Full-Newton step polynomial-time methods for LO based on locally ...

Local self-concordancy of the barrier functionWe define φ : D → R to be locally κ-SC at x ∈ D ⊆ R n if φ(x + th) is κ-SC <strong>for</strong> allh ∈ R n ; to express the dependence of κ on x we use the notation κ(x). Clearly φ is κ-SCif and only if κ(x) is bounded above by some (finite) constant on the domain of φ.It is well known that the classical logarithmic barrier function—whose kernel function is t2 −12 −ln t—is SC. But this is quite exceptional. In general kernel-function-<strong>based</strong> barrier functionsare not SC, but they are locally SC. The following table shows this <strong>for</strong> the kernel function,ψ 2 (t).iteration boundslocal value of κi kernel function ψ i (t) small-update large-update ψ(t) Φ µ (x)1t 2 −12 − ln t O ( √ ) ( )n lnnǫO n lnnǫ1 1( )2 t −1 2tO ( √ )n lnnǫ?122t √32‖v‖ ∞√3At the start of the algorithm we have v = 1, where the local value of κ is 2/ √ 3. Duringthe course of the algorithm the iterates stay so close to the central path that v stays in a verysmall neighborhood of 1, and hence the barrier function is SC <strong>for</strong> some suitable value of κ,slightly larger than 2/ √ 3.19