A Riemannian BFGS Method For Solving Under-Determinded Nonlinear Equations

Under-determined nonlinear equations arise from many important realistic problems,such as parameter-dependent systems,dynamical systems with periodic solutions,constrained optimization problems,spherical design,nonlinear eigenvalue problems,the manipulation of biological signaling pathways,and so on.For di?erent types of underdetermined nonlinear equations defined between Euclidean spaces,many scholars have proposed a variety of di?erent iterative algorithms to solve them,such as quasi-Newton method,Newton-like method,interior point method,and inexact Newton method,and so on.In this thesis,we consider the problem of solving under-determined nonlinear equations defined between general Riemannian manifolds and Euclidean spaces,which has important applications in the field of inverse eigenvalue problems.To design an algorithms with global convergence property,we transform this problem as an equivalent least squares problem defined on Riemannian manifolds.To solve this equivalent problem,we propose a Riemannian line search type optimization algorithm.At each iteration,we first construct a quasi-Newton equation using a special BFGS format,then calculate the quasi-Newton direction,a special descending direction is constructed by a linear combination of the quasi-Newton direction with the negative Riemann gradient direction of the objective function,a Riemann Armijo-type backtracking line search is applied to generate new iterative point in this direction.Under the assumption that the map is continuously di?erentiable,the algorithm has the property of global convergence,that is,any accumulation point of the iterative sequence generated by the algorithm is a stationary point of the objective function.Based on a special boundedness condition of the adjoint operator of the di?erential of the map and the Lipschitz condition of the Riemann gradient of the objective function,we show that the algorithm has a local linear convergence speed.Further,under some additional assumptions,the algorithm has a local superlinear convergence speed.Finally,numerical experiments are listed to compare the newly designed algorithm with some existing algorithms,so as to visually illustrate the stability and e?ectiveness of the algorithm in solving under-determined nonlinear equations.