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Proximal-based regularization methods and successive approximation of variational inequalities in Hilbert spaces

Kaplan A., Tichatschke R.

Control and Cybernetics

2002

Vol. 31, no 3

521-544

For variational inequalities with multi-valued, maximal monotone operators in Hilbert spaces we study proximal-based methods with an improvement of the data approximation after each (approximately performed) proximal iteration. The standard conditions on a distance functional of Bregman's type are weakened, depending on a "reserve of monotonicity" of the operator in the variational inequality, and the enlargement concept is used for approximating the operator. Weak convergence of the proxinnal iterates to a solution of tire original problem is proved. The construction of the [epsilon]-enlargement of monotone operators is analyzed for some particular cases.

Regularized penalty method for non-coercive parabolic optimal control problems

Kaplan A., Tichatschke R.

Control and Cybernetics

1998

Vol. 27, no 1

5-27

The application of a proximal point approach to illposed convex control problems governed by linear parabolic equations is studied. A stable penalty method is constructed by means of multi-step proximal regularization (only w.r.t. the control functions) in the penalized problems. For distributed control problems with state constraints convergence of the approximately determined solutions of the regularized problems to an optimal process is proved.