Error Analysis of Numerical Methods for Optimization Problems

Abstract The article discusses methods for constructing solution error estimates in optimization problems, which fall into two categories: theoretical and numerical. Theoretical estimates are based on convergence analysis and are mainly useful for qualitative insights, while numerical estimates provide explicit values but are limited to certain methods. The paper introduces two new numerical error estimation methods for a broad range of optimization problems. The first method uses a three-point scheme to derive an exact error estimate from a decreasing sequence of objective function values. The second method, called the rounding method, estimates the error by tracking the increase in significant digits of the solution as iterations progress. Numerical experiments are provided to support these methods.