We consider optimal distributed and boundary control problems
for semilinear parabolic equations, where pointwise constraints on
the control and pointwise mixed control-state constraints of bottleneck
type are given. Our main result states the existence of regular
Lagrange multipliers for the state-constraints. Under natural
assumptions, we are able to show the existence of bounded and measurable
Lagrange multipliers. The method is based on results from the theory
of continuous linear programming problems.