Research And Applications Of Continuation Methods For Equality-constrained Optimization

The robot localization and autonomous navigation technology is one of the key technologies of mobile robot perception,and it is a research hotspot for researchers.As a kind of mobile robots,unmanned aerial vehicles(UAVs)have been widely used in military and civilian fields.However,the localization and navigation accuracy of UAVs using pure inertial navigation systems fails to meet the needs of the industry.Thus,this dissertation tries to improve its localization accuracy.Firstly,this dissertation analyzes the problem of the autonomous localization of UAVs,and reconstructs a linearly equality-constrained optimization problem of visual-inertial fusion localization for horizontal flight of UAVs to improve the localization accuracy of UAVs.In order to solve the linearly equality-constrained nonlinear problem efficiently,this dissertation revises the Levenberg-Marquardt method,and gives a continuation method with the trust-region time-stepping scheme(Ptctr).This method constructs a differential-algebraic dynamical system for the linearly equality-constrained nonlinear problem.Then,it utilizes the first-order Rosenbrock continuation method to track the trajectory of the dynamical system and adaptively adjusts the time step based on trust-region updating strategy.When Ptctr solves the linearly equality-constrained problem,it only solves the system of linear equations at every iteration.However,the sequential quadratic programming method(SQP)needs to solve a quadratic programming subproblem at every iteration.In addition,Ptctr is close to the Newton method when the time step is small enough,and it is close to the gradient method when the time step is large enough.Therefore,the number of its iterations does not increase.Consequently,the computational efficiency of Ptctr is higher than that of the SQP method.Numerical results also show that Ptctr is faster than SQP and the traditional dynamic system method(ode15s).The computational time of Ptctr is one fifth of that of SQP,which performs better than ode 15s.Secondly,this dissertation further considers the case of UAVs flying with full degrees of freedom,and constructs a nonlinear optimization problem with orthogonality constraints.In order to solve this problem,the dissertation gives a continuation method(PtcOrth)for the orthogonality constrained optimization problem form the perspective of differential-algebraic equations.This method constructs a differential dynamical system for this problem,and uses a continuation method based on high-order Pade approximation to track the trajectory of the dynamical system of this problem.The continuation method has the property of preserving structure.Then the dissertation compares PtcOrth with OptStiefel,which can solve some classical orthogonality-constrained optimization problems.The numerical results show that PtcOrth has some advantages than OptStiefel in solution accuracy.