Preprint A592/2008
MATHEMATICAL PROGRAMS WITH VANISHING CONSTRAINTS: OPTIMALITY CONDITIONS, SENSITIVITY, AND A RELAXATION METHOD

Mikhail Solodov | Izmailov, Alexey

**Keywords: **

We consider a class of optimization problems with
switch-off/switch-on constraints,
which is a relatively new problem model.
The specificity of this model is that it
contains constraints that are being imposed (switched on)
at some points of the feasible region, while being
disregarded (switched off) at other points.
This seems to be a potentially useful modeling paradigm,
that has been shown to be helpful, for example, in
optimal topology design. The fact that
some constraints ``vanish'' from the problem at certain
points, gave rise to the name of
mathematical programs with vanishing constraints (MPVC).
It turns out that such problems are usually degenerate at a solution,
but are structurally different from the related class of
mathematical programs with complementarity constraints (MPCC).
In this paper, we first discuss some
known first- and second-order necessary optimality conditions
for MPVC,
giving new very short and direct justifications. We then derive some
new special second-order sufficient optimality conditions for
these problems and show that, quite remarkably, these conditions are actually
equivalent to the classical/standard second-order sufficient
conditions in optimization.
We also provide a sensitivity analysis
for MPVC. Finally, a relaxation
method is proposed. For this method, we analyze
constraints regularity and boundedness
of the Lagrange multipliers in the relaxed subproblems, derive a
sufficient condition for local uniqueness of solutions of
subproblems, and give some convergence estimates.