<p>Adjustable Robust Optimization (ARO) is a paradigm for facing uncertainty in a decision problem, in case some recourse actions are allowed after the actual value of all input parameters is revealed. While several approaches have been introduced for the linear case, little is known regarding exact methods for the convex case. In this work, we introduce a new general framework for attacking a wide class of ARO problems involving convex functions in the recourse problem. We first recall a semi-infinite reformulation of the problem and, provided that one can solve a non-convex separation problem, show how to solve it either by a generalized Benders decomposition or by a column-and-constraint generation approach. We show that, for the relevant case where the uncertainty set is a polytope, the separation problem can be reformulated as a convex Mixed-Integer Nonlinear Problem, thus allowing us to derive computationally sound exact methods. Finally, we apply the resulting algorithms to two different applications, namely a nonlinear facility location problem and a nonlinear resource allocation problem, to numerically assess their computational performance.</p>