On solving some optimization problems in stochastic and integer programming with applications in finance and banking

First, we proposed a scenario model which minimizes a regret function, and a 2-step approach to solve the scenario model. The 2-step approach generates a policy which is solution and model robust. We implemented the method for two financial models. The first model determines a portfolio which tracks the indices of mortgage backed securities. We used real data to test the 2-step approach. The results show that the portfolio tracks the indices very well and outperforms the indices. The second model is a financial trading model, which is also a multistage full recourse model. We solved it using the 2-step approach with randomly generated data. The model is solved for two different sizes of scenarios. The method had not been tested or implemented before. Second, we developed two models for check sorting and clearing. There was no previous published work done in modelling both sorting and clearing decisions, as far as we know. The two models were tested using three sets of pilot data. We found that the second model is better than the first model. We obtained a data set covering the Chicago area and generated a national data set based on it. We presented computational results for both sets of data and performed some sensitivity analysis. Due to the nature of the model, it is difficult to use the optimal solution if the data change. However, we recommend two possible ways of finding more flexible sorting and clearing decisions. Third, we designed an algorithm to solve a primal relaxation problem for linear and nonlinear mixed integer problems, introduced by Michelon and Maculan (1992) in the linear case, and by Guignard and Contesse (1995) in the nonlinear case. The algorithm combines a simplicial decomposition method and a proximal augmented Lagrangean method. It was tested for both nonlinear and linear capacitated plant location problems, and we found that it was successful in computing optimal Lagrangean bounds for a range of parameter values