# Fast algorithm for finding lattice subspaces in R n and its implementation

#### Abstract

There are known necessary and sufficient conditions for a subspace of Rm to be lattice-ordered. Let Y = {y1,…,ym} and yi are rows of the matrix X. A subspace ⟨X⟩, of linear space generated by the set X of n linearly independent positive vectors is lattice-ordered if and only the set X admits a fundamental set of indices I, which means that the subset YI ⊆ Y of vectors indexed by I is linearly independent, and every vector from Y\YI is a nonnegative linear combination of vectors form YI. In economics it is possible to prove that the minimum-cost insured portfolio exists if and only if the linear space generated by the corresponding financial instruments is lattice-ordered. In the literature there are known algorithms with exponential complexity that determine if a given subspace is lattice-ordered. In this thesis a polynomial time algorithm (serial and parallel) as well as its computer implementation will be presented. The method can be applied in economics as well as in the theory of vector lattices.

#### Subject Area

Mathematics

#### Recommended Citation

Pownuk, Andrew M, "Fast algorithm for finding lattice subspaces in R n and its implementation" (2014). *ETD Collection for University of Texas, El Paso*. AAI1583942.

https://scholarworks.utep.edu/dissertations/AAI1583942