problems demonstrate that the proposed algorithm is efficient and competitive in comparison to the existing methods.</p>