In finance, the implied volatility surface is plotted against strike price and time to maturity. The shape of this volatility surface can be identified by fitting the model to what is actually observed in the market. The metric that is used to measure the discrepancy between the model and the market is usually defined by a mean squares of error of the model prices to the market prices. A regularization term can be added to this error metric to make the solution possess some desired properties. The discrepancy that we want to minimize is usually a highly nonlinear function of a set of model parameters with the regularization term. Typically monotonic decreasing algorithm is adopted to solve this minimization problem. Steepest descent or Newton type algorithms are two iterative methods but they are local, i.e., they use derivative information around the current iterate to find the next iterate. In order to ensure convergence, line search and trust region methods are two widely used globalization techniques. Motivated by the simplicity of Barzilai-Borwein method and the convergence properties brought by globalization techniques, we propose a new Scaled Gradient (SG) method for minimizing a differentiable function plus an L1-norm. This non-monotone iterative method only requires gradient information and safeguarded Barzilai-Borwein steplength is used in each iteration. An adaptive line search with the Armijo-type condition check is performed in each iteration to ensure convergence. Coleman, Li and Wang proposed another trust region approach in solving the same problem. We give a theoretical proof of the convergence of their algorithm. The objective of this thesis is to numerically investigate the performance of the SG method and establish global and local convergence properties of Coleman, Li and Wang’s trust region method proposed in [26]. Some future research directions are also given at the end of this thesis.

In finance, the implied volatility surface is plotted against strike price and time to maturity. The shape of this volatility surface can be identified by fitting the model to what is actually observed in the market. The metric that is used to measure the discrepancy between the model and the market is usually defined by a mean squares of error of the model prices to the market prices. A regularization term can be added to this error metric to make the solution possess some desired properties. The discrepancy that we want to minimize is usually a highly nonlinear function of a set of model parameters with the regularization term. Typically monotonic decreasing algorithm is adopted to solve this minimization problem. Steepest descent or Newton type algorithms are two iterative methods but they are local, i.e., they use derivative information around the current iterate to find the next iterate. In order to ensure convergence, line search and trust region methods are two widely used globalization techniques. Motivated by the simplicity of Barzilai-Borwein method and the convergence properties brought by globalization techniques, we propose a new Scaled Gradient (SG) method for minimizing a differentiable function plus an L1-norm. This non-monotone iterative method only requires gradient information and safeguarded Barzilai-Borwein steplength is used in each iteration. An adaptive line search with the Armijo-type condition check is performed in each iteration to ensure convergence. Coleman, Li and Wang proposed another trust region approach in solving the same problem. We give a theoretical proof of the convergence of their algorithm. The objective of this thesis is to numerically investigate the performance of the SG method and establish global and local convergence properties of Coleman, Li and Wang’s trust region method proposed in [26]. Some future research directions are also given at the end of this thesis.

In this paper we combine ideas from cutting plane and interior point methods in order to solve variational inequality problems efficiently. In particular, we introduce a general framework that incorporates nonlinear as well as linear "smarter" cuts. These cuts utilize second order information on the problem through the use of a gap function. We establish convergence as well as complexity results for this framework. Moreover, in order to devise more practical methods, we consider an affine scaling method as it applies to symmetric, monotone variationalinequality problems and demonstrate its convergence. Finally, in order to further improve the computational efficiency of the methods in this paper, we combine the cutting plane approach with the affine scaling approach.