Convex Optimization for Parameter Synthesis in MDPs

Probabilistic model-checking aims to prove whether a Markov decision process (MDP) satisfies a temporal logic specification. The underlying methods rely on an often unrealistic assumption that the MDP is precisely known. Consequently, parametric MDPs (pMDPs) extend MDPs with transition probabilities...

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Bibliographic Details
Published in:IEEE transactions on automatic control 2022-12, Vol.67 (12), p.6333-6348
Main Authors: Cubuktepe, Murat, Jansen, Nils, Junges, Sebastian, Katoen, Joost-Pieter, Topcu, Ufuk
Format: Article
Language:English
Subjects:
Adaptation models
Collision avoidance
Computational geometry
Convex analysis
Convex functions
Convex optimization
Convexity
Markov processes
Mathematical programming
Model checking
Optimization
Parameters
Probabilistic logic
Probabilistic models
Quadratic programming
Scalability
Specifications
Synthesis
Temporal logic
Transition probabilities
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Summary:Probabilistic model-checking aims to prove whether a Markov decision process (MDP) satisfies a temporal logic specification. The underlying methods rely on an often unrealistic assumption that the MDP is precisely known. Consequently, parametric MDPs (pMDPs) extend MDPs with transition probabilities that are functions over unspecified parameters. The parameter synthesis problem is to compute an instantiation of these unspecified parameters such that the resulting MDP satisfies the temporal logic specification. We formulate the parameter synthesis problem as a quadratically constrained quadratic program, which is nonconvex and is NP-hard to solve in general. We develop two approaches that iteratively obtain locally optimal solutions. The first approach exploits the so-called convex-concave procedure (CCP), and the second approach utilizes a sequential convex programming (SCP) method. The techniques improve the runtime and scalability by multiple orders of magnitude compared to black-box CCP and SCP by merging ideas from convex optimization and probabilistic model-checking. We demonstrate the approaches on a satellite collision avoidance problem with hundreds of thousands of states and tens of thousands of parameters and their scalability on a wide range of commonly used benchmarks.
ISSN:0018-9286
1558-2523
DOI:10.1109/TAC.2021.3133265