Solution of linear equations and inequalities in idempotent vector spaces

Linear vector equations and inequalities are consid- ered defined in terms of idempotent mathematics. To solve the equations, we apply an approach that is based on the analysis of dis- tances between vectors in idempotent vector spaces. The approach reduces the solution of the equation to that of an optimization problem in the idempotent algebra setting. Based on the approach, existence and uniqueness conditions are established for the solution of equations, and a general solution to both linear equations and inequalities are given. Finally, a problem of simultaneous solution of equations and inequalities is also considered. With the approach, existence conditions are established and a procedure to find all solutions of the equation is described in terms of the covering sets. The maximum solution to the equation is given in the form x = A d, where A is a pseudoinverse matrix in the initial idempotent semimodule (called there extremal inverse matrix), and denotes matrix-vector multiplication in a dual semimodule. In (10), (11), the above approach is extended to investigate linear dependence in idempotent semimodules. The development of the theory and methods in (3) is aimed in particular at the solution of equations when the matrix A may have zero entries. The operation of pseu- doinversion is extended to such matrices (the matrix A is called conjugate to A). For the solution, existence conditions in the form of an equality A(A d) = d, where is the multiplication in a dual semimodule, are given and unique- ness conditions are established. A procedure is proposed to determine the linear dependence between vectors. The results are further developed in (9) to offer a combinatorial and an algebraic techniques for the solution of equations. In (4), (12), (13), a notion of a subsolution to the equation is introduced as any vector x that satisfies the condition Ax d. A residuation operation n is defined so that And represents the maximal subsolution of the equation. It is shown that when an ordinary solution exists, it can be written in terms of a dual semimodule and then And = A d. For an extended equation Ax b = d, where denotes idempotent vector addition, a necessary and sufficient condi- tion for the existence of its subsolutions is given in (4), (13) in the form of an inequality b d that however suggests only necessary conditions for the actual solution. Another approach that is based on the application of an idempotent analogue for the matrix determinant, known as dominant, is proposed in (14). A solution technique is developed which uses Cramer's rule with the dominant in place of determinant. The implementation of the approach requires, however, that some sufficient constraints for both the matrix A and the vector d to satisfy. In this paper another solution approach is described which uses the analysis of distances between vectors in idempotent vector spaces. As a metric, we take a distance function that involves only main binary operations of the semimodule supplemented with the operation of pseudoinversion. This allows to represent subsequent results in a compact vector form only in terms of the initial semimodule and give them clear and natural geometrical interpretation in the plane with the Cartesian coordinates. The results presented are based