The Study Of Theories And Algorithms For A Class Of Nonconvex Optimization Problems

This thesis studies a class of nonconvex optimization problems with special structures.We first consider a special nonconvex quartic-quadratic minimization problem over a single spherical constraint,which includes the discretized energy functional minimization problem of non-rotating Bose-Einstein condensates(BECs)as one of the important applications.Such a problem is studied by exploiting its characterization as a nonlinear eigenvalue problem with eigenvector nonlinearity(NEPv).We show that the NEPv has a unique nonnegative eigenvector,which is actually positive,corresponding to the smallest nonlinear eigenvalue of NEPv,and is exactly the global minimizer to the nonconvex optimization problem.Secondly,with these properties,we obtain that any algorithm converging to the nonnegative stationary point of this optimization problem finds its global optimum,such as the regularized Newton method.In particular,we obtain the convergence to the global optimum of the inexact alternating direction method of multipliers for this problem.Thirdly,according to the equivalence between the positive eigenvector of NEPv and the global optimum,we use two Newton-based methods to compute the positive eigenvector of NEPv.The first method comes from the Newton-Noda iteration for saturable nonlinear Schr ¨odinger equations proposed by Ching-Sung Liu,which can be transferred to the NEPv we consider.The second method combines the idea of the root-finding methods and the idea of Newton method,in which,each subproblem involving block tridiagonal linear systems can be solved easily for practical applications like solving discretized BECs problems.Numerical experiments are provided to support our theoretical results.Finally,we give some sufficient conditions,such that the results about the global optimum and the efficiency of algorithms can be extended to more general nonconvex optimization problems and the NEPv with generally locally nonlinearities accordingly.In particular,we apply our theories to the discretized nonlinear saturable energy functional minimization problem as an example.