Uncertainty is ubiquitous throughout engineering design processes. Robust optimization (RO) aims to find optimal solutions that are relatively insensitive to input uncertainty. In this paper, a new approach is presented for single-objective RO problems with an objective function and constraints that are continuous and differentiable. Both the design variables and parameters with interval uncertainties are represented as affine forms. A mixed interval arithmetic (IA)/affine arithmetic (AA) model is subsequently utilized in order to obtain affine approximations for the objective and feasibility robustness constraint functions. Consequently, the RO problem is converted to a deterministic problem, by bounding all constraints. Finally, nonlinear optimization solvers are applied to obtain a robust optimal solution for the deterministic optimization problem. Some numerical and engineering examples are presented in order to demonstrate the advantages and disadvantages of the proposed approach. The main advantage of the proposed approach lies in the simplicity of the conversion from a nonlinear RO problem with interval uncertainty to a deterministic single-looped optimization problem. Although this approach cannot be applied to problems with black-box models, it requires a minimal use of IA/AA computation and applies some widely used advanced solvers to single-looped optimization problems, making it more suitable for applications in engineering fields.