In this paper, for the global optimization problem , to minimize the proper extended real-valued function over the given subset of a normed linear space equipped with the strong convergence, well-posedness criteria are derived. The given problem is embedded into a smoothly parametrized family of minimization problems, where is a parameter belonging to a given Banach space , and is the parameter value to which the given unperturbed problem corresponds, i.e., . Defining the value function the author gives the following definition of well-posedness.
is well-posed with respect to the embedding iff , , and there exists a unique and for every sequence and every sequence such that as we have in .
This definition is stronger than the Tikhonov well-posedness. In the following, the defined well-posedness is related under suitable conditions to the differentiability properties of at . These abstract results are applied to one-dimensional problems of the calculus of variations.