Dr. Chi Yue Tan, Center for Computation and Technology, Louisiana State University, USA

Wei Gong, Associate Professor

On Lagrange multipliers of the KKT system in Hilbert spaces

9:00-10:00 September 9 (Friday), Tencent Meeting ID: 872 323 457

In this talk we investigate the Karush-Kuhn-Tucker (KKT) system of a model optimization problem in Hilbert spaces. We derive it by using an optimization procedure relaxation approach, which avoids interior point conditions that are commonly used in the infinite dimensional case. Our main tool is the classical augmented Lagrangian method. The main result is a weak form asymptotic KKT system. Based on it, we introduce the essential Lagrange multiplier and establish the existence and uniqueness results of this new type multiplier. Our results show that in the finite dimensional case, the essential Lagrange multiplier always exists, while in the infinite dimensional case, this is not always true. The close connections of the essential Lagrange multiplier and the usual proper Lagrange multiplier enable us to derive sufficient and necessary conditions for the existence and uniqueness of the usual proper Lagrange multiplier. A direct application of these results to general nonlinear and nonconvex optimization problems and variational inequalities is presented as well. We also reveal the equivalence of the weak convergence of the multipliers in the classical augmented Lagrangian algorithm for the the model problem and the existence of the essential Lagrange multiplier of the KKT system of the model problem.