Reciprocal-Manifold Annealed KKT Flows for Constrained Optimization: Application to the Nonconvex AC Optimal Power Flow

Abstract

Safety-critical optimization applications, such as real-time power system operation, maintain feasibility at every intermediate step, not merely at convergence. Existing approaches either violate constraints mid-solve (interior-point methods) or enforce feasibility through per-instant quadratic programming subproblems with cubic computational cost and unbounded worst-case execution time. We propose a continuous-time optimization framework for smooth constrained nonlinear problems that preserves feasibility throughout the optimization process without requiring projection operators, quadratic programming subproblems, or other per-iteration optimization routines. The method is built around a reciprocal multiplier manifold, which establishes an explicit relationship between inequality constraints and their associated Lagrange multipliers. By designing a continuous multiplier update law, the manifold is shown to remain forward invariant, while the resulting dynamics are equivalent to continuous-time logarithmic barrier gradient descent. The proposed framework naturally extends to multiple inequality constraints, equality constraints, nonconvex feasible sets, and infeasible initial conditions. The method is further enhanced through an augmented Uzawa flow that eliminates oscillatory transients commonly observed in classical primal-dual saddle-point dynamics. The effectiveness of the proposed approach is applied to the AC Optimal Power Flow problem of IEEE 9-bus and IEEE 57-bus systems. Numerical results show convergence to solutions within 0.4\% of the benchmark optimum while maintaining strict feasibility of all constraints. A computational complexity analysis shows that the proposed dynamics reduce the per-step computational cost from cubic to linear complexity. Finally, dynamic tracking studies under time-varying operating conditions demonstrate reliable feasibility preservation.