Lipschitz continuity of inf-projections
- Computational Optimization and Applications
- 25 (2003), 269-282
Content:
Let's consider the following family of perturbed mathematical program:
- { Min f(u,x), x in X | u in U }.
The bivariate functions `f' is extended real-valued (so as to allow for
constraints). The question raised is this article is: under which condition
is the inf-projection `p' of `f' (locally) Lipschitz continuous? Here
- p(u) = inf { f(u,x) | x in X };
and I restrict myself to the case when X and U are finite dimensional.
The answer provided is based on a continuity condition for the
function-valued mapping u |--> f(u,.). I show that if this function-valued
mapping is (locally) epi-sub-Lipschitz continuous and level bounded (to
guarantee the existence of solutions) then `p' is (locally) Lipschitz
continuous (on its effective domain).
A set-valued mapping is (locally) epi-sub-Lipschitz continuous if its
epigraphical mapping u |--> epi f(u,.) is sub-Lipschitz [see "Variational
Analysis" (R.T. Rockafellar & R.J-B Wets, Springer, 1998), Definition
9.27].
The last section considers the special case
- f(u,x) = g(u,x) if Ax = b-u, x in D
where D is a polyhedral set, i.e., the case of a nonlinear program with
linear constraints. The relationship with earlier results is brought to
the fore as well as the limitations of those results.
September 1, 1999