Xin-long Luo

Xin-long Luo, Hang Xiao, Sen Zhang

The global minimum point of an optimization problem is of interest in engineering fields and it is difficult to be found, especially for a nonconvex large-scale optimization problem. In this article, we consider a new memetic algorithm for this problem. That is to say, we use the continuation Newton method with the deflation technique to find multiple stationary points of the objective function and use those found stationary points as the initial seeds of the evolutionary algorithm, other than the random initial seeds of the known evolutionary algorithms. Meanwhile, in order to retain the usability of the derivative-free method and the fast convergence of the gradient-based method, we use the automatic differentiation technique to compute the gradient and replace the Hessian matrix with its finite difference approximation. According to our numerical experiments, this new algorithm works well for unconstrained optimization problems and finds their global minima efficiently, in comparison to the other representative global optimization methods such as the multi-start methods (the built-in subroutine GlobalSearch.m of MATLAB R2021b, GLODS and VRBBO), the branch-and-bound method (Couenne, a state-of-the-art open-source solver for mixed integer nonlinear programming problems), and the derivative-free algorithms (CMA-ES and MCS).

Xin-long Luo, Yi-yan Yao

In this article, we consider the primal-dual path-following method and the trust-region updating strategy for the standard linear programming problem. For the rank-deficient problem with the small noisy data, we also give the preprocessing method based on the QR decomposition with column pivoting. Then, we prove the global convergence of the new method when the initial point is strictly primal-dual feasible. Finally, for some rank-deficient problems with or without the small noisy data from the NETLIB collection, we compare it with other two popular interior-point methods, i.e. the subroutine pathfollow.m and the built-in subroutine linprog.m of the MATLAB environment. Numerical results show that the new method is more robust than the other two methods for the rank-deficient problem with the small noise data.

Yiyan Yao, Xin-long Luo

In this paper, a method of improving vertical positioning accuracy with the Global Positioning System (GPS) information and barometric pressure values is proposed. Firstly, we clear null values for the raw data collected in various environments, and use the 3$σ$-rule to identify outliers. Secondly, the Weighted Multinomial Logistic Regression (WMLR) classifier is trained to obtain the predicted altitude of outliers. Finally, in order to verify its effect, we compare the MLR method, the WMLR method, and the Support Vector Machine (SVM) method for the cleaned dataset which is regarded as the test baseline. The numerical results show that the vertical positioning accuracy is improved from 5.9 meters (the MLR method), 5.4 meters (the SVM method) to 5 meters (the WMLR method) for 67% test points.

Xin-long Luo, Jia-hui Lv, Geng Sun

This paper investigates the localization problem of high-speed high-altitude unmanned aerial vehicle (UAV) with a monocular camera and inertial navigation system. It proposes a navigation method utilizing the complementarity of vision and inertial devices to overcome the singularity which arises from the horizontal flight of UAV. Furthermore, it modifies the mathematical model of localization problem via separating linear parts from nonlinear parts and replaces a nonlinear least-squares problem with a linearly equality-constrained optimization problem. In order to avoid the ill-condition property near the optimal point of sequential unconstrained minimization techniques(penalty methods), it constructs a semi-implicit continuous method with a trust-region technique based on a differential-algebraic dynamical system to solve the linearly equality-constrained optimization problem. It also analyzes the global convergence property of the semi-implicit continuous method in an infinity integrated interval other than the traditional convergence analysis of numerical methods for ordinary differential equations in a finite integrated interval. Finally, the promising numerical results are also presented.

Xin-Long Luo, Geng Sun

This article explores some geometric and algebraic properties of the dynamical system which is represented by matrix differential equations arising from inertial navigation problems, such as the symplecticity and the orthogonality. Furthermore, it extends the applicable fields of symplectic geometric algorithms from the even dimensional Hamiltonian system to the odd dimensional dynamical system. Finally, some numerical experiments are presented and illustrate the theoretical results of this paper.