Quadratic Function

Bruce C. Dieffenbach ()
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Bruce C. Dieffenbach: Independent author

Chapter 28 in Conjugate Duality in Economic Analysis, 2026, pp 215-220 from Springer

Abstract: Abstract We define and analyze the quadratic function, a generalization of the quadratic $$\frac {1}{2}\left \langle \boldsymbol {x},\boldsymbol {x}\right \rangle$$ . The quadratic set having indicator $$\delta _{Q}\left ( \boldsymbol {x},y\right ) =\delta _{y\geq \frac {1}{2}\boldsymbol {x}^{\top }\boldsymbol {x}}$$ is the epigraph of the quadratic $$\frac {1}{2}\left \langle \boldsymbol {x},\boldsymbol {x}\right \rangle$$ . The quadratic function $$~\delta _{Q}^{\ast }\left ( \boldsymbol {x}^{\ast },-y^{\ast }\right )$$ is the epi-multiplication of the conjugate $$~\frac {1}{2}\left \langle \boldsymbol {x}^{\ast },\boldsymbol {x}^{\ast }\right \rangle$$ by y*. We generalize this concept. Let $$\left ( {\textit {{\sf { {A}}}}},{\textit {{\sf { {S}}}}}\right ) \in \mathbf {T}^{\mathbf {X}\rightarrow \mathbf {Y}}\oplus \mathbf {S}^{\mathbf {X}}$$ . The quadratic set has indicator $$\delta _{Q}\left ( {\textit {{\sf { {A}}}}},{\textit {{\sf { {S}}}}}\right ) =\delta _{{\textit {{\sf { {S}}}}}\succeq \frac {1}{2}{\textit {{\sf { {A}}}}}^{\top }{\textit {{\sf { {A}}}}}}$$ . The quadratic function is the support $$~\delta _{Q}^{\ast }\left ( {\textit {{\sf { {A}}}}}^{\ast },-{\textit {{\sf { {S}}}}}^{\ast }\right )$$ , which we evaluate by perturbation duality. We interpret the Fenchel factor for the indicator of the quadratic set and its support via equivalent conditions. This book applies the quadratic function to least-squares linear regression and maximum-likelihood linear regression in econometrics.

Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:spr:conchp:978-3-032-21396-9_28

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DOI: 10.1007/978-3-032-21396-9_28

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