This dissertation presents novel and efficient tensorbased methods for solving constrained tensor
equations and generalized least squares problems. Initially, constrained tensor equations along
with their concepts and applications are described. Inspired by iterative algorithms for solving
constrained matrix equations, an iterative algorithm based on the generalized least squares method
is proposed, utilizing Einstein summation for solving constrained tensor equations. This algorithm
is capable of addressing tensor equations with general constraints, and by considering the tensor
structure, it aids in reducing computational complexity. To implement the generalized least squares
algorithm efficiently in the tensor space, multilinear operators and their duals are defined and analyzed. The stability and convergence analysis of the proposed method is thoroughly presented,
demonstrating strong mathematical properties and reliability of the technique. Furthermore, a
novel approach for solving the generalized least squares problem is introduced. This problem
is an extension of the generalized least squares problem under conditions where overdetermined
linear systems AX ≈ B face significant errors in both the data matrix A and the observation
vector B. Leveraging firstorder Taylor series expansion, the generalized least squares problem
is transformed into a linear problem, allowing for the application of the generalized least squares
algorithm in a tensor framework. This transformation leads to simplification of computations,
improved accuracy of results, and reduced execution time. The efficiency and accuracy of the
proposed methods are validated through performance evaluations in the context of image reconstruction as a practical application. Results obtained from numerous numerical examples, alongside comparisons with existing methods in reputable literature, demonstrate the superiority of the
proposed methods in terms of computational efficiency and accuracy of res