2024/06/07 by Henryk Gzyl, Gzyl, Henryk
Computer Science · Mathematics · #Statistical and Computational Modeling #Rough Sets and Fuzzy Logic #Fuzzy Systems and Optimization
paper · pdf · doi:10.48550/arxiv.2406.08513
When there are no constraints upon the solutions of the equation A\mathbfξ= y, where A is a K× N-matrix, \mathbfξ∈ℝN and y∈ℝK a given vector, the description of the set of solutions as y varies in ℝK is well known. But this is not so when the solutions are required to satisfy \mathbfξ ∈ K∏i≤ j≤ N [aj,bj], for finite aj≤ bj: 1≤ j≤ N. Here we provide a description of the set of solutions as a surface in the constraint set, parameterized by the Lagrange multipliers that come up in a related optimization problem in which A\mathbfξ = y appears as a constraint. It is the dependence of the Lagrange multipliers on the data vector y that determines how the solution changes as the datum changes. The geometry on the solutions is inherited from a Riemannian geometry on the set of constraints induced by the Hessian of an entropy of the Fermi-Dirac type which is the objective in the restatement of the optimization problem mentioned above. We prove that the set of solutions is contained in ker(A)^⊥ in the metric defined as the Hessian of the entropy.