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.