Greedy algorithm and symmetric matroids

Symmetric matroids are set systems which are obtained, in some sense, by a weakening of the structure of a matroid. These set systems are characterized by a greedy algorithm and they are suitable for dealing with autodual properties of matroids. Applications are given to the eulerian tours of 4-regular graphs and the theory ofg-matroids.     

The Schur algorithm and its applications

The Schur algorithm and its time-domain counterpart, the fast Cholseky recursions, are some efficient signal processing algorithms which are well adapted to the study of inverse scattering problems. These algorithms use a layer stripping approach to reconstruct a lossless scattering medium described by symmetric two-component wave equations which model the interaction of right and left propagating waves. In this paper, the Schur and fast Chokesky recursions are presented and are used to study several inverse problems such as the reconstruction of nonuniform lossless transmission lines, the inverse problem for a layered acoustic medium, and the linear least-squares estimation of stationary stochastic processes. The inverse scattering problem for asymmetric two-component wave equations corresponding to lossy media is also examined and solved by using two coupled sets of Schur recursions. This procedure is then applied to the inverse problem for lossy transmission lines.The work of this author was supported by the Exxon Education FoundationThe work of this author was supported by the Air Force Office of Scientific Research under Grant AFOSR-82-0135A.     

On the discrete-time system with server breakdowns: Computational algorithm and optimization algorithm

This paper analyzes a discrete-time Geo/Geo/1 queueing system with the server subject to breakdowns and repairs, in which two different possible types of the server breakdowns are considered. In Type 1, the server may break down only when the system is busy, while in Type 2, the server can break down even if the system is idle. The server lifetimes are assumed to be geometrical and the server repair times are also geometric distributions. We model this system by the level-dependent quasi-birth-death (QBD) process and develop computation algorithms of the stationary distribution of the number of customers in the system using the matrix analytic method. The search algorithm for parameter optimization based on a cost model is developed and performed herein.     

Affinity genetic algorithm

Based on some phenomena from human society and nature, we propose a binary affinity genetic algorithm (aGA) by adopting the following strategies: the population is adaptively updated to avoid stagnation; the newly generated individuals will be ensured to survive for some generations in order for them to have time to show their good genes; new individuals and the old ones are balanced to have the advantages of both. In order to quantitatively analyze the selective pressure, the concept of selection degree and a simple linear control equation are introduced. We can maintain the diversity of the evolutionary population by controlling the value of the selection degree. Performance of aGA is further enhanced by incorporating local search strategies. Partially supported by a National Key Basic Research Project of China and by a USA NSF grant CCR-0201253.     

The optimal exchange algorithm and comparisons with the generalized remes algorithm † ‡

This paper is concerned with the general problem of the determination of the best uniform approximation of a given function. A special case is the calculation of the minimax solution of an overdetermined linear system. Single point exchange algorithms produce successive approximate solutions for such problems. An example is furnished by the generalized Remes algorithm, which includes both the original Remes algorithm and the Stiefel algorithm as special cases. The optimal exchange algorithm is similar, but it has the important feature that every exchange is optimal in a certain desirable sense. It is proved that eventually the optimal exchange and generalized Remes algorithms coincide. However, early exchanges in the latter algorithm can be far from optimal and quite inefficient. A combination of the optimal exchange and generalized Remes algorithms is suggested as a reasonable strategy.     

On Khachian's algorithm and minimal ellipsoids

Summary Khachian's algorithm for solving a system of linear inequalities is accelerated by choosing smaller ellipsoids than in the original version. Furthermore, certain inequalities can be successively eliminated, from the constraints, yielding a different stopping rule.     

Prime witnesses in the Shor algorithm and the Miller-Rabin algorithm

We prove that prime witnesses in the Miller-Rabin algorithm coincide with those in the Shor algorithm which satisfy the condition of Fermat’s little theorem. We describe the set of natural numbers, whose prime witnesses in the Miller-Rabin algorithm coincide with those in the Shor algorithm. We find all such numbers less than 100,000,000 and experimentally study the rate of increase of the ratio of the quantity of such numbers to the quantity of Carmichael numbers.     

Linear scaling and the DIRECT algorithm

In this paper, we discuss the performance of the DIRECT global optimization algorithm on problems with linear scaling. We show with computations that the performance of DIRECT can be affected by linear scaling of the objective function. We also provide a theoretical result which shows that DIRECT does not perform well when the absolute value of the objective function is large enough. Then we present DIRECT-a, a modification of DIRECT, to eliminate the sensitivity to linear scaling of the objective function. We prove theoretically that linear scaling of the objective function does not affect the performance of DIRECT-a. Similarly, we prove that some modifications of DIRECT are also unaffected by linear scaling of the objective function, while the original DIRECT algorithm is sensitive to linear scaling. Numerical results in this paper show that DIRECT-a is more robust than the original DIRECT algorithm, which support the theoretical results. Numerical results also show that careful choices of the parameter ε can help DIRECT perform well when the objective function is poorly linearly scaled.     

The octahedral algorithm,a new simplicial fixed point algorithm

A new variable dimension simplicial algorithm for the computation of solutions of systems of nonlinear equations or the computation of fixed points is presented. It uses the restrart technique of Merrill to improve the accuracy of the solution. The algorithm is shown to converge quadratically under certain conditions. The algorithm should be efficient and relatively easy to implement.Partially supported by the Western Michigan University Sabbatical and Faculty Research Funds.     

A combined algorithm for identification and approximation

The optimum point of an unknown function which is measured with noise and the unknown parameters in the measurement noise are estimated by a combined algorithm. The almost sure convergence and the convergence rate are established for both unconstrained and constrained problems.Project supported by the National Natural Science Foundation of China.     

Monte-Carlo parallelization algorithm

Parallelization of computer algorithms for multidimensional problems of computational physics is considered. The focus is on efficient data processing by computer systems with a large number of concurrent processors, allowing for data exchange time. The computing time is minimized by a Monte-Carlo technique.Translated from Matematicheskie Modeli Estestvoznaniya, Published by Moscow University, Moscow, 1995, pp. 150–156.     

Split Newton iterative algorithm and its application

Inspired by some implicit-explicit linear multistep schemes and additive Runge-Kutta methods, we develop a novel split Newton iterative algorithm for the numerical solution of nonlinear equations. The proposed method improves computational efficiency by reducing the computational cost of the Jacobian matrix. Consistency and global convergence of the new method are also maintained. To test its effectiveness, we apply the method to nonlinear reaction-diffusion equations, such as Burger’s-Huxley equation and fisher’s equation. Numerical examples suggest that the involved iterative method is much faster than the classical Newton’s method on a given time interval.     

Hedge algorithm and Dual Averaging schemes

We show that the Hedge algorithm, a method that is widely used in Machine Learning, can be interpreted as a particular instance of Dual Averaging schemes, which have recently been introduced by Nesterov for regret minimization. Based on this interpretation, we establish three alternative methods of the Hedge algorithm: one in the form of the original method, but with optimal parameters, one that requires less a priori information, and one that is better adapted to the context of the Hedge algorithm. All our modified methods have convergence results that are better or at least as good as the performance guarantees of the vanilla method. In numerical experiments, our methods significantly outperform the original scheme.     

Vizing's coloring algorithm and the fan number

Most upper bounds for the chromatic index of a graph come from algorithms that produce edge colorings. One such algorithm was invented by Vizing [Diskret Analiz 3 (1964), 25–30] in 1964. Vizing's algorithm colors the edges of a graph one at a time and never uses more than Δ+µ colors, where Δ is the maximum degree and µ is the maximum multiplicity, respectively. In general, though, this upper bound of Δ+µ is rather generous. In this paper, we define a new parameter fan(G) in terms of the degrees and the multiplicities of G. We call fan(G) the fan number of G. First we show that the fan number can be computed by a polynomial‐time algorithm. Then we prove that the parameter Fan(G)=max{Δ(G), fan(G)} is an upper bound for the chromatic index that can be realized by Vizing's coloring algorithm. Many of the known upper bounds for the chromatic index are also upper bounds for the fan number. Furthermore, we discuss the following question. What is the best (efficiently realizable) upper bound for the chromatic index in terms of Δ and µ ? Goldberg's Conjecture supports the conjecture that χ′+1 is the best efficiently realizable upper bound for χ′ at all provided that P ≠ NP . © 2009 Wiley Periodicals, Inc. J Graph Theory 65: 115–138, 2010     

Chaotic bat algorithm

Bat algorithm (BA) is a recent metaheuristic optimization algorithm proposed by Yang. In the present study, we have introduced chaos into BA so as to increase its global search mobility for robust global optimization. Detailed studies have been carried out on benchmark problems with different chaotic maps. Here, four different variants of chaotic BA are introduced and thirteen different chaotic maps are utilized for validating each of these four variants. The results show that some variants of chaotic BAs can clearly outperform the standard BA for these benchmarks.     

Simplices by point-sliding and the Yamnitsky-Levin algorithm

Yamnitsky and Levin proposed a variant of Khachiyan's ellopsoid method for testing feasibility of systems of linear inequalities that also runs in polynomial time but uses simplices instead of ellipsoids. Starting with then-simplexS and the half-space {x¦a ^Tx }, the algorithm finds a simplexS _YL of small volume that enclosesS {x¦a ^Tx }. We interpretS _YL as a simplex obtainable by point-sliding and show that the smallest such simplex can be determined by minimizing a simple strictly convex function. We furthermore discuss some numerical results. The results suggest that the number of iterations used by our method may be considerably less than that of the standard ellipsoid method.     

A stochastic steepest-descent algorithm

A stochastic steepest-descent algorithm for function minimization under noisy observations is presented. Function evaluation is done by performing a number of random experiments on a suitable probability space. The number of experiments performed at a point generated by the algorithm reflects a balance between the conflicting requirements of accuracy and computational complexity. The algorithm uses an adaptive precision scheme to determine the number of random experiments at a point; this number tends to increase whenever a stationary point is approached and to decrease otherwise. Two rules are used to determine the number of random experiments at a point; one, in the inner loop of the algorithm, uses the magnitude of the observed gradient of the function to be minimized; and the other, in the outer-loop, uses a measure of accumulated errors in function evaluations at past points generated by the algorithm. Once a stochastic approximation of the function to be minimized is obtained at a point, the algorithm proceeds to generate the next point by using the steepest-descent deterministic methods of Armijo and Polak (Refs. 3, 4). Convergence of the algorithm to stationary points is demonstrated under suitable assumptions.     

A quadratic programming algorithm

By using conjugate directions a method for solving convex quadratic programming problems is developed. The algorithm generates a sequence of feasible solutions and terminates after a finite number of iterations. Extensions of the algorithm for nonconvex and large structured quadratic programming problems are discussed.Sponsored by the United States Army under Contract No. DAAG29-80-C-0041 and in part by the Natural Sciences and Engineering Research Council of Canada under Grant Nos. A 8189 and E 5582.     

A circle packing algorithm

A circle packing is a configuration P of circles realizing a specified pattern of tangencies. Radii of packings in the euclidean and hyperbolic planes may be computed using an iterative process suggested by William Thurston. We describe an efficient implementation, discuss its performance, and illustrate recent applications. A central role is played by new and subtle monotonicity results for “flowers” of circles.