Iterative Optimization of Multidimensional Functions on Turing Machines under Performance Guarantees

This paper studies the effective convergence of iterative methods for solving convex minimization problems using block Gauss--Seidel algorithms. It investigates whether it is always possible to algorithmically terminate the iteration in such a way that the outcome of the iterative algorithm satisfies any predefined error bound. It is shown that the answer is generally negative. Specifically, it is shown that even if a computable continuous function which is convex in each variable possesses computable minimizers, a block Gauss--Seidel iterative method might not be able to effectively compute any of these minimizers. This means that it is impossible to algorithmically terminate the iteration such that a given performance guarantee is satisfied. The paper discusses two reasons for this behavior. First, it might happen that certain steps in the Gauss--Seidel iteration cannot be effectively implemented on a digital computer. Second, all computable minimizers of the problem may not be reachable by the Gauss--Seidel method. Simple and concrete examples for both behaviors are provided.