Adaptive Constraint Reduction for Convex Quadratic Programming and Training Support Vector Machines

Title of dissertation: Adaptive Constraint Reduction for Convex Quadratic Programming and Training Support Vector Machines Jin Hyuk Jung, Doctor of Philosophy, 2008 Dissertation directed by: Professor Dianne P. O’Leary Department of Computer Science Convex quadratic programming (CQP) is an optimization problem of minimizing a convex quadratic objective function subject to linear constraints. We propose an adaptive constraint reduction primal-dual interior-point algorithm for convex quadratic programming with many more constraints than variables. We reduce the computational effort by assembling the normal equation matrix with a subset of the constraints. Instead of the exact matrix, we compute an approximate matrix for a well chosen index set which includes indices of constraints that seem to be most critical. Starting with a large portion of the constraints, our proposed scheme excludes more unnecessary constraints at later iterations. We provide proofs for the global convergence and the quadratic local convergence rate of an affine scaling variant. A similar approach can be applied to Mehrotra’s predictor-corrector type algorithms. An example of CQP arises in training a linear support vector machine (SVM), which is a popular tool for pattern recognition. The difficulty in training a support vector machine (SVM) lies in the typically vast number of patterns used for the training process. In this work, we propose an adaptive constraint reduction primaldual interior-point method for training the linear SVM with l1 hinge loss. We reduce the computational effort by assembling the normal equation matrix with a subset of well-chosen patterns. Starting with a large portion of the patterns, our proposed scheme excludes more and more unnecessary patterns as the iteration proceeds. We extend our approach to training nonlinear SVMs through Gram matrix approximation methods. Promising numerical results are reported. Adaptive Constraint Reduction for Convex Quadratic Programming and Training Support Vector Machines by Jin Hyuk Jung Dissertation submitted to the Faculty of the Graduate School of the University of Maryland, College Park in partial fulfillment of the requirements for the degree of Doctor of Philosophy 2008 Advisory Commmittee: Professor Dianne P. O’Leary, Chair/Advisor Professor Kyu Yong Choi Professor Hanan Samet Professor G. W. Stewart Professor André L. Tits c © Copyright by Jin Hyuk Jung 2008