KKT Residual Sliding Manifolds: A Projection-Free Structured Finite-Time Solver for Nonlinear Constrained Optimisation
Abstract
Nonlinear Constrained Optimisation (NCO) requires simultaneous satisfaction of the Karush-Kuhn-Tucker (KKT) conditions. Existing methods enforce these conditions through projections, penalties, barriers, or sequential approximations, potentially introducing computational complexity and convergence difficulties. For nonconvex problems, a KKT point need not be a local optimum. This paper reformulates NCO as stabilisation of a unified KKT sliding residual manifold and develops SM-KKT, a projection-free solver. Stationarity, equality feasibility, and regularised Fischer-Burmeister complementarity are integrated into a unified residual system, with complementarity encoding inequality feasibility, dual feasibility, and complementary slackness. The manifold's zero set is shown to be equivalent to the KKT conditions. The required Jacobian saddle-point inertia is characterised and shown, under stated assumptions, to be equivalent to the simultaneous satisfaction of the Linear Independence Constraint Qualification (LICQ) and Second-Order Sufficient Condition (SOSC). This characterisation establishes local stability of the structured correction algorithm under these conditions and distinguishes nondegenerate locally optimal KKT solutions from points failing second-order optimality. SM-KKT computes structured primal--dual corrections and applies a discrete sliding reaching law to drive the KKT residual into a prescribed boundary layer in finite time. Four nonlinear benchmarks show recovery of reference constrained optima with KKT residuals below 10−6, earlier boundary-layer reaching, and favourable KKT accuracy compared with projected primal-dual, augmented-Lagrangian, quadratic-penalty, and log-barrier baselines.
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BibTeX
@misc{zenati2026residual,
title = {{KKT Residual Sliding Manifolds: A Projection-Free Structured Finite-Time Solver for Nonlinear Constrained Optimisation}},
author = {Abdelhafid Zenati and Kamal Youcef-Toumi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15545},
eprint = {2610.15545},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15545},
}
RIS
TY - UNPB
TI - KKT Residual Sliding Manifolds: A Projection-Free Structured Finite-Time Solver for Nonlinear Constrained Optimisation
AU - Abdelhafid Zenati
AU - Kamal Youcef-Toumi
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15545
UR - https://arxiv.org/abs/2610.15545
AB - Nonlinear Constrained Optimisation (NCO) requires simultaneous satisfaction of the Karush-Kuhn-Tucker (KKT) conditions. Existing methods enforce these conditions through projections, penalties, barriers, or sequential approximations, potentially introducing computational complexity and convergence difficulties. For nonconvex problems, a KKT point need not be a local optimum. This paper reformulates NCO as stabilisation of a unified KKT sliding residual manifold and develops SM-KKT, a projection-free solver. Stationarity, equality feasibility, and regularised Fischer-Burmeister complementarity are integrated into a unified residual system, with complementarity encoding inequality feasibility, dual feasibility, and complementary slackness. The manifold's zero set is shown to be equivalent to the KKT conditions. The required Jacobian saddle-point inertia is characterised and shown, under stated assumptions, to be equivalent to the simultaneous satisfaction of the Linear Independence Constraint Qualification (LICQ) and Second-Order Sufficient Condition (SOSC). This characterisation establishes local stability of the structured correction algorithm under these conditions and distinguishes nondegenerate locally optimal KKT solutions from points failing second-order optimality. SM-KKT computes structured primal--dual corrections and applies a discrete sliding reaching law to drive the KKT residual into a prescribed boundary layer in finite time. Four nonlinear benchmarks show recovery of reference constrained optima with KKT residuals below 10^−6, earlier boundary-layer reaching, and favourable KKT accuracy compared with projected primal-dual, augmented-Lagrangian, quadratic-penalty, and log-barrier baselines.
ER -