On the concepts of quality for local solutions of nonlinear programs

This thesis covers the topic of assessing the quality of local solutions in both standard and parametric nonlinear programming.

From a theoretical perspective, a formal definition of a quality criterion for local solutions of nonlinear programs is introduced, and the notions of primary and secondary quality criteria are distinguished. Since the primary criterion is inherently associated with the optimality of local solutions, the main focus is on secondary criteria. In particular, over 25 existing secondary quality criteria are systematically reviewed and analyzed. These criteria, arising in both non-parametric and parametric nonlinear programming, are derived from the fields of dynamical systems, robust optimization, stochastic optimization, and parametric sensitivity analysis.

From an applied perspective, two novel secondary quality criteria are proposed: the maximum radius of attraction (MRoA) and the parametric stability score (PSS). MRoA is a measure of quality for local solutions of non-parametric nonlinear programs. It is defined as the radius of the largest ball centered at a local solution such that a given optimization algorithm, once initialized inside that ball, is guaranteed never to escape it and is expected to converge to the same solution. PSS, in turn, quantifies the quality of local solutions of parametric nonlinear programs. It is defined as the maximum magnitude of a perturbation of the nominal parameter, such that for any smaller perturbation, the solutions and corresponding optimal costs of the nominal and perturbed problems remain within prescribed distances. The utility of the PSS concept is demonstrated using an optimal control problem for accomplishing the swing-up maneuver of the pendulum on a cart system.

For both proposed criteria, connections to the field of online optimization are established, emphasizing their potential to support informed decision-making.