Solving large scale nonlinear optimization problems requires either significant computing resources or the development of specialized algorithms. In the case of Linear Programming (LP) problems there are decomposition methods that take advantage of problem structure, gradually constructing the full problem by generating variables or constraints. There are no direct adaptations of Column Generation (CG) methods for nonlinear optimization problems. In this work we present a generic CG method for nonlinear optimization problems, such that when optimizing over a structured set X plus a moderate number of complicating constraints , we solve a succession of 1) restricted master problems on a smaller set S ⊂ X and 2) pricing problems that are Lagrangean relaxations with respect to the complicating constraints. The former provides feasible solutions and feeds dual information to the latter. In turn, the pricing problem identifies a variable of interest that is then taken into account into an updated subset S ′ ⊂ X. We show that our approach is valid not only when S is the convex hull of the generated variables as in CG for LPs, but also with a variety of subsets such as the cone hull, the linear span, and a special variable aggregation set. We discuss how the structure of the set we choose and the way it is updated influences the number of iterations required to reach optimality or near-optimality and the difficulty of solving the restricted master problems. We present linearized schemes that alleviate the computational burden of solving the pricing problem. We test our methods on Portfolio Optimization instances with up to 5 million variables including non-linear objective functions and Second Order Cone (SOC) constraints. We show that some CGs with linearized pricing are 2-3 times faster than solving the complete problem directly. Further, for the larger instances the same CG methods are able to provide solutions within 1% of optimality in 6 hours whereas solving the complete problem runs out of memory.