We present a systematic approach for constructing bar frameworks with higher-order rigidity using constrained optimization, where each local minimum corresponds to a prestress stable configuration that is not first-order rigid. By allowing certain edge lengths to vary and optimizing the length of specific free edges, we demonstrate that the resulting frameworks are prestress stable but not first-order rigid under certain weak conditions. Our approach applies to both 2D and 3D bar frameworks, producing a wide range of prestress stable but not first-order rigid examples that have not been presented in the existing literature. Additionally, we present a bifurcation method to obtain rigid but not second-order rigid bar framework. Our results highlight connections between rigidity properties and constrained optimization, offering new insights into the construction and analysis of bar frameworks with higher-order rigidity.