Control of an elliptic problem with pointwise state constraints
SIAM Journal on Control and Optimization
A nonsmooth version of Newton's method
Mathematical Programming: Series A and B
Journal of Computational and Applied Mathematics - Special issue on SQP-based direct discretization methods for practical optimal control problems
Smoothing Methods and Semismooth Methods for Nondifferentiable Operator Equations
SIAM Journal on Numerical Analysis
Semismooth Newton Methods for Operator Equations in Function Spaces
SIAM Journal on Optimization
The Primal-Dual Active Set Strategy as a Semismooth Newton Method
SIAM Journal on Optimization
A level set approach for the solution of a state-constrained optimal control problem
Numerische Mathematik
Regular Lagrange Multipliers for Control Problems with Mixed Pointwise Control-State Constraints
SIAM Journal on Optimization
Optimal Control of PDEs with Regularized Pointwise State Constraints
Computational Optimization and Applications
Path-following Methods for a Class of Constrained Minimization Problems in Function Space
SIAM Journal on Optimization
Feasible and Noninterior Path-Following in Constrained Minimization with Low Multiplier Regularity
SIAM Journal on Control and Optimization
Computational Optimization and Applications
Lagrange Multiplier Approach to Variational Problems and Applications
Lagrange Multiplier Approach to Variational Problems and Applications
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A general Moreau-Yosida-based framework for minimization problems subject to partial differential equations and pointwise constraints on the control, the state, and its derivative is considered. A range space constraint qualification is used to argue existence of Lagrange multipliers and to derive a KKT-type system for characterizing first-order optimality of the unregularized problem. The theoretical framework is then used to develop a semismooth Newton algorithm in function space and to prove its locally superlinear convergence when solving the regularized problems. Further, for maintaining the local superlinear convergence in function space it is demonstrated that in some cases it might be necessary to add a lifting step to the Newton framework in order to bridge an $L^2$-$L^r$-norm gap, with $r2$. The paper ends by a report on numerical tests.