Equivalence of Linear Complementarity Problems and Linear Programs in Vector Lattice Hilbert Spaces
Colin Cryer, M. A. H. Dempster · SIAM Journal on Control and Optimization · 1980
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 \in X^ * $ with $p > 0$. We consider the relationship between the linear complementarity problem: Find $x \in X$ such that $x \geqq 0$, $Mx + q \geqq 0$, $\langle {x,Mx + q} \rangle = 0$, and the linear programming problem: Find $x \in X$ which minimizes $\langle {x,p} \rangle $ subject to $x \geqq 0$, $Mx + q \geqq 0$. 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.