To obtain an upper bound for a μ-updated proximity function in each outer iteration in the algorithm, we use the well-known Lemma 3.1, which can be induced from the decreasing part of the kernel function, instead of using theorems which can be obtained from some properties for eligible functions (for example, Theorem 3.2 in [6] and Theorem 5.4 in [20]).
In this paper, we consider the so-called eligible kernel function [6], i.e., the kernel function satisfies four of the following five conditions, namely the first and the last three conditions: t ψ ″ ( t ) + ψ ′ ( t ) > 0, t < 1, (23a).
For our IPM, we use the following new eligible kernel function: ψ ( t ) = t 2 − 1 2 + e p ( t − q − 1 ) − 1 p q, p ≧ 1 and q ≧ 1 for t > 0. (11).
Of these genes, 3721 were eligible for function and pathway analysis with IPA, which identified transcription and modification of protein as the top functions enriched in the combined gene set (Additional file 1: Table S4).
which means that the total number of iterations is completely determined by the parameters θ, β, γ, τ, and the eligible kernel function ψ ( t ).
In addition, the appropriate values of β and γ will vary for each eligible kernel function and finding them may not always be straightforward.
Step 0: Input an eligible kernel function ψ ( t ) ; an update parameter θ, 0 < θ < 1 ; a threshold parameter τ; and an accuracy parameter ε.
Since (7) has the same matrix of coefficients as (6), also (7) has a unique solution.a It follows that the eligible kernel function ψ ′ ( t ) determines in a natural way search directions for an interior-point algorithm.
In this paper, we replace the right-hand side of the third equation in (6) by − ψ ′ ( v ), i.e., − ∇ Ψ ( v ), as defined by (26) (see Section 3), where ψ ( t ) is any eligible kernel function.
Given any eligible kernel function ψ ( t ), the parameters τ, θ and the step size α should be chosen in such a way that the algorithm is 'optimized' in the sense that the number of iterations required by the algorithm is as small as possible.
