On the Primal Duty Geometry of Level Sets in Linear And Conic Optimization

Cover 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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