Convergence of quasi-Newton matrices generated by the symmetric rank one update

Quasi-Newton algorithms for unconstrained nonlinear minimization generate a sequence of matrices that can be considered as approximations of the objective function second derivatives. This paper gives conditions under which these approximations can be proved to converge globally to the true Hessian matrix, in the case where the Symmetric Rank One update formula is used. The rate of convergence is also examined and proven to be improving with the rate of convergence of the underlying iterates. The theory is confirmed by some numerical experiments that also show the convergence of the Hessian approximations to be substantially slower for other known quasi-Newton formulae.The work of this author was supported by the National Sciences and Engineering Research Council of Canada, and by the Information Technology Research Centre, which is funded by the Province of Ontario.     

Symmetric minimum-norm updates for use in gibbs free energy calculations

This paper examines a type of symmetric quasi-Newton update for use in nonlinear optimization algorithms. The updates presented here impose additional properties on the Hessian approximations that do not result if the usual quasi-Newton updating schemes are applied to certain Gibbs free energy minimization problems. The updates derived in this paper are symmetric matrices that satisfy a given matrix equation and are least squares solutions to the secant equation. A general representation for this class of updates is given. The update in this class that has the minimum weighted Frobenius norm is also presented. This work was done at Sandia National Laboratories and supported by the US Dept. of Energy under contract no. DE-AC04-76DP00789.     

A simulated annealing based solution approach for the two-layered location registration and paging areas partitioning problem in cellular mobile networks

This paper presents a mathematical model and simulated annealing based solution approach for finding optimal location updates and paging area configuration for mobile communication networks. We use a two-layered zone-based location registration and paging scheme in which the costs of location updates and paging signaling traffic are reduced by introducing a two-step paging process. The location updates and paging procedures in a two-layered scheme are first described, and an approximation of the measure required for calculating the paging-related signaling volume is provided based on assumptions of cell shapes and mobile stations’ movement patterns. A simulated annealing (SA)-based solution method is devised along with a greedy heuristic, and computational experiments are conducted to illustrate the superiority of the proposed SA-based method over other solution methods.     

Consistent incremental approximation of dissipation pseudo-potentials in the variational formulation of thermo-mechanical constitutive updates

In this paper, we detail a consistent approximate expression for incremental dissipation pseudo-potentials which appear in the variational formulation of coupled thermo-mechanical boundary-value problems. We explain why the most intuitive expression does not work in the case of an explicit temperature dependence in the dissipation, and propose an alternative expression ensuring consistent results when reducing the time increment towards zero.     

A numerical investigation of rank-two ellipsoid algorithms for nonlinear programming

We study the performance of some rank-two ellipsoid algorithms when used to solve nonlinear programming problems. Experiments are reported which show that the rank-two algorithms studied are slightly less efficient than the usual rank-one (center-cut) algorithm. Some results are also presented concerning the growth of ellipsoid asphericity in rank-one and rank-two algorithms.     

Nonmonotone BFGS-trained recurrent neural networks for temporal sequence processing

In this paper we propose a nonmonotone approach to recurrent neural networks training for temporal sequence processing applications. This approach allows learning performance to deteriorate in some iterations, nevertheless the network’s performance is improved over time. A self-scaling BFGS is equipped with an adaptive nonmonotone technique that employs approximations of the Lipschitz constant and is tested on a set of sequence processing problems. Simulation results show that the proposed algorithm outperforms the BFGS as well as other methods previously applied to these sequences, providing an effective modification that is capable of training recurrent networks of various architectures.     

A UoI-Optimal Policy for Timely Status Updates with Resource Constraint

Timely status updates are critical in remote control systems such as autonomous driving and the industrial Internet of Things, where timeliness requirements are usually context dependent. Accordingly, the Urgency of Information (UoI) has been proposed beyond the well-known Age of Information (AoI) by further including context-aware weights which indicate whether the monitored process is in an emergency. However, the optimal updating and scheduling strategies in terms of UoI remain open. In this paper, we propose a UoI-optimal updating policy for timely status information with resource constraint. We first formulate the problem in a constrained Markov decision process and prove that the UoI-optimal policy has a threshold structure. When the context-aware weights are known, we propose a numerical method based on linear programming. When the weights are unknown, we further design a reinforcement learning (RL)-based scheduling policy. The simulation reveals that the threshold of the UoI-optimal policy increases as the resource constraint tightens. In addition, the UoI-optimal policy outperforms the AoI-optimal policy in terms of average squared estimation error, and the proposed RL-based updating policy achieves a near-optimal performance without the advanced knowledge of the system model.     

Computing Karmarkar projections quickly

The paper presents a numerical method for computing the projections for Karmarkar's new algorithm for linear programming. The method is simple to implement, fully exploits sparsity, and appears in limited experimentation to have good stability properties. Preliminary numerical experience indicates that the method promises advantages over methods that refactor a matrix at every iteration.Research sponsored by the Air Force Office of Scientific Research, Air Force Systems Command, USAF under Grant AFOSR-86-0170. The US Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon.This work was initiated while the author was at the Graduate School of Administration, University of California, Davis, Davis, California.     

Inertia-preserving secant updates

A class of rank-two, inertia-preserving updates for symmetric matricesH _c is studied. To ensure that inertia is preserved, the updates are chosen to be of the formH ₊=FH _c F ^t, whereF=I+qr ^t, withq andr selected so that the secant equation is satisfied. A characterization is given for all such updates. Using a parameterization of this family of updates, the connection between them and the Broyden class of updates is established. Also, parameter selection criteria that can be used to choose the optimally conditioned update or the update closest to the SR1 update are discussed.The work of the first author was partially supported by AFOSR Grant 84-0326. The work of the second author was partially supported by NSF Grant EAR-82-18743.