Real radical branch recovery for second order semialgebraic optimization
Abstract
We study parabolic second order tangent sets in semialgebraic optimization when the supplied polynomial representation is singular. Our focus is the representation problem: formal differentiation of singular defining equations may lose the intrinsic second order geometry. The main result is a real radical branch recovery theorem. If the local real minimal prime branches of the equality ideal are smooth, then the branches selected by a direction and their exact parabolic second order tangent sets can be recovered from a local real radical decomposition and first- and second order polynomial data, without a preassigned stratification. The formula is invariant under changes of regular branch generators, while counterexamples delimit the roles of real radicalization, branch regularity, and constraint qualification. In optimization, the recovered models reduce each directional second order necessary test to finitely many linear programs in the parabolic correction and hence to branchwise multiplier certificates. A parameterized example shows that the formal second order system of the original singular representation may falsely reject a strict local minimizer, whereas branch recovery gives the correct test. Intrinsically singular branches are separated from representation singularities and can be incorporated when an exact higher order model is available.
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BibTeX
@misc{le2026real,
title = {{Real radical branch recovery for second order semialgebraic optimization}},
author = {Cong Trinh Le},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.06849},
eprint = {2606.06849},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.06849},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Real radical branch recovery for second order semialgebraic optimization
AU - Cong Trinh Le
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2606.06849
UR - https://arxiv.org/abs/2606.06849
AB - We study parabolic second order tangent sets in semialgebraic optimization when the supplied polynomial representation is singular. Our focus is the representation problem: formal differentiation of singular defining equations may lose the intrinsic second order geometry. The main result is a real radical branch recovery theorem. If the local real minimal prime branches of the equality ideal are smooth, then the branches selected by a direction and their exact parabolic second order tangent sets can be recovered from a local real radical decomposition and first- and second order polynomial data, without a preassigned stratification. The formula is invariant under changes of regular branch generators, while counterexamples delimit the roles of real radicalization, branch regularity, and constraint qualification. In optimization, the recovered models reduce each directional second order necessary test to finitely many linear programs in the parabolic correction and hence to branchwise multiplier certificates. A parameterized example shows that the formal second order system of the original singular representation may falsely reject a strict local minimizer, whereas branch recovery gives the correct test. Intrinsically singular branches are separated from representation singularities and can be incorporated when an exact higher order model is available.
ER -