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.