Robust conic quadratic programming applied to quality improvement -a robustification of CMARS

Özmen, Ayşe
In this thesis, we study and use Conic Quadratic Programming (CQP) for purposes of operational research, especially, for quality improvement in manufacturing. In previous works, the importance and benefit of CQP in this area became already demonstrated. There, the complexity of the regression method Multivariate Adaptive Regression Spline (MARS), which especially means sensitivity with respect to noise in the data, became penalized in the form of so-called Tikhonov regularization, which became expressed and studied as a CQP problem. This was leading to the new method CMARS; it is more model-based and employs continuous, actually, well-structured convex optimization which enables the use of Interior Point Methods and their codes such as MOSEK. In this study, we are generalizing the regression problem by including uncertainty in the model, especially, in the input data, too. CMARS, recently developed as an alternative method to MARS, is powerful in overcoming complex and heterogeneous data. However, for MARS and CMARS method, data are assumed to contain fixed variables. In fact, data include noise in both output and input variables. Consequently, optimization problem’s solutions can show a remarkable sensitivity to perturbations in the parameters of the problem. In this study, we include the existence of uncertainty in the future scenarios into CMARS and robustify it with robust optimization which is dealt with data uncertainty. That kind of optimization was introduced by Aharon Ben-Tal and Arkadi Nemirovski, and used by Laurent El Ghaoui in the area of data mining. It incorporates various kinds of noise and perturbations into the programming problem. This robustification of CQP with robust optimization is compared with previous contributions that based on Tikhonov regularization, and with the traditional MARS method.