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It is shown in this paper that by exploiting properties of limiting constraints for flexibility in a design, problems for flexibility analysis can be formulated as mixed-integer optimization problems. Formulations are derived when control variables are present or not, and when equalities are eliminated or handled explicitly. These formulations do not rely on the assumption that critical parameter values are vertices, nor do they require exhaustive vertex searches. The case of linear constraints reduces to standard MILP problems, while for the nonlinear case a novel active constraint strategy is proposed and its theoretical properties are analyzed. Examples are presented for both rigorous and screening calculations.