a rank one reduction formula and its applications to matrix factorizations let a in r m times n denote an arbitrary matrix if x in r n and y in r m are vectors such that omega y t ax ne 0 then the matrix b a omega 1 axy t a a has rank exactly one less than the rank of a this wedderburn rank one reduction formula is easy to prove yet the idea is so powerful that perhaps all matrix factorizations can be derived from it the formula also appears in places such as the positive definite secant updates bfgs and dfp as well as the abs methods by repeatedly applying the formula to reduce ranks a biconjugation process analogous to the gram schmidt process with oblique projections can be developed this process provides a mechanism for constructing factorizations such as text ldm t qr and svd under a common framework of a general biconjugate decomposition v t au omega that is diagonal and nonsingular two characterizations of biconjugation provide new insight into the lanczos method and its breakdown one characterization shows that

a graph based hyper heuristic for educational timetabling problems this paper presents an investigation of a simple generic hyper heuristic approach upon a set of widely used constructive heuristics graph coloring heuristics in timetabling within the hyperheuristic framework a tabu search approach is employed to search for permutations of graph heuristics which are used for constructing timetables in exam and course timetabling problems this underpins a multi stage hyper heuristic where the tabu search employs permutations upon a different number of graph heuristics in two stages we study this graph based hyper heuristic approach within the context of exploring fundamental issues concerning the search space of the hyper heuristic the heuristic space and the solution space such issues have not been addressed in other hyper heuristic research these approaches are tested on both exam and course benchmark timetabling problems and are compared with the fine tuned bespoke state of the art approaches the results are within the range of the best results reported in the literature the approach described here represents a significantly more generally applicable approach than the current state of the art in the literature future work will extend this hyper heuristic framework by employing methodologies which are applicable on a wider range of timetabling and scheduling problems

a tabu search approach to the constraint satisfaction problem as a general problem solver many combinatorial problems including a variety of combinatorial optimization problems can be naturally formulated as a constraint satisfaction problem csp we develop in this paper a tabu search based algorithm for the csp as a foundation for a general problem solver in addition to the basic components of tabu search we develop a number of elaborations such as an automatic control mechanism for the tabu tenure modification of the penalty function to handle objective functions and enlargement of the neighborhood by allowing swap operations computational results with our algorithm are reported for various problems selected from a wide range of applications i e graph coloring generalized assignment set covering timetabling and nurse scheduling our results appear to be competitive with those of existing algorithms specially developed for the respective problem domains

fast and stable qr eigenvalue algorithms for generalized companion matrices and secular equations we introduce a class of n n structured matrices which includes three well known classes of generalized companion matrices tridiagonal plus rank one matrices comrade matrices diagonal plus rank one matrices and arrowhead matrices relying on the structure properties of we show that if a then a rq where a qr is the qr decomposition of a this allows one to implement the qr iteration for computing the eigenvalues and the eigenvectors of any a with o n arithmetic operations per iteration and with o n memory storage this iteration applied to generalized companion matrices provides new o n2 flops algorithms for computing polynomial zeros and for solving the associated rational secular equations numerical experiments confirm the effectiveness and the robustness of our approach

new studies of randomized augmentation and additive preprocessing abstract a standard gaussian random matrix hereafter referred to just as gaussian matrix has full rank with probability 1 and is well conditioned with a probability quite close to 1 and converging to 1 fast as the matrix deviates from the square shape and becomes more rectangular if we append sufficiently many gaussian rows or columns to any normalized and possibly rank deficient or ill conditioned matrix then the augmented matrix has full rank with probability 1 and is well conditioned with a probability close to 1 we specify and prove these properties of augmentation and extend them to additive preprocessing that is to adding a product of two rectangular gaussian matrices by applying our randomization techniques to a matrix that has numerical rank r we accelerate the known algorithms for the approximation of its trailing singular spaces associated with all its positive singular values except for the r largest values our algorithms use much fewer random parameters and run much faster when various random sparse and structured preprocessors replace gaussian empirically the outputs of the resulting algorithms are as accurate as the outputs under gaussian preprocessing our novel duality techniques provides formal support so far missing for these empirical observations and opens door to de randomization of our preprocessing and to further acceleration and simplification of our algorithms by using more efficient sparse and structured preprocessors our techniques and our progress can be applied to various other fundamental matrix computations such as the celebrated low rank approximation of a matrix by means of random sampling

search with evolutionary ruin and stochastic rebuild a theoretic framework and a case study on exam timetabling this paper presents a state transition based formal framework for a new search method called evolutionary ruin and stochastic recreate which tries to learn and adapt to the changing environments during the search process it improves the performance of the original ruin and recreate principle by embedding an additional phase of evolutionary ruin to mimic the survival of the fittest mechanism within single solutions this method executes a cycle of solution decomposition evolutionary ruin stochastic recreate and solution acceptance until a certain stopping condition is met the solution decomposition phase first uses some problem specific knowledge to decompose a complete solution into its components and assigns a score to each component the evolutionary ruin phase then employs two evolutionary operators namely selection and mutation to destroy a certain fraction of the solution and the next stochastic recreate phase repairs the broken solution last the solution acceptance phase selects a specific strategy to determine the probability of accepting the newly generated solution hence optimisation is achieved by an iterative process of component evaluation solution disruption and stochastic constructive repair from the state transitions point of view this paper presents a probabilistic model and implements a markov chain analysis on some theoretical properties of the approach unlike the theoretical work on genetic algorithm and simulated annealing which are based on state transitions within the space of complete assignments our model is based on state transitions within the space of partial assignments the exam timetabling problems are used to test the performance in solving real world hard problems

solving a polynomial equation some history and recent progress the classical problem of solving an nth degree polynomial equation has substantially influenced the development of mathematics throughout the centuries and still has several important applications to the theory and practice of present day computing we briefly recall the history of the algorithmic approach to this problem and then review some successful solution algorithms we end by outlining some algorithms of 1995 that solve this problem at a surprisingly low computational cost

additive preconditioning eigenspaces and the inverse iteration we incorporate our recent preconditioning techniques into the classical inverse power rayleigh quotient iteration for computing matrix eigenvectors every loop of this iteration essentially amounts to solving an ill conditioned linear system of equations due to our modification we solve a well conditioned linear system instead we prove that this modification preserves local quadratic convergence show experimentally that fast global convergence is preserved as well and yield similar results for higher order inverse iteration covering the cases of multiple and clustered eigenvalues

a new error free floating point summation algorithm summation is a basic operation in scientific computing furthermore division free arithmetic computations can be boiled down to summation we propose a new summation algorithm which consists of double precision floating point operations and outputs the error free sums the computational time is proportional to the condition number of the problem is low according to both our estimates and extensive experiments and further decreases for producing faithful rounding of the sum rather than its error free value

variable neighborhood search principles and applications abstract systematic change of neighborhood within a possibly randomized local search algorithm yields a simple and effective metaheuristic for combinatorial and global optimization called variable neighborhood search vns we present a basic scheme for this purpose which can easily be implemented using any local search algorithm as a subroutine its effectiveness is illustrated by solving several classical combinatorial or global optimization problems moreover several extensions are proposed for solving large problem instances using vns within the successive approximation method yields a two level vns called variable neighborhood decomposition search vnds modifying the basic scheme to explore easily valleys far from the incumbent solution yields an efficient skewed vns svns heuristic finally we show how to stabilize column generation algorithms with help of vns and discuss various ways to use vns in graph theory i e to suggest disprove or give hints on how to prove conjectures an area where metaheuristics do not appear to have been applied before

cluster based population initialization for differential evolution frameworks this article proposes a procedure to perform an intelligent initialization for population based algorithms the proposed pre processing procedure namely cluster based population initialization cbpi consists of three consecutive stages at the first stage the individuals belonging to a randomly sampled population undergo two subsequent local search algorithms i e a simple local search that performs moves along the axes and rosenbrock algorithm at the second stage the solutions processed by the two local searches undergo the k means clustering algorithm and are grouped into sets on the basis of their euclidean distance at the third stage the best individuals belonging to each cluster are saved into the initial population of a generic optimization algorithm if the population has not been yet filled the other individuals of the population are sampled within the clusters by using a fitness based probabilistic criterion this three stage procedure implicitly performs an initial screening of the problem features in order to roughly estimate the most interesting regions of the decision space the proposed cbpi has been tested on multiple classical and modern differential evolution variants on a wide array of test problems and dimensionality values as well as on a real world problem the proposed intelligent sampling appears to have a significant impact on the algorithmic functioning as it consistently enhances the performance of the algorithms with which it is integrated

polynomial zerofinding iterative matrix algorithms abstract newbery s method is completed to a method for the construction of a complex symmetric or nonsymmetric matrix with a given characteristic polynomial the methods of fiedler schmeisser and dorfler and schmeisser for similar constructions of symmetric matrices are reviewed polynomials found in the literature are solved iteratively by one of fiedler s methods with initial values supplied either by schmeisser s method or taken on a large circle or randomly in a region of the complex plane the determinental equations are solved by the qr algorithm fiedler s method used iteratively exhibits fast convergence to simple roots even in the presence of multiple roots if at some iteration step the values of the iterates which are converging to a multiple root are averaged according to the hull mathon procedure then fast convergence is also attained for multiple roots this combination appears to have nice features for polynomials of small to moderate degree

lower bounds for the itc 2007 curriculum based course timetabling problem abstract this paper describes an approach for generating lower bounds for the curriculum based course timetabling problem which was presented at the international timetabling competition itc 2007 track 3 so far several methods based on integer linear programming have been proposed for computing lower bounds of this minimization problem we present a new partition based approach that is based on the divide and conquer principle the proposed approach uses iterative tabu search to partition the initial problem into sub problems which are solved with an ilp solver computational outcomes show that this approach is able to improve on the current best lower bounds for 12 out of the 21 benchmark instances and to prove optimality for 6 of them these new lower bounds are useful to estimate the quality of the upper bounds obtained with various heuristic approaches

a tabu search algorithm for computing an operational timetable abstract the many constraints of different types that must be taken into account and the volume of data make the timetabling problem very hard to solve various specific requirements are considered in this paper after having described precisely the problem to be solved we present a general technique based on tabu search for finding an acceptable timetable the resulting computerized algorithm is used for constructing different real course schedules the results we obtain are very satisfactory

on the use of multi neighbourhood structures within a tabu based memetic approach to university timetabling problems finding a good university timetabling system is not a simple task for a higher educational organisation as a result many approaches to generating sufficiently good solutions have been introduced this is mainly due to the high complexity within the search landscape moreover each educational organisation has its own rules and specifications in this paper a tabu based memetic algorithm that hybridises a genetic algorithm with a tabu search algorithm is proposed as an improved algorithm for university timetabling problems this algorithm is employed on a set of neighbourhood structures during the search process with the aim of gaining significant improvements in solution quality the sequence of neighbourhood structures has been considered to understand its effect on the search space random best and general sequences of neighbourhood structures have been evaluated in this work the concept of a tabu list is embedded to control the selection of neighbourhood structures that are not dependent on the problem domains during the optimisation process after the crossover and mutation operators are applied to the selected solutions from the population pool the algorithm will penalise neighbourhood structures that are unable to generate better solutions the proposed algorithm has been applied and evaluated against the latest methodologies in the literature with respect to standard benchmark problems we demonstrate that the proposed algorithm produces some of the best known results when tested on itc2007 competition datasets

iterated tabu search and variable neighborhood descent for packing unequal circles into a circular container this paper presents an iterated tabu search and variable neighborhood descent its vnd algorithm for packing unequal circles into a circular container pucc the algorithm adapts the tabu search procedure of iterated tabu search algorithms and proposes a tabu search and variable neighborhood descent ts vnd procedure we observe there are strong complementarities between the small circles and the large vacant places and propose the insert neighborhood to match up the small circles with the large vacant places although the insert neighborhood is inefficient and time consuming it is an important supplement to the classic swap neighborhood as it could arrange the small circles properly predicated on these features we employ the insert neighborhood only at chosen local minima of the swap neighborhood that shows promise for an improvement the traditional tabu search procedure is then transformed into a hybrid procedure composed of two alternative parts namely variable neighborhood descent and tabu search respectively besides this reformed procedure its vnd also incorporates other new features such as an adaptive evaluation function a novel method for accelerating the neighborhood exploration and the collision accidents criterion for evaluating how intensively the area near the current solution has been explored the computational results on three well established benchmark sets show that the proposed algorithm not only has a good discovery capability but also can provide good results within a reasonable time for a total of 84 benchmark instances the proposed algorithm improves the best known results on 23 instances matches 60 and only misses one

adaptive tabu search for course timetabling this paper presents an adaptive tabu search algorithm denoted by ats for solving a problem of curriculum based course timetabling the proposed algorithm follows a general framework composed of three phases initialization intensification and diversification the initialization phase constructs a feasible initial timetable using a fast greedy heuristic then an adaptively combined intensification and diversification phase is used to reduce the number of soft constraint violations while maintaining the satisfaction of hard constraints the proposed ats algorithm integrates several distinguished features such as an original double kempe chains neighborhood structure a penalty guided perturbation operator and an adaptive search mechanism computational results show the high effectiveness of the proposed ats algorithm compared with five reference algorithms as well as the current best known results this paper also shows an analysis to explain which are the essential ingredients of the ats algorithm

school timetabling a case in large binary integer linear programming a timetabling problem is formulated as a large integer linear programming problem in 0 1 variables a solution method based on lagrangean relaxation coupled with subgradient optimization is presented the solution method also incorporates a branch and bound procedure which takes advantage of special ordered sets of variables we present computational results for a large timetabling problem involving 900 subjects in a one year graduate program

preconditioning techniques for large linear systems a survey this article surveys preconditioning techniques for the iterative solution of large linear systems with a focus on algebraic methods suitable for general sparse matrices covered topics include progress in incomplete factorization methods sparse approximate inverses reorderings parallelization issues and block and multilevel extensions some of the challenges ahead are also discussed an extensive bibliography completes the paper

adaptive neighborhood search for nurse rostering abstract this paper presents an adaptive neighborhood search method ans for solving the nurse rostering problem proposed for the first international nurse rostering competition inrc 2010 ans uses jointly two distinct neighborhood moves and adaptively switches among three intensification and diversification search strategies according to the search history computational results assessed on the three sets of 60 competition instances show that ans improves the best known results for 12 instances while matching the best bounds for 39 other instances an analysis of some key elements of ans sheds light on the understanding of the behavior of the proposed algorithm

analysing the effects of solution space connectivity with an effective metaheuristic for the course timetabling problem this paper provides a mathematical treatment of the np hard post enrolment based course timetabling problem and presents a powerful two stage metaheuristic based algorithm to approximately solve it we focus particularly on the issue of solution space connectivity and demonstrate that when this is increased via specialised neighbourhood operators the quality of solutions achieved is generally enhanced across a well known suite of benchmark problem instances our proposed algorithm is shown to produce results that are superior to all other methods appearing in the literature however we also make note of those instances where our algorithm struggles in comparison to others and offer evidence as to why

the amended dsesc power method for polynomial root finding cardinal s matrix version of the sebastiao e silva polynomial root finder rapidly approximates the roots as the eigenvalues of the associated frobenius matrix we preserve rapid convergence to the roots but amend the algorithm to allow input polynomials with multiple roots and root clusters as in cardinal s algorithm we repeatedly square the frobenius matrix in nearly linear arithmetic time per squaring which yields dramatic speedup versus the recent effective polynomial root finder based on the application of the inverse power method to the frobenius matrix

a hybrid model of integer programming and variable neighbourhood search for highly constrained nurse rostering problems abstract this paper presents a hybrid multi objective model that combines integer programming ip and variable neighbourhood search vns to deal with highly constrained nurse rostering problems in modern hospital environments an ip is first used to solve the subproblem which includes the full set of hard constraints and a subset of soft constrains a basic vns then follows as a postprocessing procedure to further improve the ip s resulting solutions the satisfaction of the excluded constraints from the preceding ip model is the major focus of the vns very promising results are reported compared with a commercial genetic algorithm and a hybrid vns approach on real instances arising in a dutch hospital the comparison results demonstrate that our hybrid approach combines the advantages of both the ip and the vns to beat other approaches in solving this type of problems we also believe that the proposed methodology can be applied to other resource allocation problems with a large number of constraints

matrix computations and polynomial root finding with preprocessing solution of homogeneous linear systems of equations is a basic operation of matrix computations the customary algorithms rely on pivoting orthogonalization and svd but we employ randomized preprocessing instead this enables us to accelerate the solution dramatically both in terms of the estimated arithmetic cost and the observed cpu time the approach is effective in the cases of both general and structured input matrices and we extend it and its computational advantages to the solution of nonhomogeneous linear systems of equations matrix eigen solving the solution of polynomial and secular equations and approximation of a matrix by a nearby matrix that has a smaller rank or a fixed structure e g of the toeplitz or hankel type our analysis and extensive experiments show the power of the presented algorithms

eigen solving via reduction to dpr1 matrices highly effective polynomial root finders have been recently designed based on eigen solving for dpr1 that is diagonal rank one matrices we extend these algorithms to eigen solving for the general matrix by reducing the problem to the case of the dpr1 input via intermediate transition to a tpr1 that is triangular rank one matrix our transforms use substantially fewer arithmetic operations than the qr classical algorithms but employ non unitary similarity transforms of a tpr1 matrix whose representation tends to be numerically unstable we however operate with tpr1 matrices implicitly as with the inverses of hessenberg matrices in this way our transform of an input matrix into a similar dpr1 matrix partly avoids numerical stability problems and still substantially decreases arithmetic cost versus the qr algorithm

functions on the sphere with critical points in pairs and orthogonal geodesic chords abstract using an estimate on the number of critical points for a morse even function on the sphere s m m 1 we prove a multiplicity result for orthogonal geodesic chords in riemannian manifolds with boundary that are diffeomorphic to euclidean balls this yields also a multiplicity result for brake orbits in a potential well

additive preconditioning and aggregation in matrix computations we combine our novel svd free additive preconditioning with aggregation and other relevant techniques to facilitate the solution of a linear system of equations and other fundamental matrix computations our analysis and experiments show the power of our algorithms guide us in selecting most effective policies of preconditioning and aggregation and provide some new insights into these and related subjects compared to the popular svd based multiplicative preconditioners our additive preconditioners are generated more readily and for a much larger class of matrices furthermore they better preserve matrix structure and sparseness and have a wider range of applications e g they facilitate the solution of a consistent singular linear system of equations and of the eigenproblem

adaptive linear combination of heuristic orderings in constructing examination timetables in this paper we investigate adaptive linear combinations of graph coloring heuristics with a heuristic modifier to address the examination timetabling problem we invoke a normalisation strategy for each parameter in order to generalise the specific problem data two graph coloring heuristics were used in this study largest degree and saturation degree a score for the difficulty of assigning each examination was obtained from an adaptive linear combination of these two heuristics and examinations in the list were ordered based on this value the examinations with the score value representing the higher difficulty were chosen for scheduling based on two strategies we tested for single and multiple heuristics with and without a heuristic modifier with different combinations of weight values for each parameter on the toronto and itc2007 benchmark data sets we observed that the combination of multiple heuristics with a heuristic modifier offers an effective way to obtain good solution quality experimental results demonstrate that our approach delivers promising results we conclude that this adaptive linear combination of heuristics is a highly effective method and simple to implement

additive preconditioning for matrix computations abstract our randomized additive preconditioners are readily available and regularly facilitate the solution of linear systems of equations and eigen solving for a very large class of input matrices we study the generation of such preconditioners and their impact on the rank and the condition number of a matrix we also propose some techniques for their refinement and two alternative versions of randomized preprocessing our analysis and experiments show the power of our approach even where we employ weak randomization that is generate sparse and structured preconditioners defined by a small number of random parameters

computing matrix eigenvalues and polynomial zeros where the output is real surprisingly simple corollaries from the courant fischer minimax characterization theorem enable us to devise a very effective algorithm for the evaluation of a set s interleaving the set e of the eigenvalues of an n x n real symmetric tridiagonal rst matrix tn as well as a point that splits e into two subsets of comparable cardinalities furthermore we extend this algorithm so as to approximate all the n eigenvalues of tn at nearly optimal sequential and parallel cost that is at the cost of staying within polylogarithmic factors from the straightforward lower bounds the resulting improvement of the known processor bound in nc algorithms for the rst eigenproblem is roughly by factor n our approach extends the previous works m ben or and p tiwari j complexity 6 1990 pp 417 442 and m ben or et al siam j comput 17 1988 pp 1081 1092 for the approximation of the zeros of a polynomial having only real zeros and our algorithm leads to an alternative and simplified derivation of the known record parallel and sequential complexity estimates for the latter problem

iterated tabu search for the circular open dimension problem abstract this paper investigates the circular open dimension problem codp which consists of packing a set of circles of known radii into a strip of fixed width and unlimited length without overlapping the objective is to minimize the length of the strip in this paper codp is solved by a series of sub problems each corresponding to a fixed strip length for each sub problem an iterated tabu search approach named its is proposed its starts from a randomly generated solution and attempts to gain improvements by a tabu search procedure after that if the obtained solution is not feasible a perturbation operator is subsequently employed to reconstruct the incumbent solution and an acceptance criterion is implemented to determine whether or not accept the perturbed solution as a supplementary method the length of the strip is determined in monotonously decreasing way with the aid of some post processing techniques the search terminates and returns the best found solution after the allowed computation time has been elapsed computational experiments based on numerous well known benchmark instances show that its produces quite competitive results with respect to the best known results while the computational time remains reasonable for each instance

an iterated multi stage selection hyper heuristic there is a growing interest towards the design of reusable general purpose search methods that are applicable to different problems instead of tailored solutions to a single particular problem hyper heuristics have emerged as such high level methods that explore the space formed by a set of heuristics move operators or heuristic components for solving computationally hard problems a selection hyper heuristic mixes and controls a predefined set of low level heuristics with the goal of improving an initially generated solution by choosing and applying an appropriate heuristic to a solution in hand and deciding whether to accept or reject the new solution at each step under an iterative framework designing an adaptive control mechanism for the heuristic selection and combining it with a suitable acceptance method is a major challenge because both components can influence the overall performance of a selection hyper heuristic in this study we describe a novel iterated multi stage hyper heuristic approach which cycles through two interacting hyper heuristics and operates based on the principle that not all low level heuristics for a problem domain would be useful at any point of the search process the empirical results on a hyper heuristic benchmark indicate the success of the proposed selection hyper heuristic across six problem domains beating the state of the art approach

mathematical programming models and algorithms for a class faculty assignment problem this paper presents mathematical programming models for assigning faculty members to classes including among typical academic class scheduling issues certain specialized central policies at kuwait university the time slots for classes are initially assumed to be given and an integer programming model cfam is constructed to solve the resulting problem which aims to minimize the individual and collective dissatisfaction of faculty members in a fair fashion where dissatisfaction is measured by a function of the assignment of faculty members to time slots and specific classes in order to enhance the quality of results obtained in practice the model is modified ecfam so that the time slots for the classes can be changed however with restrictions related to efficient facility utilization and permitting an administratively regulated maximum number of changes gender based modeling considerations are also introduced in order to maintain desirable class offering patterns computational results are presented based on solving the models directly by the cplex mip version 7 5 package and also using a specialized lp based heuristic the faculty schedules generated via the proposed approach based on a number of case studies related to the department of mathematics and computer science at kuwait university reveal that this approach yields improved schedules in terms of fairness and enhanced satisfaction levels among faculty members

two purposes for matrix factorization a historical appraisal matrix factorization in numerical linear algebra nla typically serves the purpose of restating some given problem in such a way that it can be solved more readily for example one major application is in the solution of a linear system of equations in contrast within applied statistics psychometrics as p a much more common use for matrix factorization is in presenting possibly spatially the structure that may be inherent in a given data matrix obtained on a collection of objects observed over a set of variables the actual components of a factorization are now of prime importance and not just as a mechanism for solving another problem we review some connections between nla and as p and their respective concerns with matrix factorization and the subsequent rank reduction of a matrix we note in particular that several results available for many decades in as p were more recently re discovered in the nla literature two other distinctions between nla and as p are also discussed briefly how a generalized singular value decomposition might be defined and the differing uses for the newer methods of optimization based on cyclic or iterative projections

randomized preprocessing of homogeneous linear systems of equations abstract our randomized preprocessing enables pivoting free and orthogonalization free solution of homogeneous linear systems of equations in the case of toeplitz inputs we decrease the estimated solution time from quadratic to nearly linear and our tests show dramatic decrease of the cpu time as well we prove numerical stability of our approach and extend it to solving nonsingular linear systems inversion and generalized moore penrose inversion of general and structured matrices by means of newton s iteration approximation of a matrix by a nearby matrix that has a smaller rank or a smaller displacement rank matrix eigen solving and root finding for polynomial and secular equations and for polynomial systems of equations some by products and extensions of our study can be of independent technical intersest e g our extensions of the sherman morrison woodbury formula for matrix inversion our estimates for the condition number of randomized matrix products and preprocessing via augmentation

new progress in real and complex polynomial root finding matrix methods are increasingly popular for polynomial root finding the idea is to approximate the roots as the eigenvalues of the companion or generalized companion matrix associated with an input polynomial the algorithms also solve secular equation qr algorithm is the most customary method for eigen solving but we explore the inverse rayleigh quotient iteration instead which turns out to be competitive with the most popular root finders because of its excellence in exploiting matrix structure to advance the iteration we preprocess the matrix and incorporate newton s linearization repeated squaring homotopy continuation techniques and some heuristics the resulting algorithms accelerate the known numerical root finders for univariate polynomial and secular equations and are particularly well suited for the acceleration by using parallel processing furthermore even on serial computers the acceleration is dramatic for numerical approximation of the real roots in the typical case where they are much less numerous than all complex roots

a generalized class teacher model for some timetabling problems abstract we consider a theoretical model which extends the basic class teacher model of timetabling and which corresponds to some situations which occur frequently in the basic training programs of universities and schools we are given m teachers t 1 t m and n classes c 1 c n the set of classes is partitioned into p disjoint subsets g 1 g p in such a way that in addition to the lectures given by one teacher to one class there are some lectures given by one teacher to the students of all classes in group g l 1 l p such lectures will be called group lectures the number a lj of one hour group lectures which teacher t j must deliver to group g l and the number b ij of one hour class teaching which t j must give to class c i are given is there a timetable of t hours or length t so that each class c i and each group g l receive all their lectures but no student is scheduled to be taught by more than one teacher in each hour and no teacher must teach to more than one group or class in each hour we show that this problem is np complete and find some sufficient conditions for the existence of a timetable of length t we also describe an algorithm for constructing a timetable corresponding to the requirement matrices a a lj and b b ij and show that under a natural assumption on a and b this algorithm finds a timetable within 7 6 of the optimum length

a memetic algorithm for graph coloring abstract given an undirected graph g v e with a set v of vertices and a set e of edges the graph coloring problem consists of partitioning all vertices into k independent sets and the number of used colors k is minimized this paper presents a memetic algorithm denoted by macol for solving the problem of graph coloring the proposed macol algorithm integrates several distinguished features such as an adaptive multi parent crossover ampax operator and a distance and quality based replacement criterion for pool updating the proposed algorithm is evaluated on the dimacs challenge benchmarks and computational results show that the proposed macol algorithm achieves highly competitive results compared with 11 state of the art algorithms the influence of some ingredients of macol on its performance is also analyzed

a methodology for determining an effective subset of heuristics in selection hyper heuristics we address the important step of determining an effective subset of heuristics in selection hyper heuristics little attention has been devoted to this in the literature and the decision is left at the discretion of the investigator the performance of a hyper heuristic depends on the quality and size of the heuristic pool using more than one heuristic is generally advantageous however an unnecessary large pool can decrease the performance of adaptive approaches our goal is to bring methodological rigour to this step the proposed methodology uses non parametric statistics and fitness landscape measurements from an available set of heuristics and benchmark instances in order to produce a compact subset of effective heuristics for the underlying problem we also propose a new iterated local search hyper heuristic using multi armed bandits coupled with a change detection mechanism the methodology is tested on two real world optimization problems course timetabling and vehicle routing the proposed hyper heuristic with a compact heuristic pool outperforms state of the art hyper heuristics and competes with problem specific methods in course timetabling even producing new best known solutions in 5 out of the 24 studied instances

a study of heuristic combinations for hyper heuristic systems for the uncapacitated examination timetabling problem research in the domain of examination timetabling is moving towards developing methods that generalise well over a range of problems this is achieved by implementing hyper heuristic systems to find the best heuristic or heuristic combination to allocate examinations when constructing a timetable for a problem heuristic combinations usually take the form of a list of low level heuristics that are applied sequentially this study proposes an alternative representation for heuristic combinations namely a hierarchical combination of heuristics furthermore the heuristics in each combination are applied simultaneously rather than sequentially the study also introduces a new low level heuristic namely highest cost a set of heuristic combinations of this format have been tested on the 13 carter benchmarks the quality of the examination timetables induced using these combinations are comparable to and in some cases better than those produced by hyper heuristic systems combining and applying heuristic combinations sequentially

a composite polynomial zerofinding matrix algorithm abstract a globally convergent matrix algorithm is presented for finding the real and complex zeros of a complex polynomial p x a combination of schmeisser s and fiedler s matrices is used in the algorithm first the greatest common divisor g x of p x and its derivative p x is obtained and used to reduce p x to a polynomial q x p x q x with simple zeros second the zeros of q x are computed either by finding the eigenvalues of the first block of schmeisser s matrix or by applying fiedler s algorithm recursively in either case the eigenvalues are obtained by the qr algorithm third the multiplicity of each zero of p x is calculated by means of lagouanelle s modified limiting formula sample numerical results are given to demonstrate the effectiveness of the algorithm

hyper heuristic approaches for the response time variability problem we propose two classes for the implementation of hyper heuristic algorithms the first is based on constructive heuristics whereas the second uses improvement methods within the latter class a general framework is designed for the use of local search procedures and metaheuristics as low level heuristics a dynamic scheme to guide the use of these approaches is also devised these ideas are tested on an np hard scheduling problem known as the response time variability problem rtvp an intensive computational experiment shows especially in the second class where the new best results are found the effectiveness of the proposed hyper heuristics

schur aggregation for linear systems and determinants we apply our recent preconditioning techniques to the solution of linear systems of equations and computing determinants we combine these techniques with the sherman morrison woodbury formula its new variations aggregation iterative refinement and advanced algorithms that rapidly compute sums and products either error free or with the desired high accuracy our theoretical and experimental study shows the power of this approach

solving linear systems of equations with randomization augmentation and aggregation abstract seeking a basis for the null space of a rectangular and possibly rank deficient and ill conditioned matrix we apply randomization augmentation and aggregation to reduce our task to computations with well conditioned matrices of full rank our algorithms avoid pivoting and orthogonalization preserve matrix structure and sparseness and in the case of an ill conditioned input perform only a small part of the computations with high accuracy we extend the algorithms to the solution of nonhomogeneous nonsingular ill conditioned linear systems of equations whose matrices have small numerical nullities our estimates and experiments show dramatic progress versus the customary matrix algorithms where the input matrices are rank deficient or ill conditioned our study can be of independent technical interest we extend the known results on conditioning of random matrices to randomized preconditioning estimate the condition numbers of randomly augmented matrices and link augmentation to aggregation as well as homogeneous to nonhomogeneous linear systems of equations

seifert conjecture in the even convex case in this paper we prove that there exist at least n geometrically distinct brake orbits on every c2 compact convex symmetric hypersurface in 2n satisfying the reversible condition n with n diag in in as a consequence we show that if the hamiltonian function is convex and even then seifert conjecture of 1948 on the multiplicity of brake orbits holds for any positive integern 2014 wiley periodicals inc

heuristic space diversity control for improved meta hyper heuristic performance this paper expands on the concept of heuristic space diversity and investigates various strategies for the management of heuristic space diversity within the context of a meta hyper heuristic algorithm in search of greater performance benefits evaluation of various strategies on a diverse set of floating point benchmark problems shows that heuristic space diversity has a significant impact on hyper heuristic performance an exponentially increasing strategy eihh obtained the best results the value of a priori information about constituent algorithm performance on the benchmark set in question was also evaluated finally eihh demonstrated good performance when compared to a popular population based algorithm portfolio algorithm and the best performing constituent algorithm

iteration theory of l index and multiplicity of brake orbits abstract in this paper we first establish the bott type iteration formulas and some abstract precise iteration formulas of the maslov type index theory associated with a lagrangian subspace for symplectic paths as an application we prove that there exist at least n 2 1 geometrically distinct brake orbits on every c 2 compact convex symmetric hypersurface in r 2 n satisfying the reversible condition n furthermore if all brake orbits on this hypersurface are nondegenerate then there are at least n geometrically distinct brake orbits on it as a consequence we show that there exist at least n 2 1 geometrically distinct brake orbits in every bounded convex symmetric domain in r n furthermore if all brake orbits in this domain are nondegenerate then there are at least n geometrically distinct brake orbits in it in the symmetric case we give a positive answer to the seifert conjecture of 1948 under a generic condition

examples with minimal number of brake orbits and homoclinics in annular potential regions abstract we use a geometric construction to exhibit examples of autonomous lagrangian systems admitting exactly two homoclinics emanating from a nondegenerate maximum of the potential energy and reaching a regular level of the potential having the same value of the maximum point similarly we show examples of hamiltonian systems that admit exactly two brake orbits in an annular potential region connecting the two connected components of the boundary of the potential well these examples show that the estimates proven in 2 are sharp