{"ID":2864540,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.23072","arxiv_id":"2509.23072","title":"A Constrained Optimization Approach for Constructing Rigid Bar Frameworks with Higher-order Rigidity","abstract":"We present a systematic approach for constructing bar frameworks that are rigid but not first-order rigid, using constrained optimization. We show that prestress stable (but not first-order rigid) frameworks arise as the solution to a simple optimization problem, which asks to maximize or minimize the length of one edge while keep the other edge lengths fixed. By starting with a random first-order rigid framework, we can thus design a wide variety of prestress stable frameworks, which, unlike many examples known in the literature, have no special symmetries. We then show how to incorporate a bifurcation method to design frameworks that are third-order rigid. Our results highlight connections between concepts in rigidity theory and constrained optimization, offering new insights into the construction and analysis of bar frameworks with higher-order rigidity.","short_abstract":"We present a systematic approach for constructing bar frameworks that are rigid but not first-order rigid, using constrained optimization. We show that prestress stable (but not first-order rigid) frameworks arise as the solution to a simple optimization problem, which asks to maximize or minimize the length of one edg...","url_abs":"https://arxiv.org/abs/2509.23072","url_pdf":"https://arxiv.org/pdf/2509.23072v2","authors":"[\"Xuenan Li\",\"Christian D. Santangelo\",\"Miranda Holmes-Cerfon\"]","published":"2025-09-27T02:54:37Z","proceeding":"math.MG","tasks":"[\"math.MG\",\"math.OC\"]","methods":"[]","has_code":false}