Well-Posedness of Optimization Problems. On the Stability of the Set of Saddle Points of the Lagrangian

Abstract

Questions of existence of solutions and how they depend on a problem’s parameters are usually important for many problems of mathematics, not only in optimization. The term well-posedness refers to the existence and uniqueness of a solution and its continuous behavior with respect to data perturbations, which is referred to as stability. In general, a problem is said to be stable if

$$\varepsilon (\delta ) \to 0\;when\;\delta \to 0,$$

where δ is a given tolerance of the problem’s data, ε(δ) is the accuracy with which the solution can be determined, and ε(δ) is a continuous function of δ. Besides these conditions, accompanying robustness properties in the convergence of sequence of approximate solutions are also required.