Event № 361
Numerical optimization algorithms are increasingly an amalgam of iterative algorithms within iterative algorithms. The theory for convergence and complexity of these schemes is often presented in the ideal world of exact arithmetic and exact evaluation of functions. When experimental results from implementations are presented two obvious issues are often swept under the rug: how do you decide to stop the inner iteration, and what is the effect of inexact evaluation of the operators on overall convergence? Put more bluntly, what relation, if any, does an actual numerical implementation have to the theory? We attempt to tackle these questions through the development and application of the theory of regularity, not only of the underlying functions and operators, but of the set of solutions themselves. Convexity is but one type of regularity which the theory encompasses, but in nonconvex settings the theory is local. We have focused in particular on local linear convergence as a way to provide meaningful stopping with reasonable error bounds. We survey progress over the last several years on sufficient conditions for local linear convergence and discuss challenges and prospects for further progress.