Granular Materials, Contact Problems, DVI, MPRGP

Active-set algorithm for solving inner optimization problem in multi-body dynamics is presented. The efficiency of our algorithm is demonstrated on the solution of simple simulation with thousands of moving spherical particles and static box obstacles. We discuss the solvability and the uniqueness of solution of the problem and the influence of solution to resulting velocity during time-stepping schema

A two-phase gradient method for quadratic programming problems with a single linear constraint and bounds on the variables

We propose a gradient-based method for quadratic programming problems with a single linear constraint and bounds on the variables. Inspired by the GPCG algorithm for bound-constrained convex quadratic programming [J.J. Mor\'e and G. Toraldo, SIAM J. Optim. 1, 1991], our approach alternates between two phases until convergence: an identification phase, which performs gradient projection iterations until either a candidate active set is identified or no reasonable progress is made, and an unconstrained minimization phase, which reduces the objective function in a suitable space defined by the identification phase, by applying either the conjugate gradient method or a recently proposed spectral gradient method. However, the algorithm differs from GPCG not only because it deals with a more general class of problems, but mainly for the way it stops the minimization phase. This is based on a comparison between a measure of optimality in the reduced space and a measure of bindingness of the variables that are on the bounds, defined by extending the concept of proportioning, which was proposed by some authors for box-constrained problems. If the objective function is bounded, the algorithm converges to a stationary point thanks to a suitable application of the gradient projection method in the identification phase. For strictly convex problems, the algorithm converges to the optimal solution in a finite number of steps even in case of degeneracy. Extensive numerical experiments show the effectiveness of the proposed approach.Comment: 30 pages, 17 figure

Minimizing Nonconvex Quadratic Functions Subject to Bound Constraints

We present an active-set algorithm for finding a local minimizer to a nonconvex bound-constrained quadratic problem. Our algorithm extends the ideas developed by Dost al and Sch oberl that is based on the linear conjugate gradient algorithm for (approximately) solving a linear system with a positive-de finite coefficient matrix. This is achieved by making two key changes. First, we perform a line search along negative curvature directions when they are encountered in the linear conjugate gradient iteration. Second, we use Lanczos iterations to compute approximations to leftmost eigen-pairs, which is needed to promote convergence to points satisfying certain second-order optimality conditions. Preliminary numerical results show that our method is e fficient and robust on nonconvex bound-constrained quadratic problems