Efficient algorithms for globally optimal trajectories

We present serial and parallel algorithms for solving a system of equations that arises from the discretization of the Hamilton-Jacobi equation associated to a trajectory optimization problem of the following type. A vehicle starts at a prespecified point x_o and follows a unit speed trajectory x(t) inside a region in ℛ^m until an unspecified time T that the region is exited. A trajectory minimizing a cost function of the form ∫₀^T r(x(t))dt+q(x(T)) is sought. The discretized Hamilton-Jacobi equation corresponding to this problem is usually solved using iterative methods. Nevertheless, assuming that the function r is positive, we are able to exploit the problem structure and develop one-pass algorithms for the discretized problem. The first algorithm resembles Dijkstra's shortest path algorithm and runs in time O(n log n), where n is the number of grid points. The second algorithm uses a somewhat different discretization and borrows some ideas from a variation of Dial's shortest path algorithm (1969) that we develop here; it runs in time O(n), which is the best possible, under some fairly mild assumptions. Finally, we show that the latter algorithm can be efficiently parallelized: for two-dimensional problems and with p processors, its running time becomes O(n/p), provided that p=O(√n/log n)     

QAOA最大切割问题的类Dijkstra优化及实现

最大切割问题是可以用量子近似优化算法（QAOA）来解决的典型问题,Ansatz线路构造为该算法的重要组成部分。为了减少多种图结构在QAOA中的构造代价和提高其稳定性,从线路的可优化性出发进行分析,结合 Dijkstra算法的点边存放特点,提出了该线路的类Dijkstra优化算法,并将其应用于QAOA最大切割问题。使用qiskit量子框架来模拟优化算法的正确性,并用IBM Quantum composer的真实环境进行对比实验来验证优化的稳定性。与未优化的线路相比,此优化算法下的CNOT门能减少约40%左右,其稳定性也得到了明显的提高。结果表明类Dijkstra优化算法可以适用于QAOA最大切割问题的多种图结构优化。     

An Upper Bound on the Loss from Approximate Optimal-Value Functions

Many reinforcement learning approaches can be formulated using the theory of Markov decision processes and the associated method of dynamic programming (DP). The value of this theoretical understanding, however, is tempered by many practical concerns. One important question is whether DP-based approaches that use function approximation rather than lookup tables can avoid catastrophic effects on performance. This note presents a result of Bertsekas (1987) which guarantees that small errors in the approximation of a task's optimal value function cannot produce arbitrarily bad performance when actions are selected by a greedy policy. We derive an upper bound on performance loss that is slightly tighter than that in Bertsekas (1987), and we show the extension of the bound to Q-learning (Watkins, 1989). These results provide a partial theoretical rationale for the approximation of value functions, an issue of great practical importance in reinforcement learning.     

An efficient Dijkstra-like labeling method for computing shortest odd/even paths

We show how the problem of determining shortest paths of even or odd length between two specified vertices in a graph G = (V, E) can be reduced to the problem of finding a shortest alternating path with respect to a specific matching in a related graph H. This problem can be solved by a Dijkstra-like labeling procedure of complexity O(|V|²) respectively O(|E|log|V|). Interpreting this procedure appropriately the method can then be applied directly on the given graph G.     

Efficient processing of label-constraint reachability queries in large graphs

In this paper, we study a variant of reachability queries, called label-constraint reachability (LCR) queries. Specifically, given a label set S and two vertices u₁ and u₂ in a large directed graph G, we check the existence of a directed path from u₁ to u₂, where edge labels along the path are a subset of S. We propose the path-label transitive closure method to answer LCR queries. Specifically, we t4ransform an edge-labeled directed graph into an augmented DAG by replacing the maximal strongly connected components as bipartite graphs. We also propose a Dijkstra-like algorithm to compute path-label transitive closure by re-defining the “distance” of a path. Comparing with the existing solutions, we prove that our method is optimal in terms of the search space. Furthermore, we propose a simple yet effective partition-based framework (local path-label transitive closure+online traversal) to answer LCR queries in large graphs. We prove that finding the optimal graph partition to minimize query processing cost is a NP-hard problem. Therefore, we propose a sampling-based solution to find the sub-optimal partition. Moreover, we address the index maintenance issues to answer LCR queries over the dynamic graphs. Extensive experiments confirm the superiority of our method.     

A new fast heuristic for labeling points

In the point site labeling problem, we are given a set P={p₁,p₂,…,pn} of point sites in the plane. The label of a point pi is an axis-parallel rectangle of specified size. The objective is to label the maximum number of points on the map so that the placed labels are mutually non-overlapping. Here, we investigate a special class of the point site labeling problem where (i) height of the labels of all the points are same but their lengths may differ, (ii) the label of a point pi touches the point at one of its four corners, and (iii) the label of one point does not obscure any other point in P. We describe an efficient heuristic algorithm for this problem which runs in time in the worst case. We run our algorithm as well as the algorithm Rules proposed by Wagner et al. on randomly generated point sets and on the available benchmarks. The results produced by our algorithm are almost the same as Rules in most of the cases. But our algorithm is faster than Rules in dense instance. We have also computed the optimum solutions for all the examples we have considered by designing an algorithm, which performs an exhaustive search in the worst case. We found that the exhaustive search algorithm runs reasonably fast for most of the examples we have considered.     

Multiple-Instance Learning of Real-Valued Geometric Patterns

Recently there has been significant research in multiple-instance learning, yet most of this work has only considered this model when there are Boolean labels. However, in many of the application areas for which the multiple-instance model fits, real-valued labels are more appropriate than Boolean labels. We define and study a real-valued multiple-instance model in which each multiple-instance example (bag) is given a real-valued label in [0, 1] that indicates the degree to which the bag satisfies the target concept. To provide additional structure to the learning problem, we associate a real-valued label with each point in the bag. These values are then combined using a real-valued aggregation operator to obtain the label for the bag. We then present on-line agnostic algorithms for learning real-valued multiple-instance geometric concepts defined by axis-aligned boxes in constant-dimensional space and describe several possible applications of these algorithms. We obtain our learning algorithms by reducing the problem to one in which the exponentiated gradient or gradient descent algorithm can be used. We also give a novel application of the virtual weights technique. In typical applications of the virtual weights technique, all of the concepts in a group have the same weight and prediction, allowing a single representative concept from each group to be tracked. However, in our application there are an exponential number of different weights and possible predictions. Hence, boxes in each group have different weights and predictions, making the computation of the contribution of a group significantly more involved. However, we are able to both keep the number of groups polynomial in the number of trials and efficiently compute the overall prediction.     

多标签分类器准确性评估方法的研究

分类是数据挖掘领域研究的核心技术之一,分类器性能评估方法也是众多学者的研究热点之一。以往的分类器性能评估方法一般针对于单标签数据集,对于多标签问题并未涉及。文中主要针对多标签分类问题中的单实例情况,提出了一种多标签分类器准确性评估方法（EMOSIML）。该方法的思路是：如果分类器对一个多标签对象预测的类别标签是其属于的多个类别标签中的任何一个,则分类结果都是正确的。该方法用C#编程实现,并对朴素贝叶斯分类器进行分类器性能评估实验,实验结果表明,EMOSIML评估方法较传统的准确率评估方法更合理。     

Discernible visualization of high dimensional data using label information

Visualization methods could significantly improve the outcome of automated knowledge discovery systems by involving human judgment. Star coordinate is a visualization technique that maps k-dimensional data onto a circle using a set of axes sharing the same origin at the center of the circle. It provides the users with the ability to adjust this mapping, through scaling and rotating of the axes, until no mapped point-clouds (clusters) overlap one another. In this state, similar groups of data are easily detectable. However an effective adjustment could be a difficult or even an impossible task for the user in high dimensions. This is specially the case when the input space dimension is about 50 or more.In this paper, we propose a novel method toward automatic axes adjustment for high dimensional data in Star Coordinate visualization method. This method finds the best two-dimensional view point that minimizes intra-cluster distances while keeping the inter-cluster distances as large as possible by using label information. We call this view point a discernible visualization, where clusters are easily detectable by human eye. The label information could be provided by the user or could be the result of performing a conventional clustering method over the input data. The proposed approach optimizes the Star Coordinate representation by formulating the problem as a maximization of a Fisher discriminant. Therefore the problem has a unique global solution and polynomial time complexity. We also prove that manipulating the scaling factor alone is effective enough for creating any given visualization mapping. Moreover it is showed that k-dimensional data visualization can be modeled as an eigenvalue problem. Using this approach, an optimal axes adjustment in the Star Coordinate method for high dimensional data can be achieved without any user intervention. The experimental results demonstrate the effectiveness of the proposed approach in terms of accuracy and performance.     

On the Optimal Linear Convergence Rate of a Generalized Proximal Point Algorithm

The proximal point algorithm (PPA) has been well studied in the literature. In particular, its linear convergence rate has been studied by Rockafellar in 1976 under certain condition. We consider a generalized PPA in the generic setting of finding a zero point of a maximal monotone operator, and show that the condition proposed by Rockafellar can also sufficiently ensure the linear convergence rate for this generalized PPA. Indeed we show that these linear convergence rates are optimal. Both the exact and inexact versions of this generalized PPA are discussed. The motivation of considering this generalized PPA is that it includes as special cases the relaxed versions of some splitting methods that are originated from PPA. Thus, linear convergence results of this generalized PPA can be used to better understand the convergence of some widely used algorithms in the literature. We focus on the particular convex minimization context and specify Rockafellar’s condition to see how to ensure the linear convergence rate for some efficient numerical schemes, including the classical augmented Lagrangian method proposed by Hensen and Powell in 1969 and its relaxed version, the original alternating direction method of multipliers (ADMM) by Glowinski and Marrocco in 1975 and its relaxed version (i.e., the generalized ADMM by Eckstein and Bertsekas in 1992). Some refined conditions weaker than existing ones are proposed in these particular contexts.     

Neural networks with multidimensional transfer functions.

We present a new type of neural network (NN) where the data for the input layer are the value x in R, the vector y in R(m ) associated to an initial value problem (IVP) with y'(x)= f(y(x)) and a steplength h. Then the stages of a Runge-Kutta (RK) method with trainable coefficients are used as hidden layers for the integration of the IVP using f as transfer function. We take as output two estimations y*, y;* of IVP at the point x+h. Training the RK method at some test problems and counting the cost of the method under the coefficients used, we may achieve coefficients that help the method to perform better at a wider class of problems.     

Regular Model Checking using Widening Techniques

In this paper, we consider symbolic model checking of safety properties of linear parametrized systems. Sets of configurations are represented by regular languages and actions by regular relations. Since the verification problem amounts to the computation of the reachability set, we focus on the computation of R^*(φ) for a regular relation R and a regular language φ. We present a technique called regular widening that allows, when it terminates, the computation of either the reachability set R^*(φ) of a system or the transitive closure R^* of a regular relation. We show that our method can be uniformly applied to several parametrized systems. Furthermore, we show that it is powerful enough to simulate some existing methods that compute either R^* or R^*(φ) for each R (resp. φ) belonging to a subclass of regular relations (resp. belonging to a subclass of regular languages).     

Analysis of Parallel Algorithms for Matrix Chain Product and Matrix Powers on Distributed Memory Systems

Given N matrices A₁, A_2,...,A_N of size NtimesN, the matrix chain product problem is to compute A₁timesA₂times...timesA_N. Given an NtimesN matrix A, the matrix powers problem is to calculate the first N powers of A, that is, A, A², A³,..., A^N. We solve the two problems on distributed memory systems (DMSs) with p processors that can support one-to-one communications in T(p) time. Assume that the fastest sequential matrix multiplication algorithm has time complexity O(N^alpha), where the currently best value of a is less than 2.3755. Let p be arbitrarily chosen in the range 1lesplesN^alpha+1/(log N)². We show that the two problems can be solved by a DMS with p processors in T_chain(N,p)=O((N^alpha+1/p)+T(p))((N^2(2+1/alpha/p^2/alpha)(log+p/N)^1-2/alpha+log+((p log N)/N^alpha)) and T_power (N,p)=O(N^alpha+1/p+T(p)((N^2(1+1/alpha)/p^2/alpha)(log+p/2 log N)^1-2/alpha+(log N)²))) times, respectively, where the function log+ is defined as follows: log+ x=log x if xges1 and log+ x=1 if 0相似文献   

ON COMPUTING THE MINIMAL LABELS IN TIME POINT ALGEBRA NETWORKS

We analyze the problem of computing the minimal labels for a network of temporal relations in point algebra. Van Beek proposes an algorithm for accomplishing this task, which takes O (max( n ³, n² m) ) time (for n points and m ≠ -relations). We show that the proof of the correctness of this algorithm given by van Beek and Cohen is faulty, and we provide a new proof showing that the algorithm is indeed correct.     

Tabu Search Heuristic for Point-Feature Cartographic Label Placement

The generation of better label placement configurations in maps is a problem that comes up in automated cartographic production. The objective of a good label placement is to display the geographic position of the features with their corresponding label in a clear and harmonious fashion, following accepted cartographic conventions. In this work, we have approached this problem from a combinatorial optimization point of view, and our research consisted of the evaluation of the tabu search (TS) heuristic applied to cartographic label placement. When compared, in real and random test cases, with techniques such as simulated annealing and genetic algorithm (GA), TS has proven to be an efficient choice, with the best performance in quality. We concluded that TS is a recommended method to solve cartographic label placement problem of point features, due to its simplicity, practicality, efficiency and good performance along with its ability to generate quality solutions in acceptable computational time.     

On fast planning of suboptimal paths amidst polygonal obstacles in plane

The problem of planning a path for a point robot from a source point s to a destination point d so as to avoid a set of polygonal obstacles in plane is considered. Using well-known methods, a shortest path from s to d can be computed with a time complexity of O(n²) where n is the total number of obstacle vertices. The focus here is in

1. (a) planning paths faster at the expense of setting for suboptimal path lengths and

2. (b) performance analysis of simple and/or well-known suboptimal methods.

A method that enables a hierarchical implementation of any path planning algorithm with no increase in the worst-case time complexity, is presented; this implementation enables fast planning of simple paths. Then methods are presented based on the Voronoi diagrams, trapezoidal decomposition and triangulation, which compute (suboptimal) paths in O(n√log n) time with the preprocessing costs of O(n log n), O(n²) and O(n log n), respectively. Using existing navigational algorithms for unknown terrains, algorithms that run in O(n log n) time (after preprocessing) and yield suboptimal paths, are presented. For all these algorithms, upper bounds on the path lengths are estimated in terms of the shortest of the obstacles, etc.     

ASYNCHRONOUS MONOTONE NEWTON ITERATIVE METHOD ON DISTRIBUTED COMPUTERS

In this paper, according to the asynchronous iteration model presented by G. M. Baudet [4] and D.P. Bertsekas [5], we propose an asynchronous monotone Newton iterative method for solving the nonlinear system of equations F(x) = 0 on a distributed computer, and prove its convergence.     

Proximity Queries Between Convex Objects: An Interior Point Approach for Implicit Surfaces

This paper presents a general method for exact distance computation between convex objects represented as intersections of implicit surfaces. Exact distance computation algorithms are particularly important for applications involving objects that make intermittent contact, such as in dynamic simulations and in haptic interactions. They can also be used in the narrow phase of hierarchical collision detection. In contrast to geometric approaches developed for polyhedral objects, we formulate the distance computation problem as a convex optimization problem. We use an interior point method to solve the optimization problem and demonstrate that, for general convex objects represented as implicit surfaces, interior point approaches are globally convergent, and fast in practice. Further, they provide polynomial-time guarantees for implicit surface objects when the implicit surfaces have self-concordant barrier functions. We use a primal-dual interior point algorithm that solves the Karush-Kuhn-Tucker (KKT) conditions obtained from the convex programming formulation. For the case of polyhedra and quadrics, we establish a theoretical time complexity of O(n^1.5), where n is the number of constraints. We present implementation results for example implicit surface objects, including polyhedra, quadrics, and generalizations of quadrics such as superquadrics and hyperquadrics, as well as intersections of these surfaces. We demonstrate that in practice, the algorithm takes time linear in the number of constraints, and that distance computation rates of about 1 kHz can be achieved. We also extend the approach to proximity queries between deforming convex objects. Finally, we show that continuous collision detection for linearly translating objects can be performed by solving two related convex optimization problems. For polyhedra and quadrics, we establish that the computational complexity of this problem is also O(n^1.5).     

Finding optimal paths in MREP routing

Maximum Residual Energy Path (MREP) routing has been shown an effective routing scheme for energy conservation in battery powered wireless networks. Past studies on MREP routing are based on the assumption that the transmitting node consumes power, but the receiving node does not. This assumption is false if acknowledgment is required as occurs, for example, in some Bluetooth applications.If the receiving node does not consume power then the MREP routing problem for a single message is easily solvable in polynomial time using a simple Dijkstra-like algorithm. We further show in that when the receiving node does consume power the problem becomes NP-complete and is even impossible to approximate with an exponential approximation factor in polynomial time unless P=NP.     

Positively invariant sets for constrained continuous-time systemswith cone properties

This note deals with some properties of particular bounded sets w.r.t. linear continuous-time systems described by x˙(t)=A(0)x(t)+c(t), where c(t)∈Ω⊂Rⁿ, Ω a compact set, and matrix e/sup tA(0/) has the property of leaving a proper cone K positively invariant, that is e/sup tA(0/)K⊂K. The considered bounded sets 𝒟(K; a, b) are described as the intersection of shifted cones. Necessary and sufficient conditions are given. They guarantee that such sets are positively invariant w.r.t. the considered system. The trajectories starting from x ₀∈Rⁿ/𝒟(K; a, b) (respectively to x₀ ∈Rⁿ) are studied in terms of attractivity and contractivity of the set 𝒟(K; a, b). The results are applied to the study of the constrained state feedback regulator problem