We perform the first tight convergence analysis of the gradient method with fixed step sizes applied to the class of smooth hypoconvex (weakly-convex) functions, i.e., smooth nonconvex functions whose curvature belongs to the interval [µ, L] with µ < 0. The results fill the gap between convergence rates for smooth nonconvex and smooth convex functions.The convergence rates were identified using the performance estimation framework adapted to hypoconvex functions. We provide mathematical proofs for a large range of step sizes and prove that the convergence rates are tight when all step sizes are smaller or equal to 1/L. Finally, we identify the optimal constant step size that minimizes the worst-case of the gradient method applied to hypoconvex functions. P( f, x0 ) subject to f ∈ F Optimality conditions on x * Initial conditions on x 0 (1.1)