A Class of Iterative Methods for Determining p-Solutions of Linear Interval Parametric Systems∗

We consider a square linear interval parametric (LIP) system of size n whose elements are affine linear functions of the m-dimensional parameter vector p. Recently, a new type of solution to the LIP system considered (called parameterized or p-solution) has been introduced, which is of a corresponding linear interval (LI) form. An iterative method for determining the linear p-solution has also been suggested. The objective of the present paper is to generalize the above approach in two directions. First, a new type of p-solution in a corresponding quadratic interval (QI) form is suggested. Second, it is shown that any known iterative method for determining an outer solution to the LIP system given can be modified in a unified manner to produce a corresponding method yielding a linear or quadratic p-solution. Thus, a class of iterative methods for determining p-solutions can be constructed, depending on the iterative scheme chosen and the form, linear or quadratic, of the solution sought. As an illustration, two specific methods for determining a p-solution, based on a simple iterative process and respective LI and QI forms, are suggested. The proposed p-solutions seem to be useful in solving global optimization problems where the constraint is given as a LIP system. As an example, a parametric linear programming problem is considered.