Lebesgue Infinite Sums of Convex Functions: Subdifferential Calculus

Abstract

<jats:p> We present a subdifferential analysis for a general concept of infinite sum <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">f:=\sum_{i\in I}f_{i}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:msub> <mml:mo>∑</mml:mo> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>I</mml:mi> </mml:mrow> </mml:msub> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> of arbitrary collections of convex functions <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">f_{i}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> , called Lebesgue infinite sum. Since this problem cannot be addressed, at least directly, through classical arguments from the theory of normal convex integrands, we perform a reduction analysis showing that the <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\varepsilon</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ε</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -subdifferential of <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">f</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> reduces to that of countable/finite subsums via appropriate lower limit and closure processes. Then, the usual calculus rules of (countable) integral functions give rise to characterizations of the <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\varepsilon</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ε</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -subdifferential of <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">f</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> , which are written exclusively by means of <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\varepsilon</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ε</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -subdifferentials of the data <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">f_{i}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> . The resulting characterizations do not assume any qualification or boundedness condition. </jats:p>