Randomized Search Methods for Solving Markov Decision Processes and Global Optimization
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Markov decision process (MDP) models provide a unified framework for modeling and describing sequential decision making problems that arise in engineering, economics, and computer science. However, when the underlying problem is modeled by MDPs, there is a typical exponential growth in the size of the resultant MDP model with the size of the original problem, which makes practical solution of the MDP models intractable, especially for large problems. Moreover, for complex systems, it is often the case that some of the parameters of the MDP models cannot be obtained in a feasible way, but only simulation samples are available. In the first part of this thesis, we develop two sampling/simulation-based numerical algorithms to address the computational difficulties arising from these settings. The proposed algorithms have somewhat different emphasis: one algorithm focuses on MDPs with large state spaces but relatively small action spaces, and emphasizes on the efficient allocation of simulation samples to find good value function estimates, whereas the other algorithm targets problems with large action spaces but small state spaces, and invokes a population-based approach to avoid carrying out an optimization over the entire action space. We study the convergence properties of these algorithms and report on computational results to illustrate their performance.
The second part of this thesis is devoted to the development of a general framework called Model Reference Adaptive Search (MRAS) for solving global optimization problems. The method iteratively updates a parameterized probability distribution on the solution space, so that the sequence of candidate solutions generated from this distribution will converge asymptotically to the global optimum. We provide a particular instantiation of the framework and establish its convergence properties in both continuous and discrete domains. In addition, we explore the relationship between the recently proposed Cross-Entropy (CE) method and MRAS, and show that the model reference framework can also be used to describe the CE method and study its properties. Finally, we formally discuss the extension of the MRAS framework to stochastic optimization problems and carry out numerical experiments to investigate the performance of the method.