Interior point method: history and prospects

Two mutually dual families of interior point algorithms are considered. The history of creating the algorithms, the main theoretical results on their justification, the experience of practical use, possible directions of development, and methods for counteracting calculation errors are presented. Subsets of algorithms with various special properties are distinguished, including those that necessarily lead to relatively interior points of optimal solutions. An algorithm for finding the Chebyshev projection onto a linear manifold is presented, in which the properties of relatively interior points of optimal solutions are efficiently employed. This algorithm always elaborates a unique projection and allows one to dispense with the hard-to-verify and sometimes violated Haar condition.