A Comparative Study of Kernel Functions for Primal-Dual Interior-Point Algorithms in Linear Optimization

Infeasible Interior-Point Methods for Linear Optimization Based on Large Neighborhood

“…As in [5], we call the kernel function ψ eligible iff it satisfies the following technical conditions.…”

“…Properties of eligible kernel functions will be recalled from [5] without repeating their proofs. We would like to mention that a kernel function ψ, introduced and discussed in [5], has the following format…”

“…In this section we recall from [5] some results for a large-update FIPM for solving (P) and (D) using a kernel function-based barrier function.…”

“…Any such barrier function is based on a univariate function, named its kernel function. 1 Such functions can be found in [5] and are closely related to the so-called self-regular functions introduced in [6]. In these references only FIPMs are considered, and it is shown that these functions are much more efficient for the process of recentering, which is a crucial part in every FIPM, especially when an iterate is far from the central path.…”

“…In Sect. 4, as a preparation to our large neighborhood-based IIPM, we briefly recall the use of kernel-based barrier functions in large-update FIPMs, as presented in [5]. It will become clear in this section that the convergence rate highly depends on the underlying kernel function.…”