Complementarity eigenvalue problems for nonlinear matrix pencils

This work deals with a class of nonlinear complementarity eigenvalue problems that, from a mathematical point of view, can be written as an equilibrium model [A(λ)B(λ)C(λ)D(λ)][uw]=[v0],u≥0,v≥0,uTv=0,where the vectors u and v are subject to complementarity constraints. The block structured matrix appearing in this partially constrained equilibrium model depends continuously on a real scalar λ ∈ Λ. Such a scalar plays the role of a non-dimensional load parameter, but it may have also other physical meanings. The symbol Λ stands for a given bounded interval, possibly non-closed. The numerical problem at hand is to find all the values of λ (and, in particular, the smallest one) for which the above equilibrium model admits a nontrivial solution. By using the so-called Facial Reduction Technique, we solve efficiently such a numerical problem in various randomly generated test examples and in two mechanical examples of unilateral buckling of columns.