AE solutions and AE solvability to general interval linear systems

Abstract

We consider linear systems of equations and inequalities with coefficients varying inside given intervals. We define their solutions (so called AE solutions) and solvability (so called AE solvability) by using forall-exists quantification of interval parameters. We present an explicit description of the AE solutions, and discuss complexity issues as well. For AE solvability, we propose a sufficient condition only, but for a specific sub-class of problems, a complete characterization is developed. Moreover, we investigate inequality systems for which AE solvability is equivalent to existence of an AE solution.

Necessary and sufficient conditions for unified optimality of interval linear program in the general form

“…It is well known that x * is (α 1 φ 1 , · · · , α 6 φ 6 , c 1 φ 7 , c 2 φ 8 )-optimal if and only if x * is (α 1 φ 1 , · · · , α 6 φ 6 )-feasible (feasibility) and (4) (optimality) should be satisfied with the given forall-exists quantifiers. For efficient methods of checking feasibility of various linear interval systems, we refer the readers to [5,10,11,[16][17][18][19][20][21]. In the last section, we analyse two different types of feasible solutions with mixed constraints, which have not yet been studied, and obtain the full characterization of such special cases of UO-optimality.…”

“…Two different kinds of feasibility corresponding to the interval linear programming problems discussed in Section 4 have already been studied [11,21]. In this section, we analyse some different types of feasible solutions which have not yet been studied and obtain the full characterization of some special cases of UO-optimality.…”

“…Hladík [11] utilized the concept of AE solutions for interval inequalities. Recently, the concepts of quantified optimal solutions to IvLP were introduced in [12][13][14][15], and some necessary and sufficient conditions were proposed for checking such new optimality.…”

AE Regularity of Interval Matrices

“…Obviously, if the interval system has an AE solution, then it is AE solvable, but the converse implication does not hold in general [5].…”

The best, the worst and the semi-strong: optimal values in interval linear programming

“…However, besides the traditional solution concept of weak feasibility used in interval linear systems, other concepts that were more suited towards particular applications have been proposed. These concepts include tolerance solutions, control solutions [20] or AE solutions [8], which were later unified and extended under the notion of generalized feasibility [12,22]. Recently, a similar concept was also introduced to interval optimization in the form of semi-strong optimality [9,13,14], which serves as a generalization of weak and strong optimality.…”