Convergence Analysis of Generalized Primal-Dual Interior-Point Algorithms for Linear Optimization
We study the zeroth-, first-, and second-order algorithms proposed by Tuncel. The zeroth-order algorithms are the generalization of the classic primal-dual affine-scaling methods, and have a strong connection with the quasi-Newton method. Although the zeroth-order algorithms have the property of strict monotone decrease in both primal and dual objective values, they may not converge. We give an illustrative example as well as an algebraic proof to show that the zeroth-order algorithms do not converge to an optimal solution in some cases. The second-order algorithms use the gradients and Hessians of the barrier functions. Tuncel has shown that all second-order algorithms have a polynomial iteration bound. The second-order algorithms have a range of primal-dual scaling matrices to be chosen. We give a method to construct such a primal-dual scaling matrix. We then analyze a new centrality measure. This centrality measure appeared in both first- and second-order algorithms. We compare the neighbourhood defined by this centrality measure with other known neighbourhoods. We then analyze how this centrality measure changes in the next iteration in terms of the step length and some other information of the current iteration.