One-Shot Optimization with Additional Inequality Constraints

Abstract

The one-shot approach is a powerful simultaneous optimization framework for design tasks governed by computationally expensive steady-state systems. While previous formulations mainly focused on additional equality constraints, this work extends the one-shot framework to optimization problems with inequality constraints using slack variables embedded into a doubly augmented Lagrangian formulation. After elimination of the slack variables, the resulting formulation contains nonsmooth active-set dependent terms while preserving the characteristic coupled one-shot matrix structure. The resulting generalized gradient system admits a representation of the form $ \nabla \DLag = -Ms, $ analogous to the equality-constrained one-shot framework. To analyze the nonsmooth active-set transitions, generalized second-order subdifferentials in the sense of Clarke and Rockafellar--Wets are employed. We derive explicit positivity conditions ensuring positive definiteness of all admissible generalized Hessians and hence strict local optimality. Furthermore, we give an outlook on the construction of preconditioners based on generalized Hessian approximations.