Semi-Infinite Optimization with Implicit Functions

In this work, equality-constrained bilevel optimization problems, arising from engineering design, economics, and operations research problems, are reformulated as an equivalent semi-infinite program (SIP) with implicit functions embedded, which are defined by the original equality constraints that model the system. Using recently developed theoretical tools for bounding implicit functions, a recently developed algorithm for global optimization of implicit functions, and a recently developed algorithm for solving standard SIPs with explicit functions to global optimality, a method for solving SIPs with implicit functions embedded is presented. The method is guaranteed to converge to ε-optimality in finitely many iterations given the existence of a Slater point arbitrarily close to a minimizer. Besides the Slater point assumption, it is assumed only that the functions are continuous and factorable and that the model equations are once continuously differentiable. ■ INTRODUCTION Many engineering design feasibility and reliability problems give rise to optimization programs whose feasible sets are parametrized. This is because it is often of great interest to study performance and/or safety of engineering systems under parametric uncertainty. Particularly, it is important to study the performance and/or safety in the face of the worst case, giving rise to equality-constrained bilevel programs of the following form: