Vertical Linear Complementarity Research Articles

Equivalence of vertical block linear complementarity problems and multi-objective programming problems

In this paper, Multi-objective Programming (MOP) approaches are proposed to deal with the feasible Vertical Block Linear Complementarity Problems (VLCP) and to look into the solvability and unsolvability of the VLCP. The characterization of an unsolvable VLCP is obtained via the existence of nonzero efficient point of the MOP problem. Also a perturbed problem is proposed if the VLCP is unsolvable for small disturbances in data. This perturbed problem is useful to construct the solvable VLCP corresponding to the original unsolvable VLCP. Examples are given to demonstrate the effectiveness of the results.

Solution of parametric vertical block linear complementarity problems

The parametric vertical block linear complementarity problem is introduced where the associated vertical block matrix and the column vector are parametric. Assuming the matrix to be a vertical block P-matrix, an algorithm is developed for solving vertical block linear complementarity problems. A necessary and sufficient condition is also derived to ensure the same solution basis for any parametric input of the vertical block P-matrix. Based on these, an algorithm is proposed and the solution of the parametric vertical block linear complementarity problem is obtained. A numerical example is presented to illustrate the use of the method.

Equivalence of the Generalized Vertical Block Linear Complementarity Problems and Linear Complementarity Problems

In this paper, generalization of a vertical block linear complementarity problem associated with two different types of matrices, one of which is a square matrix and the other is a vertical block matrix, is proposed. The necessary and sufficient conditions for the existence of the solution of the generalized vertical block linear complementarity problem is derived and the relationship between the solution set of the generalized vertical block linear complementarity problem and the linear complementarity problem is established. It is proved that the generalized vertical block linear complementarity problem has the P-property if and only if the vertical block linear complementarity problem has the P-property.