Generalized solutions in optimization

An extension of optimization problems on non-compact domains is constructed by means of methods of the uniform and proximity space theory, the topology and even the uniformity of the original domains being preserved. The solution of the extended problems is considered as a generalized solution of the original problem. The existence and stability (which generally do not take place for the classical solutions) are ensured for the generalized solutions of various abstract optimization problems. Moreover, the convergence (in the sense of the extended spaces) of approximate classical solutions of perturbed or penalized problems to the generalized solutions is shown.