METAMODEL-BASED GLOBAL OPTIMIZATION AND GENERALIZED NASH EQUILIBRIUM PROBLEMS

Abstract

This thesis consists of two topics: metamodel-based global optimization algorithms of black-box problems and global sensitivity analysis on generalized Nash equilibriums.

Metamodels have been widely used in optimizing many complex design problems from natural science and engineering. In this thesis, we first propose a new adaptive framework for metamodel-based global optimization of computationally intensive blackbox problems. A new switching criterion between global search and local search is proposed based on their potential performance. Numerical results demonstrate that algorithms based on this framework well balance global search and local search, resulting in better performance over existing benchmark methods. Next, we focus on developing global optimization algorithms using radial basis function (RBF) metamodels. By exploiting the structure of RBF models, we develop several techniques to greatly improve the efficiency and the robustness of RBF models in global optimization. They include a rank-one update method to construct RBF models, a closed-form leave-one-out cross validation error of RBF models to be used for model selection, and a new flexible and efficient prediction error estimate of RBF models. In addition, we propose a novel sampling criterion called the weighted improvement, which can balance between global search and local search with a tunable parameter. Incorporating all these results into the proposed framework yields the new adaptive radial basis function method, WIRBF. Extensive numerical experiments demonstrate that the algorithm outperforms several benchmark algorithms. Lastly, we investigate an extension of RBF interpolation models to the problem of stochastic simulations with heteroscedastic noise, named the regularized RBF model. Our mathematical analysis on the proposed model unveils connections with the stochastic kriging model from a novel perspective.

The second topic concerns global sensitivity analysis on generalized Nash equilibrium for an economic network of interlinked oligopolistic markets, where firms compete by selling a homogeneous product. The generalized Nash equilibrium problem (GNEP), originating from economic science, has been widely used in various fields. GNEP allows each player¿s decision set dependent on other players¿ decisions, which endows GNEP more descriptive power to model competitions among self-interested players. We prove the existence and uniqueness of the market equilibrium. Under a set of general economic assumptions, we develop a search algorithm to perform global sensitive analysis on the equilibrium with respect to certain parameters, which can be useful for firms and market authorities in their decision-making. The significance of this result is that it represents the equilibrium as a function of certain parameters rather than a single equilibrium for fixed parameters.