The present invention relates to methods and systems for generating solutions to global optimization problems. In one aspect, the invention allows for determining whether models of optimization problems are infeasible. In another aspect, the invention allows for identifying relatively small sets of constraints that make a model infeasible.In yet another aspect, the invention provides methods and systems for creating one or more artificial infeasibilities in a model so as to seek improvement upon a known value of the objective function. Creation of artificial constraints in this manner may also permit identification of a relatively small set of constraints that may be limiting the value of the objective function.