Low Complexity Homeomorphic Projection to Ensure Neural-Network Solution Feasibility for Optimization over (Non-)Convex Set

There has been growing interest in employing neural networks (NNs) to directly solve constrained optimization problems with low run-time complexity. However, it is non-trivial to ensure NN solutions strictly satisfy problem constraints due to inherent NN prediction errors. Existing feasibility-ensuring methods are either computationally expensive or lack performance guarantee. In this paper, we propose <i>Homeomorphic Projection</i> as a low- complexity scheme to guarantee NN solution feasibility for optimization over a general set homeomorphic to a unit ball, covering all compact convex sets and certain classes of non- convex sets. The idea is to (i) learn a minimum distortion homeomorphic mapping between the constraint set and a unit ball using a bi-Lipschitz invertible NN (INN), and then (ii) perform a simple bisection operation concerning the unit ball such that the INN-mapped final solution is feasible with respect to the constraint set with minor distortion-induced optimality loss. We prove the feasibility guarantee and bounded optimality loss under mild conditions. Simulation results, including those for non-convex AC-OPF problems in power grid operation, show that homeomorphic projection outperforms existing methods in solution feasibility and run-time complexity while achieving similar optimality loss. © 2024 Enming Liang, Minghua Chen, and Steven H. Low.