Parametric Stability Score for Local Solutions of Constrained Parametric Nonlinear Programs

Abstract

AbstractNonlinear optimization problems arise in the context of many engineering processes and other real‐world applications. Depending on the complexity of modeled phenomena and constraints, such problems can feature multiple local solutions of different nature. Primarily, the quality of a solution is measured by the value of an objective function, and hence, one aims to find a global optimum. In many cases, however, additional criteria arise. For instance, nonlinear programs, in addition to decision variables, might depend on parameters undergoing perturbations. Such disturbances might be caused by noisy measurement data, changes in the environment in a real‐time setting or other factors. In this case, apart from good optimality, a practitioner might be interested in the behaviour of a solution under changes in parameters. Recently, the parametric stability score was introduced as a quality measure for solutions of unconstrained problems with respect to the influence of parameter perturbations. In this paper, we extend the concept to constrained optimization problems. The score is defined as the largest magnitude of a perturbation for which the perturbed solution does not deviate from the nominal one beyond an allowed tolerance. Our work is based on the results of parametric sensitivity analysis. This framework enables us to give a formal definition of the parametric stability score and provides tools for its efficient numerical approximation. We illustrate the approach with an example and discuss possible application scenarios for the concept.