The authors study a new algorithm for the solution of smooth inequality constrained minimization problems for which global Lipschitz constants with respect to the Euclidean norm can be determined for the gradients of the objective and each constraint function. The algorithm is a feasible point algorithm which uses first order data information only and which, in each iteration, requires the computation of the Euclidean projection of a vector onto the intersection of balls approximating the feasible set. Theoretical and computational properties of this “moving balls approximation algorithm” (MBA) are provided for its primal and its dual form, and convergence and global rate of convergence results are established for nonconvex as well as convex problems. Moreover, an extension of MBA is developed for a class of variational inequalities, and a variant of MBA with an active set strategy for the original problem is presented which possesses the same convergence properties as MBA, is particularly suited for problems with a large number of constraints, and turns out to be significantly faster than MBA. The efficiency of the proposed methods is demonstrated by numerical experiments on some generated convex quadratically constrained quadratic problems, where the methods are compared with state-of-the-art optimization methods and software respectively, as a SQP solver from the IMSL library and the CVXOPT package for convex programming.