The high-level motivation of the research is decision making via learning and prediction, mainly the decision making problems that can be formulated as continuous optimization, where applications can be found in machine learning and statistical learning. In these learning problems, aim to represent the data with certain explicit mathematical model (function) which can be used to predict future behavior later on. Learning the parameters in such mathematical model to improve how good it can reflect the data behavior can naturally be formulated as an (continuous) optimization problem. Solving this optimization problem (approximately) can ensure the quality of the model and that it provides prediction that can be trusted. The methodology undertaken in the research is the design of efficient iterative algorithms to solve the aforementioned optimization problem. The efficiency of an algorithm here is mainly characterized theoretically, where an upper bound of the total computational effort required to obtain an approximated solution is established. The total computational effort is usually measured as the total number of iterations in the context of iterative (approximated) algorithms, which is commonly referred to as “iteration complexity”.
As a specific branch in mathematical programming and an extension of the above-mentioned optimization, I’m particularly interested in a mathematical model so-called “variational inequalities” (or simply VI). In VI, find a solution in a given convex set, where a given mapping of that solution forms acute angles with all the directions in this set with such solution as the origin. While optimization is mainly used to model the decision(s) of a single decision maker, VI is capable of modeling two or more decision makers’ (optimal) decisions at the same time due to its general formulation. Therefore, it is often used to model and predict the static state behavior of a system where multiple agents make optimal decisions selfishly. The applications naturally come from strategic games such as two-player zero-sum game and multi-player general-sum game, and further extends to general equilibrium problems arising from economics and applied engineering. This research mainly focuses on developing iterative algorithms for VI problems of under various settings or with different structures and establishing theoretical performance guarantee for these algorithms in terms of iteration complexity bounds.
