Error bounds for infinite systems of convex inequalities without Slater’s condition

What is it about?

The feasible set of a convex semi–infinite program is described by a possibly infinite system of convex inequality constraints. The idea of error bounds is to find an upper bound for the distance between a given point and the feasible set in terms of the value of the constraint function that is maximally violated. If such an error bound holds for a convex optimization problem with the corresponding inequalities as constraints that is calm at a minimal point, we have exact penalization if the penalty parameter is sufficiently large for the nonsmooth penalty function that contains the residual function.

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Photo by Ryan Spencer on Unsplash

Photo by Ryan Spencer on Unsplash

Why is it important?

Exact Penalty is a nice tool both for the theory and for numerical algorithms. In optimal control it can be used to obtain optimal exact controls that reach a desired target state in finite time without terminal conditions. Of course this is only possible if the corresponding System is exactly controllable, so there is also an interesting link to exact controllability.

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