Continuous second order minimization method with variable metric projection operator

The paper examines a new continuous projection second order method of minimization of continuously Frechet differentiable convex functions on the convex closed simple set in separable, normed Hilbert space with variable metric. This method accelerates common continuous projection minimization method by means of quasi-Newton matrices. In the method, apart from variable metric operator, vector of search direction for motion to minimum, constructed in auxiliary extrapolated point, is used. By other word, complex continuous extragradient variable metric method is investigated. Short review of allied methods is presented and their connections with given method are indicated. Also some auxiliary inequalities are presented which are used for theoretical reasoning of the method. With their help, under given supplemental conditions, including requirements on operator of metric and on method parameters, convergence of the method for convex smooth functions is proved. Under conditions completely identical to those in convergence theorem, without additional requirements to the function, estimates of the method's convergence rate are obtained for convex smooth functions. It is pointed out, that one must execute computational implementation of the method by means of numerical methods for ODEs solution and by taking into account the conditions of proved theorems.

Dynamic Optimization of Multibody System Using Multithread Calculations and a Modification of Variable Metric Method

The paper presents dynamic optimization methods used to calculate the optimal braking torques applied to wheels of an articulated vehicle in the lane following/changing maneuver in order to prevent a vehicle rollover. In the case of unforeseen obstacles, the nominal trajectory of the articulated vehicle has to be modified, in order to avoid collisions. Computing the objective function requires an integration of the equation of motions of the vehicle in each optimization step. Since it is rather time-consuming, a modification of the classical gradient method—variable metric method (VMM)—was proposed by implementing parallel computing on many cores of computing unit processors. Results of optimization calculations providing stable motion of a vehicle while performing a maneuver and a description and results of parallel computing are presented in this paper.

Improved DDA Method Based on Explicitly Solving Contact Constraints

This paper proposes an improved DDA method based on explicitly solving contact constraints. The potential energy function generated by contacting, which contains only displacement variables as an unknown, is deduced based on the approximated step function and Lagrange interpolation, and the displacement variables and contact constraints are obtained via the variable metric method by analyzing the potential energy extremum. There is no need to conduct the open-close iteration during the process of calculation. The improved DDA method based on explicitly solving contact constraints has high precision and a more stable and more robust computational convergence. The accuracy and iterative stability of the improved DDA method are verified using two numerical examples.

Optimal Algorithms and the BFGS Updating Techniques for Solving Unconstrained Nonlinear Minimization Problems

To solve an unconstrained nonlinear minimization problem, we propose an optimal algorithm (OA) as well as a globally optimal algorithm (GOA), by deflecting the gradient direction to the best descent direction at each iteration step, and with an optimal parameter being derived explicitly. An invariant manifold defined for the model problem in terms of a locally quadratic function is used to derive a purely iterative algorithm and the convergence is proven. Then, the rank-two updating techniques of BFGS are employed, which result in several novel algorithms as being faster than the steepest descent method (SDM) and the variable metric method (DFP). Six numerical examples are examined and compared with exact solutions, revealing that the new algorithms of OA, GOA, and the updated ones have superior computational efficiency and accuracy.