Pseudoconvex Problems in Operational Decision Systems: Algorithms for Joint Learning and Optimization

Abstract

We consider joint optimization and learning problems arising in real-time decision systems. While most existing work focuses primarily on convex, revenue-based objectives, we extend this line of research to multi-objective formulations. In energy systems, for instance, we incorporate metrics such as renewable penetration and generation costs. Our key focus, however, is on a class of problems with a pseudoconvex structure - a natural relaxation of convexity. Representative examples include fractional objectives in energy management and logit-based revenue models in retail. The outer-level problem optimizes these pseudoconvex objectives, while the inner-level problem involves training a machine learning model using historical data. Our contributions are twofold. First, we propose a simultaneous learning-and-optimization framework that iteratively updates both inner- and outer-level variables. Second, we develop convergent algorithms for these problem classes under realistic mathematical assumptions. Using real-world datasets, we evaluate the computational performance of our methods and highlight an important observation: there exist clear trade-offs between inexact learning and computational time when assessing final solution quality.