An infeasible-interior-point predictor-corrector algorithm for theP *-geometric LCP

AP _*-geometric linear complementarity problem (P _*GP) as a generalization of the monotone geometric linear complementarity problem is introduced. In particular, it contains the monotone standard linear complementarity problem and the horizontal linear complementarity problem. Linear and quadratic programming problems can be expressed in a “natural” way (i.e., without any change of variables) asP _*GP. It is shown that the algorithm of Mizunoet al. [6] can be extended to solve theP _*GP. The extended algorithm is globally convergent and its computational complexity depends on the quality of the starting points. The algorithm is quadratically convergent for problems having a strictly complementary solution. The work of F. A. Potra was supported in part by NSF Grant DMS 9305760     

A linear assignment approach for the least-squares protein morphing problem

This work addresses the computation of free-energy differences between protein conformations by using morphing (i.e., transformation) of a source conformation into a target conformation. To enhance the morphing procedure, we employ permutations of atoms: we seek to find the permutation σ that minimizes the mean-square distance traveled by the atoms. Instead of performing this combinatorial search in the space of permutations, we show that the best permutation can be found by solving a linear assignment problem. We demonstrate that the use of such optimal permutations significantly improves the efficiency of the free-energy computation.     

Optimal control of systems with discontinuous differential equations

In this paper we discuss the problem of verifying and computing optimal controls of systems whose dynamics is governed by differential systems with a discontinuous right-hand side. In our work, we are motivated by optimal control of mechanical systems with Coulomb friction, which exhibit such a right-hand side. Notwithstanding the impressive development of nonsmooth and set-valued analysis, these systems have not been closely studied either computationally or analytically. We show that even when the solution crosses and does not stay on the discontinuity, differentiating the results of a simulation gives gradients that have errors of a size independent of the stepsize. This means that the strategy of “optimize the discretization” will usually fail for problems of this kind. We approximate the discontinuous right-hand side for the differential equations or inclusions by a smooth right-hand side. For these smoothed approximations, we show that the resulting gradients approach the true gradients provided that the start and end points of the trajectory do not lie on the discontinuity and that Euler’s method is used where the step size is “sufficiently small” in comparison with the smoothing parameter. Numerical results are presented for a crude model of car racing that involves Coulomb friction and slip showing that this approach is practical and can handle problems of moderate complexity.     

Parallel distributed-memory simplex for large-scale stochastic LP problems

We present a parallelization of the revised simplex method for large extensive forms of two-stage stochastic linear programming (LP) problems. These problems have been considered too large to solve with the simplex method; instead, decomposition approaches based on Benders decomposition or, more recently, interior-point methods are generally used. However, these approaches do not provide optimal basic solutions, which allow for efficient hot-starts (e.g., in a branch-and-bound context) and can provide important sensitivity information. Our approach exploits the dual block-angular structure of these problems inside the linear algebra of the revised simplex method in a manner suitable for high-performance distributed-memory clusters or supercomputers. While this paper focuses on stochastic LPs, the work is applicable to all problems with a dual block-angular structure. Our implementation is competitive in serial with highly efficient sparsity-exploiting simplex codes and achieves significant relative speed-ups when run in parallel. Additionally, very large problems with hundreds of millions of variables have been successfully solved to optimality. This is the largest-scale parallel sparsity-exploiting revised simplex implementation that has been developed to date and the first truly distributed solver. It is built on novel analysis of the linear algebra for dual block-angular LP problems when solved by using the revised simplex method and a novel parallel scheme for applying product-form updates.     

A preconditioning technique for Schur complement systems arising in stochastic optimization

Deterministic sample average approximations of stochastic programming problems with recourse are suitable for a scenario-based parallelization. In this paper the parallelization is obtained by using an interior-point method and a Schur complement mechanism for the interior-point linear systems. However, the direct linear solves involving the dense Schur complement matrix are expensive, and adversely affect the scalability of this approach. We address this issue by proposing a stochastic preconditioner for the Schur complement matrix and by using Krylov iterative methods for the solution of the dense linear systems. The stochastic preconditioner is built based on a subset of existing scenarios and can be assembled and factorized on a separate process before the computation of the Schur complement matrix finishes on the remaining processes. The expensive factorization of the Schur complement is removed from the parallel execution flow and the scaling of the optimization solver is considerably improved with this approach. The spectral analysis indicates an exponentially fast convergence in probability to 1 of the eigenvalues of the preconditioned matrix with the number of scenarios incorporated in the preconditioner. Numerical experiments performed on the relaxation of a unit commitment problem show good performance, in terms of both the accuracy of the solution and the execution time.     

A fast contraction mapping for solving multibody systems

The simulation of complex multibody systems with contacts and friction requires a fast and robust solver for complementarity problems. This work presents an efficient method which can solve large cone-complementarity problems by means of a fixed point iteration. Our method performs like a contractive mapping, providing a monotonic approximation to the exact solution. The algorithm features high computational performance even if thousands of unilateral constraints are added to the system. Also, this scheme fits well in a real-time simulation context because it can be terminated prematurely. As a benchmark for testing the method in challenging situations, we propose the simulation of the granular flow in a fourth-generation bebble-bed nuclear reactor, including 150'000 rigid bodies and more than a million of constraint multipliers. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimization-based simulation of nonsmooth rigid multibody dynamics

We present a time-stepping method to simulate rigid multibody dynamics with inelastic collision, contact, and friction. The method progresses with fixed time step without backtracking for collision and solves at every step a strictly convex quadratic program. We prove that a solution sequence of the method converges to the solution of a measure differential inclusion. We present numerical results for a few examples, and we illustrate the difference between the results from our scheme and previous, linear-complementarity-based time-stepping schemes.