Decision Making Under Cumulative Prospect Theory: An Alternating Direction Method of Multipliers.

This article focuses on developing a novel numerical method to solve decision-making problems under cumulative prospect theory (CPT) with finite scenario realizations, aiming to maximize CPT utility subject to constraints. The authors propose an alternating direction method of multipliers (ADMM) framework that decomposes the problem into two subproblems: a convex portfolio weight update and a challenging nonconvex, nonsmooth optimization involving CPT utility with chain constraints. To solve the latter, they introduce two algorithms—a dynamic programming (DP) method that guarantees global optimality but is computationally intensive, and a modified pooling-adjacent-violators (PAV) algorithm that efficiently finds stationary points. Numerical experiments on portfolio optimization demonstrate that the ADMM combined with the PAV algorithm outperforms the DP method, MATLAB's fmincon solver, and recent heuristics, providing both theoretical convergence guarantees and practical efficiency.