Equivalence of Linear Complementarity Problems and Linear Programs in Vector Lattice Hilbert Spaces

Let X be a vector lattice Hilbert space with dual X^ *. Let M be a continuous linear mapping of X onto X^ *. Let p,q ∈ X^ * with $p > 0$. We consider the relationship between the linear complementarity problem: Find x ∈ X such that x 0, Mx + q 0, x,Mx + q = 0, and the linear programming problem: Find x ∈ X which minimizes x,p subject to x 0, Mx + q 0. For the problem of a cavitating journal bearing, which is used as an example, the linear program requires the minimization of a linear functional which is proportional to the load borne by the bearing.