Nonlinear problems such as transient dynamic problems exhibit structural responses that can be discontinuous due to numerous bifurcations. This hinders gradient-based or response surface-based optimization. This paper proposes a novel approach to split the design space into regions where the response is continuous. This makes traditional optimization viable. A convex hull approach is adopted to isolate the points corresponding to unwanted bifurcations in the design space. The proposed approach is applied to a tube impacting a rigid wall representing a transient dynamic problem. Since nonlinear behavior is highly sensitive to small variations in design, reliability-based design optimization is performed. The proposed method provides the designer an optimal design with a prescribed dynamic behavior.