Recursive Algorithms and Regime Switching Models for Stochastic Optimization

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This research project is to design stochastic approximation and optimization algorithms, and to develop regime switching dynamic system models for solving problems arising from existing and emerging applications. Several stochastic iterative algorithms featuring non-smooth dynamics or multi-time scales, or leading to non-autonomous limit ordinary differential equations or limit systems given by differential inclusions are proposed. Their asymptotic properties such as convergence and rates of convergence will be examined through the associated dynamic systems. Variants, improvements, and efficient procedures will be developed. The proposed algorithms for tracking time-varying parameters will lead to limit regime-switching ordinary and stochastic differential equations, which are not obtainable using the existing methods in stochastic approximation. Research on regime switching models modulated by Markov chains for both discrete-time and continuous-time systems will be conducted. Aiming at reducing complexity, hierarchical structure of the dynamic systems and time-scale separation will be used. Properties of these systems will be investigated through aggregation and decomposition methods and singular perturbation methodology. These properties will further be used to guide the design and development of procedures for optimization and control of dynamic systems.To meet the growing demand for efficient computational algorithms and methods for optimal decision making in wireless communications, manufacturing systems, financial engineering, signal processing, and queueing networks, this project aims to design mathematical models useful for existing and emerging applications, and to develop algorithms applicable to such problems as CDMA communication systems, production planning, mean-variance portfolio selections, and communication networks. To accommodate systems in the real world, shifts in regimes need to be taken into consideration. Take for instance, the situation in a stock market, the market parameters depend on the market mode that jumps between the "bullish" and "bearish" states. In these states, the corresponding market parameters are quite different resulting in markedly different behavior. In addition, the stock market also exhibits multi-time-scale structure. Such models and time-scale separations also appear in communication networks, production planning and other applications. The proposed research work aims to design feasible models and procedures for the aforementioned systems. The proposed research will yield new insight, and advance state of the art of stochastic optimization methods.