On the Primal Duty Geometry of Level Sets in Linear And Conic Optimization
On the Primal Duty Geometry of Level Sets in Linear And Conic Optimization
Robert Michael Freund
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1 we have Tc-e 5R is a i3-normal barrier for C, see [5]. Then F'{-) : intC* -> 5R, the conjugate function of F(-), is also a iJ-normal barrier for C", see [5] as well. Let s° € intC* be given. The norm induced by the (^-normal barrier F(-) at . 5° is defined as follows: =o)v \\v\\, ^, o := ^vTH*[s^ where H*{s°) is the Hessian of F*{-) evaluated at s = s°. It then follows from Theorem 2. 1. 1 of [5] that B'{s\l) C C* and from Proposition 2. 3. 4 of [5] that ||s°||, _50 = \A?. Therefore under the... dual norm ||i;||, := ||v||t, sO we have Tc- > l/\/^. 3. 2 A Family of Norms on X for which tc* = 1 For every s° G intC*, there is a norm analogous to the Loo-norm for the nonnegative orthant, for which the associated min-width is tc — 1.
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