Fuzzy linear systems

Abstract

For a system of linear algebraic equations (SLAE) 𝐴𝑥 = 𝑏 in a finite-dimensional Euclidean
space 𝐸, a constructive description of the manifold of its solutions Φ(𝐴, 𝑏) is obtained using
the Gram-Schmidt orthogonalization. This description consists of an unconditional linear
parameterization.
This circumstance opens up entirely new possibilities for using SLAEs, as it allows one to
theoretically take into account a priori information about the properties of the true solution
𝑥и in its search on the manifold Φ(𝐴, 𝑏). Technically, this looks like this: the expert opinion
on the solution 𝑥и is formalized by a non-negative functional 𝐹 on Φ(𝐴, 𝑏), and the solution
𝑥и minimizes it. Thanks to the linear parameterization of Φ(𝐴, 𝑏), the minimization of 𝐹 is
unconditional.
The paper pays special attention to the case where expert information about the solution 𝑥и is formally represented by a fuzzy structure 𝜇 of coordinate weights in the space 𝐸, expressing their role in the SLAE 𝐴𝑥 = 𝑏. We call the pair (𝐴𝑥 = 𝑏, 𝜇) a fuzzy SLAE. The formation of its solutions Φ(𝐴, 𝑏, 𝜇) ⊆ Φ(𝐴, 𝑏) is associated with nonlinear optimization, for which polynomial descent algorithms are developed in the paper.
The research results are illustrated with examples.