<p>This paper proposes a depth-first directional search (DFDS) algorithm for global optimization of nonconvex problems. DFDS performs a full-depth stepping line search along each sampled direction before proceeding to the next, contrasting with existing directional search methods that prioritize broad exploratory coverage. We establish the convergence and computational complexity of DFDS through a novel geometric framework that models the success probability of finding a global optimizer as the surface area of a spherical cap. Numerical experiments on benchmark problems demonstrate that DFDS achieves a higher success rate and exhibits greater stability in locating the global optimal region compared to other random search methods under a uniform function evaluation budget, with its performance advantage growing more distinct as the problem dimension increases.</p>