Large Language Models (LLMs) are very large deep learning models, which are pre-trained on vast amounts of data, able to achieve general-purpose language understanding and generation. GPT-4 is an increasingly popular example from OpenAI. We introduce a novel framework called Generative Heuristics, which hybridize (meta)heuristics for its quantitative capabilities with the qualitative proficiency of LLMs. To illustrate this framework, we address a rich version of the Project Portfolio Selection Problem with a Generative Heuristic relying on a Simheuristic Algorithm. Our solving approach explores candidate portfolios, optimizing quantitative outcomes, such as the Net Present Value, while maintaining a focus on non-quantifiable strategic contributions. To maximize realism and applicability, the strategic plan from the LIFE program (EU’s funding instrument for environment and climate action) and project summaries from actual LIFE-funded projects are utilized. A computational experiment is carried out to illustrate the feasibility and potential of our approach, which enhance quantitative search procedures with cognitive capabilities, offering decision-makers both a more comprehensive and versatile set of options.