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

There has been growing interest in employing neural network (NN) to directly solve constrained optimization problems with low run-time complexity. However, it is non-trivial to ensure NN solutions strictly satisfying problem constraints due to inherent NN prediction errors. Existing feasibility-ensuring methods either are computationally expensive or lack performance guarantee. In this paper, we propose homeomorphic projection 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 nonconvex sets. The idea is to (i) learn a minimum distortion homeomorphic mapping between the constraint set and a unit ball using an invertible NN (INN), and then (ii) perform a simple bisection operation concerning the unit ball so 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 bound the 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.