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

Abstract

When there are no constraints upon the solutions of the equation 𝑨 ⁢ 𝝃 = 𝒚 , where 𝑨 is a K × N - matrix, 𝝃 ∈ ℝ N and 𝒚 ∈ ℝ K a given vector, the description of the set of solutions as 𝒚 varies in ℝ K is well known. But this is not so when the solutions are required to satisfy 𝝃 ∈ 𝒦 ⁢ ∏ i ≤ j ≤ N [ a j , b j ] , for finite a j ≤ b j : 1 ≤ j ≤ N . To solve this problem we bring in a strictly convex, Fermi-Dirac entropy function Ψ ⁢ ( 𝝃 ) , and find the solution as a ⁢ r ⁢ g ⁢ m ⁢ i ⁢ n ⁢ { Ψ ⁢ ( 𝝃 ) : 𝝃 ∈ 𝒦 , 𝑨 ⁢ 𝝃 = y } . If λ denotes the Lagrange multipliers of the optimization problem, we study the properties of the parametric surface 𝝀 → 𝝃 ⁢ ( 𝝀 ) in the geometry on 𝒦 defined by the Hessian metric derived from Ψ ⁢ ( 𝝃 ) . In particular, we prove that the surface 𝝀 → 𝝃 ⁢ ( 𝝀 ) is contained in ker ( 𝑨 ) ⊥ in the Hessian metric derived from Ψ.