Abstract
A class of quadratic optimization problems in Hilbert spaces is considered, where pointwise box-constraints and constraints of bottleneck type are given. The main focus is on proving the existence of regular Lagrange multipliers in $L^2$-spaces. This question is solved by investigating the solvability of a Lagrange dual quadratic problem. The theory is applied to different optimal control problems for elliptic and parabolic partial differential equations with mixed pointwise control-state constraints.
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