A Full-Newton Step Interior Point Method for Fractional Programming Problem Involving Second Order Cone Constraint

Mansour Saraj,Nezam Mahdavi Amiri,Ali Sadeghi

doi:10.18187/pjsor.v17i2.2431

Abstract

Some efficient interior-point methods (IPMs) are based on using a self-concordant barrier function related to the feasibility set of the underlying problem.Here, we use IPMs for solving fractional programming problems involving second order cone constraints. We propose a logarithmic barrier function to show the self concordant property and present an algorithm to compute $\varepsilon-$solution of a fractional programming problem. Finally, we provide a numerical example to illustrate the approach.

Highlights

Since the seminal work of Karmarkar (1984) in 1984, interior-point methods (IPMs) have shown to be very useful in many areas of optimization
Using the theory of self-concordant (SC) functions and SC barrier functions (SCBFs), IPMs were introduced by Nesterov and Nemirovski (1989)
The best known iteration bound for linear optimization (LO) is attained by a full-Newton step IPM (FNS-IPM) based on the logarithmic barrier function, mathematically given by

Summary

Introduction

Since the seminal work of Karmarkar (1984) in 1984, interior-point methods (IPMs) have shown to be very useful in many areas of optimization. Roos (2006) investigated an O(n) full Newton step for linear optimization (LO) His algorithm constructs strictly feasible iterates for a sequence of perturbations of the given problem. A Full-Newton Step Interior Point Method for Fractional Programming Problem Involving Second Order Cone Constraint. The main source in such methods is usually some variant of the Newton method Another key element in the design and analysis of IPMs is a barrier function for the domain of the problem. The best known iteration bound for LO is attained by a full-Newton step IPM (FNS-IPM) based on the logarithmic barrier function, mathematically given by. Feng et al (2014) proposed a predictor-corrector path-following int√erior-point algorithm for semi-definite optimization problems Their algorithm contains the low iteration bound of O( nL) as compared to O(nL) due to the usual wide neighborhood algorithm. We note that we have restricted ourselves to fractional programming problem with second order cone constraint, the extension to the more general class of symmetric optimization problems is nowadays more or less straightforward

Preliminaries

Properties of Newton’s Method

Logarithmic barrier algorithm with full Newton’s steps

Numerical Example

Conclusion