Linear inverse problems with box constraints

We solve the problem <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>𝑨</m:mi> <m:mo>⁢</m:mo> <m:mi>X</m:mi> </m:mrow> <m:mo>=</m:mo> <m:mi>𝒚</m:mi> </m:mrow> </m:math> {{A}X={y}} , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>𝑨</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mi mathvariant="script">𝒦</m:mi> <m:mo>→</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>N</m:mi> </m:msup> </m:mrow> </m:mrow> </m:math> {{A}:{K}→{R}N} is a linear, continuous operator, and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒦</m:mi> </m:math> {{K}} is a class of functions with prescribed bounds defined on a compact domain in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℝ</m:mi> <m:mi>m</m:mi> </m:msup> </m:math> {{R}ᵐ} . We restate the problem as consisting of minimizing an entropy of Fermi–Dirac type, subject to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>𝑨</m:mi> <m:mo>⁢</m:mo> <m:mi>X</m:mi> </m:mrow> <m:mo>=</m:mo> <m:mi>𝒚</m:mi> </m:mrow> </m:math> {{A}X={y}} as a constraint. We prove that the solution is robust respect to small perturbations of the data and converges to a true solution as the number of data points increases.