We introduce a novel exact approach for addressing a broad spectrum of optimization problems with robust nonlinear constraints. These constraints are defined as sums of products of linear times concave (SLC) functions with respect to the uncertain parameters. Our approach synergizes a cutting set method with reformulation-perspectification techniques and branch and bound. We further extend the applicability of our approach to robust convex optimization, which can be reformulated as a problem involving a sum of linear times linear functions in the uncertain parameters, thus broadening the scope of existing literature. Numerical experiments on a robust convex geometric optimization problem and a robust linear optimization problem with data uncertainty and implementation error show that our approach can solve robust nonlinear problems that cannot be solved by existing methods in the literature. Moreover, a numerical experiment on a lot-sizing problem on a network demonstrates the efficiency of our method for two-stage ARO problems.