Computational marketing has become increasingly important in today's digital world, facing challenges such as massive heterogeneous data, multi-channel customer journeys, and limited marketing budgets. In this paper, we propose a general framework for marketing AI systems, the Neural Optimization with Adaptive Heuristics (NOAH) framework. NOAH is the first general framework for marketing optimization that considers both to-business (2B) and to-consumer (2C) products, as well as both owned and paid channels. We describe key modules of the NOAH framework, including prediction, optimization, and adaptive heuristics, providing examples for bidding and content optimization. We then detail the successful application of NOAH to LinkedIn's email marketing system, showcasing significant wins over the legacy ranking system. Additionally, we share details and insights that are broadly useful, particularly on: (i) addressing delayed feedback with lifetime value, (ii) performing large-scale linear programming with randomization, (iii) improving retrieval with audience expansion, (iv) reducing signal dilution in targeting tests, and (v) handling zero-inflated heavy-tail metrics in statistical testing.