A Complete Solution to Quadratic Programming with One Inequality Quadratic Constraint

Abstract

This dissertation focused on nonconvex quadratic optimization subject to one nonconvex quadratic constraint The problem arises from the trust region method which since 1990 has been the central issue for finding a local minimum of an unconstrained nonlinear programming Traditional Newton’s method has dominated for hundreds of years for the same type of problems but it suffers from being divergent when the initial guess is unluckily far off the target The trust region method however enjoys the global convergence without depending on the location of the initial point The trust region method is often solved in theory by a semi-definite programming reformulation followed by a rank-one decomposition Both are state of the art optimization technique for the most recent decade Nevertheless the SDP relaxation suffers from expensive matrix computation due to the necessity to square the dimension of the original problem Our analysis toward the problem uses the matrix pencil as the tool and solves the problem in the original space Therefore it has the advantage of saving computational cost We also applied the same analytical method to solve the double (energy) well potential problem with a complete characterization to the global optimal set and to the duality The double well problem has appeared in many natural phenomena such as elastic mechanics and van der Waals’ force in chemistry

Date of Award	2018 Feb 7
Original language	English
Supervisor	Ruey-Lin Sheu (Supervisor)