Logarithmic-Barrier Decomposition Interior-Point Methods for Stochastic Linear Optimization in a Hilbert Space

Abstract

Several logarithmic-barrier interiors-point methods are now available for solving two-stage stochastic optimization problems with recourse in the finite-dimensional setting. However, despite the genuine need for studying such methods in general spaces, there are no infinite-dimensional analogs of these methods. Inspired by this evident gap in the literature, in this paper, we propose logarithmic-barrier decomposition-based interior-point algorithms for two-stage stochastic linear optimization problems with recourse in a Hilbert space. We study the fundamental properties of the logarithmic barrier associated with the recourse function of our problem setting. The novelty of our algorithms is that their iteration complexity results are independent on the choice of the underlying Hilbert space. In other words, after applying the obtained fundamental properties to our problem setting, the iteration complexity results obtained for the short- and long-step algorithms coincide with the best-known estimates in the finite-dimensional case.