A geometry in the set of solutions to ill-posed linear problems with box constraints: Applications to probabilities on discrete sets

When there are no constraints upon the solutions of the equation Aξ= y, where A is a K× N-matrix, ξN and yK 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 ξ ∈ 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 Aξ = 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 (A)^⊥ in the metric defined as the Hessian of the entropy.