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 푨 ⁢ 흃 = 풚 {{\boldsymbol{A}}{\boldsymbol{\xi}}={\boldsymbol{y}}} , where 푨 {{\boldsymbol{A}}} is a K × N - {K\times N-} matrix, 흃 ∈ ℝ N {{\boldsymbol{\xi}}\in{\mathbb{R}}^{N}} and 풚 ∈ ℝ K {{\boldsymbol{y}}\in{\mathbb{R}}^{K}} a given vector, the description of the set of solutions as 풚 {{\boldsymbol{y}}} varies in ℝ K {{\mathbb{R}}^{K}} is well known. But this is not so when the solutions are required to satisfy 흃 ∈ 풦 ⁢ ∏ i ≤ j ≤ N [ a j , b j ] {{\boldsymbol{\xi}}\in{\mathcal{K}}\prod_{i\leq j\leq N}[a_{j},b_{j}]} , for finite a j ≤ b j : 1 ≤ j ≤ N {a_{j}\leq b_{j}:1\leq j\leq N} . To solve this problem we bring in a strictly convex, Fermi-Dirac entropy function Ψ ⁢ ( 흃 ) {\Psi({\boldsymbol{\xi}})} , and find the solution as a ⁢ r ⁢ g ⁢ m ⁢ i ⁢ n ⁢ { Ψ ⁢ ( 흃 ) : 흃 ∈ 풦 , 푨 ⁢ 흃 = y } {argmin\{\Psi({\boldsymbol{\xi}}):{\boldsymbol{\xi}}\in{\mathcal{K}},\,{% \boldsymbol{A}}{\boldsymbol{\xi}}=y\}} . If λ denotes the Lagrange multipliers of the optimization problem, we study the properties of the parametric surface 흀 → 흃 ⁢ ( 흀 ) {{\boldsymbol{\lambda}}\to{\boldsymbol{\xi}}({\boldsymbol{\lambda}})} in the geometry on 풦 {{\mathcal{K}}} defined by the Hessian metric derived from Ψ ⁢ ( 흃 ) {\Psi({\boldsymbol{\xi}})} . In particular, we prove that the surface 흀 → 흃 ⁢ ( 흀 ) {{\boldsymbol{\lambda}}\to{\boldsymbol{\xi}}({\boldsymbol{\lambda}})} is contained in ker ( 푨 ) ⊥ {\ker({\boldsymbol{A}})^{\perp}} in the Hessian metric derived from Ψ. [ABSTRACT FROM AUTHOR]