High‐performance direct algorithms for computing the sign function of triangular matrices

Algorithms and implementations for computing the sign function of a triangular matrix are fundamental building blocks for computing the sign of arbitrary square real or complex matrices. We present novel recursive and cache‐efficient algorithms that are based on Higham's stabilized specialization of Parlett's substitution algorithm for computing the sign of a triangular matrix. We show that the new recursive algorithms are asymptotically optimal in terms of the number of cache misses that they generate. One algorithm that we present performs more arithmetic than the nonrecursive version, but this allows it to benefit from calling highly optimized matrix multiplication routines; the other performs the same number of operations as the nonrecursive version, suing custom computational kernels instead. We present implementations of both, as well as a cache‐efficient implementation of a block version of Parlett's algorithm. Our experiments demonstrate that the blocked and recursive versions are much faster than the previous algorithms and that the inertia strongly influences their relative performance, as predicted by our analysis.     

Parallel geometric algorithms for multi-core computers

Computers with multiple processor cores using shared memory are now ubiquitous. In this paper, we present several parallel geometric algorithms that specifically target this environment, with the goal of exploiting the additional computing power. The algorithms we describe are (a) 2-/3-dimensional spatial sorting of points, as is typically used for preprocessing before using incremental algorithms, (b) d-dimensional axis-aligned box intersection computation, and finally (c) 3D bulk insertion of points into Delaunay triangulations, which can be used for mesh generation algorithms, or simply for constructing 3D Delaunay triangulations. For the latter, we introduce as a foundational element the design of a container data structure that both provides concurrent addition and removal operations and is compact in memory. This makes it especially well-suited for storing large dynamic graphs such as Delaunay triangulations.We show experimental results for these algorithms, using our implementations based on the Computational Geometry Algorithms Library (CGAL). This work is a step towards what we hope will become a parallel mode for CGAL, where algorithms automatically use the available parallel resources without requiring significant user intervention.     

Computational acceleration of projection algorithms for the linear best approximation problem

This is an experimental computational account of projection algorithms for the linear best approximation problem. We focus on the sequential and simultaneous versions of Dykstra’s algorithm and the Halpern-Lions-Wittmann-Bauschke algorithm for the best approximation problem from a point to the intersection of closed convex sets in the Euclidean space. These algorithms employ different iterative approaches to reach the same goal but no mathematical connection has yet been found between their algorithmic schemes. We compare these algorithms on linear best approximation test problems that we generate so that the solution will be known a priori and enable us to assess the relative computational merits of these algorithms. For the simultaneous versions we present a new component-averaging variant that substantially accelerates their initial behavior for sparse systems.     

Shortest paths algorithms: Theory and experimental evaluation

We conduct an extensive computational study of shortest paths algorithms, including some very recent algorithms. We also suggest new algorithms motivated by the experimental results and prove interesting theoretical results suggested by the experimental data. Our computational study is based on several natural problem classes which identify strengths and weaknesses of various algorithms. These problem classes and algorithm implementations form an environment for testing the performance of shortest paths algorithms. The interaction between the experimental evaluation of algorithm behavior and the theoretical analysis of algorithm performance plays an important role in our research. This work was done while Boris V. Cherkassky was visiting Stanford University Computer Science Department and supported by the NSF and Powell Foundation grants mentioned below. Andrew V. Goldberg was supported in part by ONR Young Investigator Award N00014-91-J-1855, NSF Presidential Young Investigator Grant CCR-8858097 with matching funds from AT&T, DEC, and 3M, and a grant from Powell Foundation. Corresponding author. This work was done while Tomasz Radzik was a Postdoctoral Fellow at SORIE, Cornell University, and supported by the National Science Foundation, the Air Force Office of Scientific Research, and the Office of Naval Research, through NSF grant DMS-8920550, and by the Packard Fellowship of éva Tardos.     

Efficient Parallel Algorithms for the Minimum Cost Flow Problem

In this paper, we propose efficient parallel implementations of the auction/sequential shortest path and the -relaxation algorithms for solving the linear minimum cost flow problem. In the parallel auction algorithm, several augmenting paths can be found simultaneously, each of them starting from a different node with positive surplus. Convergence results of an asynchronous version of the algorithm are also given. For the -relaxation method, there exist already parallel versions implemented on CM-5 and CM-2; our implementation is the first on a shared memory multiprocessor. We have obtained significant speedup values for the algorithms considered; it turns out that our implementations are effective and efficient.     

The 4-triangles longest-side partition of triangles and linear refinement algorithms

In this paper we study geometrical properties of the iterative 4-triangles longest-side partition of triangles (and of a 3-triangles partition), as well as practical algorithms based on these partitions, used both directly for the triangulation refinement problem, and as a basis for point insertion strategies in Delaunay refinement algorithms. The 4-triangles partition is obtained by joining the midpoint of the longest side with the opposite vertex and the midpoints of the two remaining sides. By means of simple geometrical arguments we show that the iterative partition of obtuse triangles systematically improves the triangles (while they remain obtuse) in the following sense: the sequence of smallest angles monotonically increases while the sequence of largest angles monotonically decreases in an amount (at least) equal to the smallest angle of each iteration. This allows us to improve the known bound on the smallest angle (without making use of previous results), and to obtain a better a priori bound on the number of similarly distinct triangles, as a function of the geometry of the initial triangle. Numerical evidence, showing that the practical behavior of the 4-triangles partition is in complete agreement with this theory, is included. A 4-triangles refinement algorithm is also discussed and illustrated. Furthermore, we show that the time cost of the algorithm is linear independently of the size of the triangulation.

     

A Comparison of Large Scale Mixed Complementarity Problem Solvers

This paper provides a means for comparing various computercodes for solving large scale mixed complementarity problems. Wediscuss inadequacies in how solvers are currently compared, andpresent a testing environment that addresses these inadequacies. Thistesting environment consists of a library of test problems, along withGAMS and MATLAB interfaces that allow these problems to be easilyaccessed. The environment is intended for use as a tool byother researchers to better understand both their algorithms and theirimplementations, and to direct research toward problem classes thatare currently the most challenging. As an initial benchmark, eightdifferent algorithm implementations for large scale mixedcomplementarity problems are briefly described and tested with defaultparameter settings using the new testing environment.     

Probabilistic proximity searching algorithms based on compact partitions

The main bottleneck of the research in metric space searching is the so-called curse of dimensionality, which makes the task of searching some metric spaces intrinsically difficult, whatever algorithm is used. A recent trend to break this bottleneck resorts to probabilistic algorithms, where it has been shown that one can find 99% of the relevant objects at a fraction of the cost of the exact algorithm. These algorithms are welcome in most applications because resorting to metric space searching already involves a fuzziness in the retrieval requirements. In this paper, we push further in this direction by developing probabilistic algorithms on data structures whose exact versions are the best for high dimensions. As a result, we obtain probabilistic algorithms that are better than the previous ones. We give new insights on the problem and propose a novel view based on time-bounded searching. We also propose an experimental framework for probabilistic algorithms that permits comparing them in offline mode.     

Using convex envelopes to solve the interactive fixed-charge linear programming problem

In the well-known fixed-charge linear programming problem, it is assumed, for each activity, that the value of the fixed charge incurred when the level of the activity is positive does not depend upon which other activities, if any, are also undertaken at a positive level. However, we have encountered several practical problems where this assumption does not hold. In an earlier paper, we developed a new problem, called the interactive fixed-charge linear programming problem (IFCLP), to model these problems. In this paper, we show how to construct the convex envelopes and other convex underestimating functions for the objective function for problem (IFCLP) over various rectangular subsets of its domain. Using these results, we develop a specialized branch-and-bound algorithm for problem (IFCLP) which finds an exact optimal solution for the problem in a finite number of steps. We also discuss the main properties of this algorithm.The authors would like to thank an anonymous referee for his helpful suggestions.     

Homogeneous algorithms for multiextremal optimization

The class of homogeneous algorithms for multiextremal optimization is defined, and a number of theorems are proved, including a sufficient condition for the convergence of homogeneous algorithms to a global minimizer. An approach to the synthesis of homogeneous algorithms based on model multi-peak functions is proposed. The existing algorithms are reviewed, and a new efficient multidimensional algorithm based on the Delaunay triangulation is constructed. Some numerical results are presented.     

A new approximation algorithm for the multilevel facility location problem

In this paper we propose a new integer programming formulation for the multilevel facility location problem and a novel 3-approximation algorithm based on LP-rounding. The linear program that we use has a polynomial number of variables and constraints, thus being more efficient than the one commonly used in the approximation algorithms for these types of problems.     

A Hybrid Genetic Algorithm for Nonconvex Function Minimization

In this paper, we consider the problem of minimizing a function in severalvariables which could be multimodal and may possess discontinuities. A newalgorithm for the problem based on the genetic technique is developed. Thealgorithm is hybrid in nature in the sense that it utilizes the genetictechnique to generate search directions, which are used in an optimizationscheme and is thus different from any other methods in the literature.The algorithm has been tested on the Rosenbrock valley functions in 2 and 4dimensions, and multimodal functions in 2 and 4 dimensions, which are of ahigh degree of difficulty. The results are compared with the Adaptive RandomSearch, and Simulated Annealing algorithms. The performance of the algorithmis also compared to recent global algorithms in terms of the number offunctional evaluations needed to obtain a global minimum and results show thatthe proposed algorithm is better than these algorithms on a set of standardtest problems. It seems that the proposed algorithm is efficient and robust.     

Solving Convex MINLP Optimization Problems Using a Sequential Cutting Plane Algorithm

In this article we look at a new algorithm for solving convex mixed integer nonlinear programming problems. The algorithm uses an integrated approach, where a branch and bound strategy is mixed with solving nonlinear programming problems at each node of the tree. The nonlinear programming problems, at each node, are not solved to optimality, rather one iteration step is taken at each node and then branching is applied. A Sequential Cutting Plane (SCP) algorithm is used for solving the nonlinear programming problems by solving a sequence of linear programming problems. The proposed algorithm generates explicit lower bounds for the nodes in the branch and bound tree, which is a significant improvement over previous algorithms based on QP techniques. Initial numerical results indicate that the described algorithm is a competitive alternative to other existing algorithms for these types of problems.     

An interior-point method for multifractional programs with convex constraints

We present an interior-point method for a family of multi-fractional programs with convex constraints. The programs under consideration consist of minimizing the maximum of a finite number of linear fractions over some convex set. First, we present a simple shortstep algorithm for solving such multifractional programs, and we show that, under suitable assumptions, the convergence of the short-step algorithm is weakly polynomial in a sense specified below. Then, we describe a practical implementation of the proposed method, and we report results of numerical experiments with this algorithm. These results suggest that the proposed method is a viable alternative to the standard Dinkelbach-type algorithms for solving multifractional programs.The authors would like to thank Professor A. S. Nemirovsky for stimulating discussions via electronic mail. We are grateful to two anonymous referees for comments and suggestions that improved our paper.     

Random backtracking in backtrack search algorithms for satisfiability

This paper proposes the utilization of randomized backtracking within complete backtrack search algorithms for propositional satisfiability (SAT). In recent years, randomization has become pervasive in SAT algorithms. Incomplete algorithms for SAT, for example the ones based on local search, often resort to randomization. Complete algorithms also resort to randomization. These include state-of-the-art backtrack search SAT algorithms that often randomize variable selection heuristics. Moreover, it is plain that the introduction of randomization in other components of backtrack search SAT algorithms can potentially yield new competitive search strategies. As a result, we propose a stochastic backtrack search algorithm for SAT, that randomizes both the variable selection and the backtrack steps of the algorithm. In addition, we relate randomized backtracking with a more general form of backtracking, referred to as unrestricted backtracking. Finally, experimental results for different organizations of randomized backtracking are described and compared, providing empirical evidence that the new search algorithm for SAT is a very competitive approach for solving hard real-world instances.     

A feasible direction algorithm for convex optimization: Global convergence rates

At each iteration, the algorithm determines a feasible descent direction by minimizing a linear or quadratic approximation to the cost on the feasible set. The algorithm is easy to implement if the approximation is easy to minimize on the feasible set, which happens in some important cases. Convergence rate information is obtained, which is sufficient to enable deduction of the number of iterations needed to achieve a specified reduction in the distance from the optimum (measured in terms of the cost). Existing convergence rates for algorithms for solving such convex problems are either asymptotic (and so do not enable the required number of iterations to be deduced) or decrease as the number of constraints increases. The convergence rate information obtained here, however, is independent of the number of constraints. For the case where the quadratic approximation to the cost is not strictly convex (which includes the linear approximation case), the diameter is the only property of the feasible set which affects the convergence rate information. If the quadratic approximation is strictly convex, the convergence rate is independent of the size and geometry of the feasible set. An application to a control-constrained optimal control problem is outlined.     

Verified convex hull and distance computation for octree-encoded objects

This paper discusses algorithms for computing verified convex hull and distance enclosure for objects represented by axis-aligned or unaligned octrees. To find a convex enclosure of an octree, the concept of extreme vertices of boxes on its boundary has been used. The convex hull of all extreme vertices yields an enclosure of the object. Thus, distance algorithms for convex polyhedra to obtain lower bounds for the distance between two octrees can be applied. Since using convex hulls makes it possible to avoid the unwanted wrapping effect that results from repeated decompositions, it also opens a way to dynamic distance algorithms for moving objects.     

Viscosity Approximation Methods for a Class of Generalized Split Feasibility Problems with Variational Inequalities in Hilbert Space

Strong convergence theorem of viscosity approximation methods for nonexpansive mapping have been studied. We also know that CQ algorithm for solving the split feasibility problem (SFP) has a weak convergence result. In this paper, we use viscosity approximation methods and some related knowledge to solve a class of generalized SFP’s with monotone variational inequalities in Hilbert space. We propose some iterative algorithms based on viscosity approximation methods and get strong convergence theorems. As applications, we can use algorithms we proposed for solving split variational inequality problems (SVIP), split constrained convex minimization problems and some related problems in Hilbert space.     

A class of methods for solving large,convex quadratic programs subject to box constraints

In this paper we analyze conjugate gradient-type algorithms for solving convex quadratic programs subject only to box constraints (i.e., lower and upper bounds on the variables). Programs of this type, which we denote by BQP, play an important role in many optimization models and algorithms. We propose a new class of finite algorithms based on a nonfinite heuristic for solving a large, sparse BQP. The numerical results suggest that these algorithms are competitive with Dembo and Tulowitzski's (1983) CRGP algorithm in general, and perform better than CRGP for problems that have a low percentage of free variables at optimality, and for problems with only nonnegativity constraints.     

Multiple UAVs path planning algorithms: a comparative study

Unmanned aerial vehicles (UAVs) are used in team for detecting targets and keeping them in its sensor range. There are various algorithms available for searching and monitoring targets. The complexity of the search algorithm increases if the number of nodes is increased. This paper focuses on multi UAVs path planning and Path Finding algorithms. Number of Path Finding and Search algorithms was applied to various environments, and their performance compared. The number of searches and also the computation time increases as the number of nodes increases. The various algorithms studied are Dijkstra’s algorithm, Bellman Ford’s algorithm, Floyd-Warshall’s algorithm and the AStar algorithm. These search algorithms were compared. The results show that the AStar algorithm performed better than the other search algorithms. These path finding algorithms were compared so that a path for communication can be established and monitored.