Lower-bound Computation Research Articles

Robust Semisupervised Graph Classifier Learning With Negative Edge Weights

In a semisupervised learning scenario, (possibly noisy) partially observed labels are used as input to train a classifier in order to assign labels to unclassified samples. In this paper, we construct a complete graph-based binary classifier given only samples’ feature vectors and partial labels. Specifically, we first build appropriate similarity graphs with positive and negative edge weights connecting all samples based on internode feature distances. By viewing a binary classifier as a piecewise constant graph signal, we cast classifier learning as a signal restoration problem via a classical maximum a posteriori (MAP) formulation. One unfortunate consequence of negative edge weights is that the graph Laplacian matrix $\mathbf {L}$ can be indefinite, and previously proposed graph-signal smoothness prior $\mathbf {x}^T \mathbf {L}\mathbf {x}$ for candidate signal $\mathbf {x}$ can lead to pathological solutions. In response, we derive a minimum-norm perturbation matrix $\boldsymbol{\Delta }$ that preserves $\mathbf {L}$ 's eigen-structure—based on a fast lower-bound computation of $\mathbf {L}$ 's smallest negative eigenvalue via a novel application of the Haynsworth inertia additivity formula—so that $\mathbf {L}+ \boldsymbol{\Delta }$ is positive semidefinite, resulting in a stable signal prior. Further, instead of forcing a hard binary decision for each sample, we define the notion of generalized smoothness on graphs that promotes ambiguity in the classifier signal. Finally, we propose an algorithm based on iterative reweighted least squares that solves the posed MAP problem efficiently. Extensive simulation results show that our proposed algorithm outperforms both SVM variants and previous graph-based classifiers using positive-edge graphs noticeably.

Numerical Computation of Lower Bounds of Structured Singular Values

In this article we consider the numerical approximation of lower bounds of Structured Singular Values, SSV. The SSV is a wellknown mathematical quantity which is widely used to analyse and syntesize the robust stability and instability analysis of linear feedback systems in control theory. It links a bridge between numerical linear algebra and system theory. The computation of lower bounds of SSV by means of ordinary differential equations based technique is presented. The obtained numerical results for the lower bounds of SSV are compared with the well-known MATLAB function mussv available in MATLAB control toolbox.

A model for multi-class road network recovery scheduling of regional road networks

In this paper, an optimisation model for recovery planning of road networks is presented in which both social and economic resilience is aimed to be achieved. The model is formulated as a bi-level multi-objective discrete network design problem which forms a non-convex mixed integer non-linear problem. Solved by a Branch and Bound method, the solution algorithm employs an outer approximation method to estimate the lower bound of each node in the Branch and Bound search tree. The solution algorithm exploits a unique approach for lower-bound computation dealing with a disrupted multi-class network that may not be able to satisfy the demand between all OD pairs due to damaged links. The model is assessed by applying it on the Sioux Falls network. It is also illustrated how the Pareto-optimal solutions achieved by the multi-objective optimisation can vary depending on the emphasis placed on different classes of vehicles.

Computation of Lower Bounds for a Multiple Depot, Multiple Vehicle Routing Problem With Motion Constraints

This paper considers the problem of planning paths for a collection of identical vehicles visiting a given set of targets, such that the total lengths of their paths are minimum. Each vehicle starts at a specified location (called a depot) and it is required that each target to be on the path of at least one vehicle. The path of every vehicle must satisfy the motion constraints of every vehicle. In this paper, we develop a method to compute lower bound to the minimum total path lengths by relaxing some of the constraints and posing it as a standard multiple traveling salesmen problem (MTSP). A lower bound is often important to ascertain suboptimality bounds for heuristics and for developing stopping criterion for algorithms computing an optimal solution. Simulation results are presented to show that the proposed method can be used to improve the lower bounds of instances with four vehicles and 40 targets by approximately 39%.

A Cluster-on-a-Chip Architecture for High-Throughput Phylogeny Search

In this paper, we describe an FPGA-based coprocessor architecture that performs a high-throughput branch-and-bound search of the space of phylogenetic trees corresponding to the number of input taxa. Our coprocessor architecture is designed to accelerate maximum-parsimony phylogeny reconstruction for gene-order and sequence data and is amenable to both exhaustive and heuristic tree searches. Our architecture exposes coarse-grain parallelism by dividing the search space among parallel processing elements (PEs) and each PE exposes fine-grain memory parallelism for their lower-bound computation, the kernel computation performed by each PE. Inter-PE communication is performed entirely on-chip. When using this coprocessor for maximum-parsimony reconstruction for gene-order data, our coprocessor achieves a 40X improvement over software in search throughput, corresponding to a 14X end-to-end application improvement when including all communication and systems overheads.

Flow Trees: A Lower Bound Computation Tool with Applications to Rearrangeable Multihop Lightwave Network Optimization

This paper presents a new method for computing the lower bounds for multihop network design problems which is particularly well suited to optical networks.More specifically, given N stations each with d transceivers and pairwise average traffic values of the stations, the method provides a lower bound for the combined problem of finding optimum (i) allocation of wavelengths to the stations to determine a configuration, and (ii) routing of the traffic on this configuration while minimizing congestion – defined as the maximum flow assigned on any link.The lower bounds can be computed in time polynomial in the network size. Consequently, the results in this work yield a tool which can be used in (i) evaluating the quality of heuristic design algorithms, and (ii) determining a termination criteria during minimization.The lower bound computation is based on first building flow trees to find a lower bound on the total flow, and then distributing the total flow over the links to minimize the congestion.

Improved lower bounds on the reliability of hypercube architectures

The hypercube topology, also known as the Boolean n-cube, has recently been used for multiprocessing systems. The paper considers two structural-reliability models, namely, terminal reliability (TR) and network reliability (NR), for the hypercube. Terminal (network) reliability is defined as the probability that there exists a working path connecting two (all) nodes. There are no known polynomial time algorithms for exact computation of TR or NR for the hypercube. Thus, lower-bound computation is a better alternative, because it is more efficient computationally, and the system will be at least as reliable as the bound. The paper presents algorithms to compute lower bounds on TR and NR for the hypercube considering node and/or link failures. These algorithms provide tighter bounds for both TR and NR than known results and run in time polynomial in the cube dimension n, specifically, within time O(n/sup 2/). >

Review of Lower-Bound Limit Analysis for Pressure Vessel Intersections

On the basis of a study of several recent papers concerned with the lower-bound computation of the collapse load of pressure vessel intersections, a review is made of the satisfaction, or nonsatisfaction, of the requirements of the lower-bound theorem of limit analysis. It is shown that, whereas for rotationally symmetric structures true lower bounds have been obtained, for a nonsymmetric case such as a right cylinder-cylinder intersection it is difficult to avoid so me approximations. If attempts are to be made to develop general purpose limit analysis programs, the consequences of approximations of the type used so far must be evaluated with care if significant results are to be obtained.

Improving the Computation of Lower Bounds for Optimal Schedules

Ways of decreasing the number of operations needed to compute the lower bounds of optimal schedules, by reducing the number of time intervals that must be considered, are presented. The bounds apply to a system of identical processors executing a partially ordered set of tasks, with known execution times, using a non-preemptive scheduling strategy. In one approach we find that the required number of intervals depends on the graph. In our other approach, which subsumes the first, the number of intervals is decreased to at most min[D2/2, n2], where D is the deadline to complete the tasks and n is the number of tasks. The actual number of intervals for a particular graph can be considerably smaller than this worst case.

Computation of Lower Bounds for Multiprocessor Schedules

A multiprocessing system composed of identical units is considered. This system is executing a set of partially ordered tasks, with known execution times, using a non-preemptive scheduling strategy. Lower bounds on the number of processors required to compute the tasks before a deadline, and on the minimum time to execute the tasks with a fixed number of processors, are of great value for the determination of the corresponding optimal schedules. In this paper, methods for the efficient computation of the lower bounds obtained by Fernindez and Bussell are discussed. Computational improvements for the case of general partial orders are reported, and further reductions of the number of operations are shown to be possible for special graphs (trees, independent chains, independent tasks).