For some computable functions convex in each variable, block Gauss-Seidel coordinate descent has no Turing-computable argmin step and no effective stopping rule, so performance guarantees are impossible.
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Iterative Optimization of Multidimensional Functions on Turing Machines under Performance Guarantees
For some computable functions convex in each variable, block Gauss-Seidel coordinate descent has no Turing-computable argmin step and no effective stopping rule, so performance guarantees are impossible.