Discrete-time robust optimal control problems generally take a min-max structure
over continuous variable spaces, which can be difficult to solve in practice. In this paper, we
extend the class of such problems that can be solved through a previously proposed local
reduction method to consider those with existence constraints on the uncountable variables.
We also consider the possibility of non-unique trajectories that satisfy equality and inequality
constraints. Crucially, we show that the problems of interest can be cast into a standard semi-
infinite program and demonstrate how to generate optimal uncertainty scenario sets in order to
obtain numerical solutions. We also include examples on model predictive control for obstacle
avoidance with logical conditions, control with input saturation affected by uncertainty, and
optimal parameter estimation to highlight the need for the proposed extension. Our method
solves each of the examples considered, producing violation-free and locally optimal solutions.