Algebraic optimization of sequential decision problems

• Published in: Journal of symbolic computation Vol. 121; p. 102241
• Main Authors: Dressler, Mareike, Garrote-López, Marina, Montúfar, Guido, Müller, Johannes, Rose, Kemal
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.03.2024
• Subjects: Algebraic degree; Partially observable Markov decision process; Polynomial optimization; State aggregation; State-action frequencies; 90C40; Partially observable Markov decision process; Polynomial optimization; 62R01; 90C23; State-action frequencies; Algebraic degree; State aggregation
• ISSN: 0747-7171
• DOI: 10.1016/j.jsc.2023.102241

We study the optimization of the expected long-term reward in finite partially observable Markov decision processes over the set of stationary stochastic policies. In the case of deterministic observations, also known as state aggregation, the problem is equivalent to optimizing a linear objective subject to quadratic constraints. We characterize the feasible set of this problem as the intersection of a product of affine varieties of rank one matrices and a polytope. Based on this description, we obtain bounds on the number of critical points of the optimization problem. Finally, we conduct experiments in which we solve the KKT equations or the Lagrange equations over different boundary components of the feasible set, and we compare the result to the theoretical bounds and to other constrained optimization methods.