  Mathematical programs with vanishing constraints (MPVCs) are a class of
nonlinear optimization problems with applications to various engineering
problems such as truss topology design and robot motion planning. MPVCs are
difficult problems from both a theoretical and numerical perspective: the
combinatorial nature of the vanishing constraints often prevents standard
constraint qualifications and optimality conditions from being attained;
moreover, the feasible set is inherently nonconvex, and often has no interior
around points of interest. In this paper, we therefore study and compare four
regularization methods for the numerical solution of MPVCS. Each method depends
on a single regularization parameter, which is used to embed the original MPVC
into a sequence of standard nonlinear programs. Convergence results for these
methods based on both exact and approximate stationary of the subproblems are
established under weak assumptions. The improved regularity of the subproblems
is studied by providing sufficient conditions for the existence of KKT
multipliers. Numerical experiments, based on applications in truss topology
design and an optimal control problem from aerothermodynamics, complement the
theoretical analysis and comparison of the regularization methods. The
computational results highlight the benefit of using regularization over
applying a standard solver directly, and they allow us to identify two
promising regularization schemes.
