Abstract
Abstract We define an irreducible inconsistent system (IIS) such that every one of its proper subsystems is consistent. This study investigates the irreducibly inconsistent structures of all types of linear systems including equalities, inequalities and/or strict inequalities. By extending Loon's method, a quasi-simplex method is proposed to find the necessary and sufficient conditions on IIS's of these linear systems. Bused on these conditions, the criteria of IIS's are drawn and their relationship with IIS's of different linear systems can be surmised. Furthermore, the necessary and sufficient conditions of the inconsistency and. therefore, the criteria of an irreducibly inconsistent subsystem (IISS) of these linear systems are also proposed. In addition to the theoretical proofs, numerical examples are provided. During the study, in order to reduce the computation efforts and memory spaces, particular attention is paid to the natural constraints where the variables are restricted on the positive orthant. It is expected that this study may provide an access to the resolution of an inconsistent linear system.
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