General non-convex problems often present multiple optimal solutions for identical inputs, signifying a complex, multi-valued input-solution mapping. Conventional learning techniques, primarily tailored to learn single-valued mappings, struggle to train neural networks (NN) to accurately decipher multi-valued ones, leading to inferior solutions. We address this fundamental issue by developing a generative learning approach using a rectified flow (RectFlow) model built upon ordinary differential equations. In contrast to learning input-solution mapping, we learn the mapping from input to solution-distribution, exploiting the universal approximation capability of the RectFlow model. Upon receiving a new input, we employ the trained RectFlow model to sample high-quality solutions from the input-dependent distribution it has learned. Our approach outperforms conceivable GAN and Diffusion models in terms of training stability and run-time complexity. We provide a detailed characterization of the optimality loss and runtime complexity associated with our generative approach. Simulation results for solving non-convex problems show that our method achieves significantly better solution optimality than recent NN schemes, with comparable feasibility and speedup performance.
Generative Learning for Solving Non-Convex Problem with Multi-Valued Input-Solution Mapping
Enming Liang Minghua Chen