Primal and Dual Characterizations for Farkas Type Lemmas in Terms of Closedness Criteria

Abstract This paper deals with the characterization, in terms of closedness of certain sets regarding other sets, of Farkas lemmas determining when the upperlevel set of a convex function $f$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>f</mml:mi> </mml:math> contains a set of the form $C\cap \mathbb{A}^{-1}\left ( D\right ) $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> <mml:mo>∩</mml:mo> <mml:msup> <mml:mi>A</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>D</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , where $C$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> </mml:math> and $D$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>D</mml:mi> </mml:math> are convex sets (not necessarily cones) in locally convex spaces $X$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> (with topological dual $X^{\prime }$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>X</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> ) and $Y$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Y</mml:mi> </mml:math> , respectively, while $\mathbb{A}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> is a continuous linear operator from $X$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> to $Y$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Y</mml:mi> </mml:math> . More in detail, each of the mentioned characterizations of Farkas type lemmas consists in the closedness of certain subset of either one of the “primal” spaces $X\times Y\times \mathbb{R}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> <mml:mo>×</mml:mo> <mml:mi>Y</mml:mi> <mml:mo>×</mml:mo> <mml:mi>R</mml:mi> </mml:math> and $Y\times \mathbb{R}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Y</mml:mi> <mml:mo>×</mml:mo> <mml:mi>R</mml:mi> </mml:math> , or of the “dual” space $X^{\prime }\times \mathbb{R}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>X</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>×</mml:mo> <mml:mi>R</mml:mi> </mml:math> , regarding some singleton set of the corresponding space. Moreover, the paper also provides an existence theorem for the feasible set $C\cap \mathbb{A}^{-1}\left ( D\right ) $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> <mml:mo>∩</mml:mo> <mml:msup> <mml:mi>A</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>D</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> in terms of the closedness of certain subset of the dual space $X^{\prime }\times \mathbb{R}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>X</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>×</mml:mo> <mml:mi>R</mml:mi> </mml:math> regarding the singleton set formed by the null element. These results are illustrated with significant applications to constrained convex minimization problems and to functional approximation by polynomials.