  We analyze worst-case convergence guarantees of first-order optimization
methods over a function class extending that of smooth and convex functions.
This class contains convex functions that admit a simple quadratic upper bound.
Its study is motivated by its stability under minor perturbations. We provide a
thorough analysis of first-order methods, including worst-case convergence
guarantees for several algorithms, and demonstrate that some of them achieve
the optimal worst-case guarantee over the class. We support our analysis by
numerical validation of worst-case guarantees using performance estimation
problems. A few observations can be drawn from this analysis, particularly
regarding the optimality (resp. and adaptivity) of the heavy-ball method (resp.
heavy-ball with line-search). Finally, we show how our analysis can be
leveraged to obtain convergence guarantees over more complex classes of
functions. Overall, this study brings insights on the choice of function
classes over which standard first-order methods have working worst-case
guarantees.
