Fractional programming (FP) is a branch of mathematical optimization that deals with the optimization of ratios. It is an invaluable tool for signal processing and machine learning, because many key metrics in these fields are fractionally structured, e.g., the signal-to-interference-plus-noise ratio (SINR) in wireless communications, the Cramér-Rao bound (CRB) in radar sensing, the normalized cut in graph clustering, and the margin in support vector machine (SVM).

With the advent of large language models (LLMs), the concept of artificial intelligence (AI) agents capable of decision making and action execution has gained prominence, particularly as LLMs demonstrate increasing proficiency in tool use and task planning. Following these advancements, the established methods used for task-oriented dialogue (TOD) systems have undergone a paradigm shift by integrating LLMs’ revolutionary language understanding and reasoning skills with enhanced instruction following and response generation abilities.