Preservation of Structural Properties in Optimization with Decisions Truncated by Random Variables and Its Applications

A common technical challenge encountered in many operations management models is that decision variables are truncated by some random variables and the decisions are made before the values of these random variables are realized, leading to non-convex minimization problems. To address this challenge, we develop a powerful transformation technique that converts a nonconvex minimization problem to an equivalent convex minimization problem. We show that such a transformation enables us to prove the preservation of some desired structural properties, such as convexity, submodularity, and L-(sic)-convexity, under optimization operations, that are critical for identifying the structures of optimal policies and developing efficient algorithms. We then demonstrate the applications of our approach to several important models in inventory control and revenue management: dual sourcing with random supply capacity, assemble-to-order systems with random supply capacity, and capacity allocation in network revenue management.