Solution sensitivity from general principles

We present a generic approach for the sensitivity analysis of solutions to parameterized finite-dimensional optimization problems. We study differenti ability and continuity properties of quasi-solutions ( stationary points or stationary point-multiplier pairs), as well as their existence and uniquen ess, and the issue of when quasi-solutions are actually optimal solutions. Our approach is founded on a few general rules that can all be viewed as ge neralizations of the classical inverse mapping theorem, and sensitivity ana lyses of particular optimization models can be made by computing certain ge neralized derivatives in order to translate the general rules into the term inology of the particular model. The useful application of this approach hi nges on an inverse mapping theorem that allows us to compute derivatives of solution mappings without computing solutions, which is crucial since nume rical solutions to sensitive problems are fundamentally unreliable. We illu strate how this process works for parameterized nonlinear programs, but the generality of the rules on which our approach is based means that a simila r sensitivity analysis is possible for practically any finite-dimensional o ptimization problem. Our approach is distinguished not only by its broad ap plicability but by its separate treatment of different issues that are freq uently treated in tandem. In particular, meaningful generalized derivatives can be computed and continuity properties can be established even in cases of multiple or no quasi-solutions ( or optimal solutions) for some paramet ers. This approach has not only produced unprecedented and computable condi tions for traditional properties in well-studied situations, but has also c haracterized interesting new properties that might otherwise have remained unexplored.