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Table 6 Framework for analyzing the iteration bounds

From: Kernel function based interior-point methods for horizontal linear complementarity problems

Step 0

Define the kernel function ψ(t) and input initial values: τ ≥ 1, ϵ>0, 0<θ<1, ( x 0 ; s 0 )>0, and μ 0 >0 such that Ψ( x 0 ; s 0 , μ 0 )τ.

Step 1

Solve the equation 1 2 ψ (t)=z to find ρ(z), the inverse function of 1 2 ψ (t), 0<t ≤ 1. If the equation is hard to solve, compute a lower bound for ρ(z).

Step 2

Solve the equation ψ(t)=u to find ϱ(u), the inverse function of ψ(t), t ≥ 1. If the equation is hard to solve, compute an upper bound for ϱ(u).

Step 3

Compute a lower bound for δ with respect to Ψ.

Step 4

Compute the upper bound Ψ 0 for Ψ(v).

Step 5

Using Step 3, Step 4 and the default step size α ˜ in (22), find λ>0 and γ, 0<γ ≤ 1, as small as possible such that f( α ˜ )λΨ ( v ) 1 γ .

Step 6

Derive an upper bound for the total number of iterations from Ψ 0 γ θ λ γ log n μ 0 ϵ .

Step 7

Let τ=O(n) and θ = Θ(1) to compute an iteration bound for large-update method, and let τ=O(1) and θ=Θ( 1 n ) to get an iteration bound for small-update method.