Normal Cone

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

Chapter 22 in Conjugate Duality in Economic Analysis, 2026, pp 163-167 from Springer

Abstract: Abstract We analyze the directional derivative and subdifferential for the indicator of a set. After first considering the general case, we specialize to the contour set for a function, the vectors for which the function is less than or equal to a specified value. Translate the epigraph of the function to the origin. The translated contour set has indicator$$\delta _{C}\left ( \boldsymbol {x}\right ) =\delta _{f\left ( \boldsymbol {x}\right ) \leq 0}=\delta _{\mathrm {epi\ }f}\left ( \boldsymbol {x},0\right ) \!.$$ Define the projection $${\textit {{\sf { {X}}}}}:\left ( \boldsymbol {x},y\right ) \longmapsto \boldsymbol {x}$$. Under general conditions the subdifferential of the indicator of the contour set is the projection of the normal cone of the epigraph,$$\partial \delta _{f\leq 0}={\textit {{\sf { {X\,}}}}}\partial \delta _{\mathrm {epi\ }f}.$$ This subdifferential is also$$\partial \delta _{f\leq 0}=\mathbf {R}_{+ }\partial f\cup \partial \delta _{\,\mathrm {dom\ }f},$$ in which the two terms derive from the non-vertical and vertical supporting hyperplanes, respectively.

Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:spr:conchp:978-3-032-21396-9_22

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DOI: 10.1007/978-3-032-21396-9_22

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