Focusing on modeling and optimization for systems under uncertainty, this proposal consists of four parts. Part I proposes two types of algorithms. The first one is an approximation of an analog diffusion machine; the second one also takes measurement errors into consideration. Our goal is to develop asymptotic properties of such algorithms. By using weak convergence methods, suitably scaled sequences will be shown to converge to appropriate diffusions. Part II treats a class of hybrid models. Approximation schemes for systems involving singularly perturbed Markov chains with weak and strong interactions will be developed, which are useful for natural time-scale separation and reduction of complexity for large-scale systems. Part III investigates asymptotic properties of solutions of Cauchy problems arising from null-recurrent diffusions. Our focus is on obtaining convergence and rate of convergence of the solutions. One of the primary motivations comes from the investigation of singularly perturbed systems. The results will be useful to the ever expanding applications in optimization, controlled Markov systems, hierarchical decision making, production planning, telecommunication, queueing networks, and system reliability. Part IV models single-machine scheduling problems under random processing time, and/or under random machine breakdowns and repairs, and/or subject to random compression of processing times. Our objectives are to develop feasible models and to obtain optimal scheduling policies for the underlying systems. These results will allow us to design scheduling models and strategies for more complex jobshops by considering integrated processes as single-machine systems.
Nontechnical explanation:
To bridge the gap between theory and applications, this research project includes three components: modeling, asymptotic analysis, and simulation. The ultimate goals are to provide useful models, to investigate their basic properties, and to develop sound and feasible algorithms. Part 1 proposes two classes of algorithms with applications to machine learning, image segmentation, and various global optimization tasks. To meet the increasing demand on robust design and control of systems in speech and pattern recognition, signal processing, telecommunications, and manufacturing, Part 2 aims to reduce the complexity of a large-scale system of complex structure by using a simple system via approximation schemes. The origin of the planned work for Part 3 stems from the effort of modeling uncertainties due to random influence such as demands for a product in a manufacturing system or fluctuation in the stock market. To control the underlying system, it is imperative to understand the system's long-term behavior, which is our primary goal. In production planning, it is vital to provide good strategy in sequencing the parts to be processed by the machines. Part 4 proposes single-machine scheduling models in uncertain environment. The proposed work aims to develop optimal scheduling policies. The overall planned work represents a continuation of the PI's recent preliminary exploration in these areas. It is expected that the results will be applicable in the further improvements of optimization methods.
