Polyhedra

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

Chapter 24 in Conjugate Duality in Economic Analysis, 2026, pp 181-184 from Springer

Abstract: Abstract A polyhedron is the intersection of a finite number of closed half spaces, the solutions to linear inequalities. That the polar of a polyhedral cone is polyhedral is fundamental. A polyhedral function is subdifferentiable throughout its effective domain, and the subdifferential is polyhedral. We derive a formula for the subdifferential in terms of the polyhedral representation. The conjugate of a polyhedral function is polyhedral. A primal having a polyhedral perturbation function has a polyhedral value function and a polyhedral dual. The polyhedral condition for no duality gap applies. A cone is finitely generated if its elements are the nonnegative linear combinations of a finite set of generators. The Minkowski-Weyl theorem is intuitive geometrically, but the proof is subtle: a cone is finitely generated if and only if it is polyhedral. No simple proof by duality is possible. No simple, general formula relates a finitely generated representation and a polyhedral representation. For this reason, sophisticated algorithms are put forward for numerical polyhedral optimization.

Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:spr:conchp:978-3-032-21396-9_24

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DOI: 10.1007/978-3-032-21396-9_24

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