Well-posedness and optimization under perturbations.

*(English)*Zbl 0996.90081From the introduction: We consider global optimization problems subject to parametric perturbations of a general nature in an abstract setting. The decision variable and the parameter take values in Banach spaces. The parameter models either perturbations acting on the original problem, or approximations of it used to simplify the solution process. Also deterministic uncertainty acting on problem’s data can be modeled this way.

We assume that the original problem is well-posed in the sense of Tikhonov or under perturbations. We derive estimates about the size of the set of approximate solutions of the original problem (ignoring modifications of it) based on its well-posedness features. Then, by taking into account the effect of perturbations, we estimate the size of the approximate solutions to nearby problems having fixed the maximum amount of perturbations acting on the original problem. These estimates are metric in nature, making use of the modulus of well-posedness of the given problem. Moreover we estimate the size of those smooth perturbations acting on the original problem which guarantee that its approximate solutions (of a prescribed level) can be used to solve approximately all nearby problems. To this aim we exploit the differentiability properties of the optimal value function, with respect to the parameter, which follow from well-posedness under perturbations.

Such estimates can be of interest in at least two cases. In the first, the original problem is approximately solvable and its solutions can be used to obtain suboptimal solutions for some perturbed problems. In the second case, the original problem is difficult to be solved even approximatively, while suboptimal solutions of the perturbed problem are available, to be used as approximate solutions to the original one.

By this approach we prove a well-posedness criterion making use of a particular set of perturbed suboptimal solutions.

We assume that the original problem is well-posed in the sense of Tikhonov or under perturbations. We derive estimates about the size of the set of approximate solutions of the original problem (ignoring modifications of it) based on its well-posedness features. Then, by taking into account the effect of perturbations, we estimate the size of the approximate solutions to nearby problems having fixed the maximum amount of perturbations acting on the original problem. These estimates are metric in nature, making use of the modulus of well-posedness of the given problem. Moreover we estimate the size of those smooth perturbations acting on the original problem which guarantee that its approximate solutions (of a prescribed level) can be used to solve approximately all nearby problems. To this aim we exploit the differentiability properties of the optimal value function, with respect to the parameter, which follow from well-posedness under perturbations.

Such estimates can be of interest in at least two cases. In the first, the original problem is approximately solvable and its solutions can be used to obtain suboptimal solutions for some perturbed problems. In the second case, the original problem is difficult to be solved even approximatively, while suboptimal solutions of the perturbed problem are available, to be used as approximate solutions to the original one.

By this approach we prove a well-posedness criterion making use of a particular set of perturbed suboptimal solutions.