Parallel Lagrange-Newton-Krylov-Schur Methods for PDE-Constrained Optimization. Part II: The Lagrange-Newton Solver and Its Application to Optimal Control of Steady Viscous Flows

In part I of this article, we proposed a Lagrange–Newton–Krylov–Schur (LNKS) method for the solution of optimization problems that are constrained by partial differential equations. LNKS uses Krylov iterations to solve the linearized Karush–Kuhn–Tucker system of optimality conditions in the full space of states, adjoints, and decision variables, but invokes a preconditioner inspired by reduced space sequential quadratic programming (SQP) methods. The discussion in part I focused on the (inner, linear) Krylov solver and preconditioner. In part II, we discuss the (outer, nonlinear) Lagrange–Newton solver and address globalization, robustness, and efficiency issues, including line search methods, safeguarding Newton with quasi-Newton steps, parameter continuation, and inexact Newton ideas. We test the full LNKS method on several large-scale three-dimensional configurations of a problem of optimal boundary control of incompressible Navier–Stokes flow with a dissipation objective functional. Results of numerical experiments on up to 256 Cray T3E-900 processors demonstrate very good scalability of the new method. Moreover, LNKS is an order of magnitude faster than quasi-Newton reduced SQP, and we are able to solve previously intractable problems of up to 800,000 state and 5,000 decision variables at about 5 times the cost of a single forward flow solution.