Analyses Lagrange multiplier theory and demonstrates its impact on the development of numerical algorithms for variational problems in function spaces.

Preface; 1. Existence of Lagrange multipliers; 2. Sensitivity analysis; 3. First Order augmented Lagrangians for equality and finite rank inequality constraints; 4. Augmented Lagrangian methods for nonsmooth, convex optimization; 5. Newton and SQP methods; 6. Augmented Lagrangian-SQP methods; 7. The primal-dual active set method; 8. Semismooth Newton methods I; 9. Semismooth Newton methods II: applications; 10. Parabolic variational inequalities; 11. Shape optimization; Bibliography; Index.

Kazufumi Ito is Professor in the Department of Mathematics and an affiliate of the Center for Research in Scientific Computation at North Carolina State University. He was co-recipient of the SIAM Outstanding Paper Award in 2006.