a discrete dynamic convexized method for nonlinear integer programming in this paper we consider the box constrained nonlinear integer programming problem we present an auxiliary function which has the same discrete global minimizers as the problem the minimization of the function using a discrete local search method can escape successfully from previously converged discrete local minimizers by taking increasing values of a parameter we propose an algorithm to find a global minimizer of the box constrained nonlinear integer programming problem the algorithm minimizes the auxiliary function from random initial points we prove that the algorithm can converge asymptotically with probability one numerical experiments on a set of test problems show that the algorithm is efficient and robust

accurate solution of dense linear systems part i algorithms in rounding to nearest we investigate how extra precise accumulation of dot products can be used to solve ill conditioned linear systems accurately for a given p bit working precision extra precise evaluation of a dot product means that the products and summation are executed in 2p bit precision and that the final result is rounded into the p bit working precision denote by u 2 p the relative rounding error unit in a given working precision we treat two types of matrices first up to condition number u 1 and second up to condition number u 2 for both types of matrices we present two types of methods first for calculating an approximate solution and second for calculating rigorous error bounds for the solution together with the proof of non singularity of the matrix of the linear system in the first part of this paper we present algorithms using only rounding to nearest in part ii we use directed rounding to obtain better results all algorithms are given in executable matlab code and are available from my homepage

lattice basis reduction improved practical algorithms and solving subset sum problems we report on improved practical algorithms for lattice basis reduction we present a variant of the l3 algorithm with deep insertions and a practical algorithm for blockwise korkine zolotarev reduction a concept extending l3 reduction that has been introduced by schnorr 1987 empirical tests show that the strongest of these algorithms solves almost all subset sum problems with up to 58 random weights of arbitrary bit length within at most a few hours on a unisys 6000 70 or within a couple of minutes on a sparc 2 computer

on quadratic integral equation of fractional orders abstract we present an existence theorem for a nonlinear quadratic integral equations of fractional orders arising in the queuing theory and biology in the banach space of real functions defined and continuous on a bounded and closed interval the concept of measure of noncompactness and a fixed point theorem due to darbo are the main tool in carrying out our proof

on the existence of solutions to the quadratic mixed integer mean variance portfolio selection problem abstract in the standard mean variance portfolio selection approach several operative features are not taken into account among these neglected aspects one of particular interest is the finite divisibility of the stock assets i e the obligation to buy sell only integer quantities of asset lots whose number is pre established in order to consider such a feature we deal with a suitably defined quadratic mixed integer programming problem in particular we formulate this problem in terms of quantities of asset lots instead of as usual in terms of capital per cent quotas secondly we provide necessary and sufficient conditions for the existence of a non empty mixed integer feasible set of the considered programming problem thirdly we present some rounding procedures for finding in a finite number of steps a feasible mixed integer solution which is better than the one detected by the necessary and sufficient conditions in terms of the value assumed by the portfolio variance finally we perform an extensive computational experiment by means of which we verify the goodness of our approach

on a perturbed quadratic fractional integral equation of abel type we present an existence theorem for monotonic solutions of a perturbed quadratic fractional integral equation in c 0 1 our equation contains the famous chandrasekhar integral equation as a special case the concept of the measure of noncompactness related to monotonicity introduced by j banas and l olszowy and a fixed point theorem due to darbo are the main tools in carrying out our proof moreover we give an example for indicating the natural realizations of our abstract result presented in this paper

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

asymptotic strong duality for bounded integer programming a logarithmic exponential dual formulation a logarithmic exponential dual formulation is proposed in this paper for bounded integer programming problems this new dual formulation possesses an asymptotic strong duality property and guarantees the identification of an optimal solution of the primal problem these prominent features are achieved by exploring a novel nonlinear lagrangian function deriving an asymptotic zero duality gap investigating the unimodality of the associated dual function and ensuring the primal feasibility of optimal solutions in the dual formulation one other feature of the logarithmicexponential dual formulation is that no actual dual search is needed when parameters are set above certain threshold values

transmutation method for solving erdelyi kober fractional differintegral equations abstract by means of fractional calculus techniques we find explicit solutions of volterra integral equations of the second kind and fractional differential and differintegral equations involving erdelyi kober fractional integrals and derivatives also some hypergeometric integral equations have been considered and solved as double erdelyi kober equations of the second kind we use the transmutation method to reduce the solutions of all these equations to known solutions of simpler riemann liouville equations of the same type some examples are given

computing singular values of diagonally dominant matrices to high relative accuracy for a row diagonally dominant matrix if all of its off diagonal entries and its diagonally dominant parts which are defined for each row as the absolute value of the diagonal entry subtracted by the sum of the absolute values of off diagonal entries in that row are accurately known we develop an algorithm that computes all the singular values including zero ones if any with relative errors in the order of the machine precision when the matrix is also symmetric with positive diagonals i e a symmetric positive semi definite diagonally dominant matrix our algorithm computes all eigenvalues to high relative accuracy rounding error analysis will be given and numerical examples will be presented to demonstrate the high relative accuracy of the algorithm

accurate floating point summation part i faithful rounding given a vector of floating point numbers with exact sum s we present an algorithm for calculating a faithful rounding of s i e the result is one of the immediate floating point neighbors of s if the sum s is a floating point number we prove that this is the result of our algorithm the algorithm adapts to the condition number of the sum i e it is fast for mildly conditioned sums with slowly increasing computing time proportional to the logarithm of the condition number all statements are also true in the presence of underflow the algorithm does not depend on the exponent range our algorithm is fast in terms of measured computing time because it allows good instruction level parallelism it neither requires special operations such as access to mantissa or exponent it contains no branch in the inner loop nor does it require some extra precision the only operations used are standard floating point addition subtraction and multiplication in one working precision for example double precision certain constants used in the algorithm are proved to be optimal

the accuracy of floating point summation the usual recursive summation technique is just one of several ways of computing the sum of n floating point numbers five summation methods and their variations are analyzed here the accuracy of the methods is compared using rounding error analysis and numerical experiments four of the methods are shown to be special cases of a general class of methods and an error analysis is given for this class no one method is uniformly more accurate than the others but some guidelines are given on the choice of method in particular cases

numerical experience with lower bounds for miqp branch and bound the solution of convex mixed integer quadratic programming miqp problems with a general branch and bound framework is considered it is shown how lower bounds can be computed efficiently during the branch and bound process improved lower bounds such as the ones derived in this paper can reduce the number of quadratic programming qp problems that have to be solved the branch and bound approach is also shown to be superior to other approaches in solving miqp problems numerical experience is presented which supports these conclusions

the nonlinear resource allocation problem in this paper we study the nonlinear resource allocation problem defined as the minimization of a convex function over one convex constraint and bounded integer variables this problem is encountered in a variety of applications including capacity planning in manufacturing and computer networks production planning capital budgeting and stratified sampling despite its importance to these and other applications the nonlinear resource allocation problem has received little attention in the literature therefore we develop a branch and bound algorithm to solve this class of problems first we present a general framework for solving the continuous variable problem then we use this framework as the basis for our branch and bound method we also develop reoptimization procedures and a heuristic that significantly improve the performance of the branch and bound algorithm in addition we show how the algorithm can be modified to solve nonconvex problems so that a concave objective function can be handled

on erdelyi kober type quadratic integral equation with linear modification of the argument abstract in this paper we study the existence of solutions for a nonlinear erdelyi kober fractional quadratic integral equations with linear modification of the argument the main tool used in our considerations is the technique associated to measures of noncompactness we compare our results with the ones obtained in recent papers finally we present some examples illustrating our results

a survey of methods for pure nonlinear integer programming the subject of this paper is a classification and discussion of algorithms for solution of nonlinear pure integer programming problems the survey is organized by characterizing the mathematical form of the nonlinear optimization problems addressed by the various algorithms if any method can be used without changes to solve mixed integer nonlinear programming problems that fact is mentioned in the description of the algorithm however algorithms which must be applied only to mixed integer problems are not surveyed

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

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

on erdelyi kober fractional urysohn volterra quadratic integral equations a very general nonlinear singular integral equation is introduced namely u f 1 u f 2 u 0 s 1 k s a u s s 1 d s 0 1 0 and 0 1 the above equation is called erdelyi kober fractional urysohn volterra quadratic integral equation the main goal is to show that the above equation has solutions in c0 1 and these solutions are nonnegative and nondecreasing on 0 1 by means of a measure of noncompactness and darbo fixed point theorem we prove our main results in the end of the paper we give an example to show that our assumptions of our abstract results are rather easy to verify

reducing floating point error in dot product using the superblock family of algorithms this paper discusses both the theoretical and statistical errors obtained by various well known dot products from the canonical to pairwise algorithms and introduces a new and more general framework that we have named superblock which subsumes them and permits a practitioner to make trade offs between computational performance memory usage and error behavior we show that algorithms with lower error bounds tend to behave noticeably better in practice unlike many such error reducing algorithms superblock requires no additional floating point operations and should be implementable with little to no performance loss making it suitable for use as a performance critical building block of a linear algebra kernel

john von neumann s analysis of gaussian elimination and the origins of modern numerical analysis just when modern computers digital electronic and programmable were being invented john von neumann and herman goldstine wrote a paper to illustrate the mathematical analyses that they believed would be needed to use the new machines effectively and to guide the development of still faster computers their foresight and the congruence of historical events made their work the first modern paper in numerical analysis von neumann once remarked that to found a mathematical theory one had to prove the first theorem which he and goldstine did for the accuracy of mechanized gaussian elimination but their paper was about more than that von neumann and goldstine described what they surmized would be the significant questions once computers became available for computational science and they suggested enduring ways to answer them

monotonic solutions of a quadratic integral equation of volterra type abstract we study a nonlinear quadratic integral equation of volterra type in the banach space of real functions defined and continuous on a bounded and closed interval with the help of a suitable measure of noncompactness we show that the mentioned integral equation has monotonic solutions

accurate summation dot product and polynomial evaluation in complex floating point arithmetic several different techniques and softwares intend to improve the accuracy of results computed in a fixed finite precision here we focus on methods to improve the accuracy of summation dot product and polynomial evaluation such algorithms exist real floating point numbers in this paper we provide new algorithms which deal with complex floating point numbers we show that the computed results are as accurate as if computed in twice the working precision the algorithms are simple since they only require addition subtraction and multiplication of floating point numbers in the same working precision as the given data

on ergodic and mixing properties of the triangle map abstract in this paper we study in detail both analytically and numerically the dynamical properties of the triangle map a piecewise parabolic automorphism of the two dimensional torus for different values of the two independent parameters defining the map the dynamics is studied numerically by means of two different symbolic encoding schemes both relying on the fact that it maps polygons to polygons in the first scheme we consider dynamically generated partitions made out of suitable sets of disjoint polygons in the second we consider the standard binary partition of the torus induced by the discontinuity set these encoding schemes are studied in detail and shown to be compatible although not equivalent the ergodic properties of the triangle map are then investigated in terms of the markov transition matrices associated to the above schemes and furthermore compared to the spectral properties of the koopman operator in l 2 t 2 finally a stochastic version of the triangle map is introduced and studied a simple heuristic analysis of the latter yields the correct statistical and scaling behaviours of the correlation functions of the original map

a hierarchy of relaxations for nonlinear convex generalized disjunctive programming we propose a framework to generate alternative mixed integer nonlinear programming formulations for disjunctive convex programs that lead to stronger relaxations we extend the concept of basic steps defined for disjunctive linear programs to the nonlinear case a basic step is an operation that takes a disjunctive set to another with fewer number of conjuncts we show that the strength of the relaxations increases as the number of conjuncts decreases leading to a hierarchy of relaxations we prove that the tightest of these relaxations allows in theory the solution of the disjunctive convex program as a nonlinear programming problem we present a methodology to guide the generation of strong relaxations without incurring an exponential increase of the size of the reformulated mixed integer program finally we apply the theory developed to improve the computational efficiency of solution methods for nonlinear convex generalized disjunctive programs gdp this methodology is validated through a set of numerical examples

an approximate algorithm for nonlinear integer programming abstract this paper suggests an approximate algorithm designed to solve nonlinear integer problems this algorithm belongs to the class of component algorithms of feasible integer directions the search for a feasible integer direction is done on the basis of a linear approximation of the objective function and the constraints at the integer points under consideration theoretical analysis is presented as well as experimental investigation using the algorithm for test examples taken from the literature

when is rounding allowed in integer nonlinear optimization abstract in this paper we consider nonlinear integer optimization problems nonlinear integer programming has mainly been studied for special classes such as convex and concave objective functions and polyhedral constraints in this paper we follow an other approach which is not based on convexity or concavity studying geometric properties of the level sets and the feasible region we identify cases in which an integer minimizer of a nonlinear program can be found by rounding up or down the coordinates of a solution to its continuous relaxation we call this property rounding property if it is satisfied it enables us for fixed dimension to solve an integer programming problem in the same time complexity as its continuous relaxation we also investigate the strong rounding property which allows rounding a solution to the continuous relaxation to the next integer solution and in turn yields that the integer version can be solved in the same time complexity as its continuous relaxation for arbitrary dimensions

computation and storage of the bianisotropic scalar green s function and its derivatives a set of algorithms is proposed for the accurate and efficient computation and storage of the bianisotropic scalar green s function the computation is based on an expansion of the green s function into chebyshev polynomials the analytical properties of these polynomials are exploited to allow the accurate computation of the derivatives of the green s function as well as the green s function itself for lossy materials the proposed computation strategy is provably robust in addition a multilevel storage scheme with a favorable complexity based on the chebyshev polynomial expansion is proposed for the storage of the expansion coefficients numerical results showcase the accuracy and computational complexity of the proposed algorithms

a new distillation algorithm for floating point summation the summation of n floating point numbers is ubiquitous in numerical computations we present a new distillation algorithm for floating point summation which is stable efficient and accurate the algorithm iteratively distills the summands without discarding any significant digit until the partial sums cannot change the whole sum it uses standard floating point arithmetic and does not rely on the choice of radix or any other specific assumption furthermore the error bound of our algorithm is independent of n and less than 1 ulp

monotonic solutions of a quadratic integral equation of fractional order abstract in the paper we indicate an error made in the proof of the main result of the paper m a darwish on quadratic integral equation of fractional orders j math anal appl 311 2005 112 119 moreover we provide correct proof of a slightly modified version of the mentioned result the main tool used in our proof is the technique associated with the hausdorff measure of noncompactness

an algorithm for indefinite integer quadratic programming abstract we present an algorithm for finding the global minimum of an indefinite quadratic function over the integers contained in a compact convex set to find this minimum the algorithm first transforms the problem into an equivalent problem with a separable objective function it then uses a branch and bound search on the values of the constraints rather than the variables of the transformed problem

branch and bound experiments in convex nonlinear integer programming the branch and bound principle has long been established as an effective computational tool for solving mixed integer linear programming problems this paper investigates the computational feasibility of branch and bound methods in solving convex nonlinear integer programming problems the efficiency of a branch and bound method often depends on the rules used for selecting the branching variables and branching nodes among others the concepts of pseudo costs and estimations are implemented in selecting these parameters since the efficiency of the algorithm also depends on how fast an upper bound on the objective minimum is attained heuristic rules are developed to locate an integer feasible solution to provide an upper bound the different criteria for selecting branching variables branching nodes and heuristics form a total of 27 branch and bound strategies these strategies are computationally tested and the empirical results are presented

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

ultimately fast accurate summation we present two new algorithms fastaccsum and fastprecsum one to compute a faithful rounding of the sum of floating point numbers and the other for a result as if computed in k fold precision faithful rounding means the computed result either is one of the immediate floating point neighbors of the exact result or is equal to the exact sum if this is a floating point number the algorithms are based on our previous algorithms accsum and precsum and improve them by up to 25 the first algorithm adapts to the condition number of the sum i e the computing time is proportional to the difficulty of the problem the second algorithm does not need extra memory and the computing time depends only on the number of summands and k both algorithms are the fastest known in terms of flops they allow good instruction level parallelism so that they are also fast in terms of measured computing time the algorithms require only standard floating point addition subtraction and multiplication in one working precision for example double precision

polyhedral approximation in mixed integer convex optimization generalizing both mixed integer linear optimization and convex optimization mixed integer convex optimization possesses broad modeling power but has seen relatively few advances in general purpose solvers in recent years in this paper we intend to provide a broadly accessible introduction to our recent work in developing algorithms and software for this problem class our approach is based on constructing polyhedral outer approximations of the convex constraints resulting in a global solution by solving a finite number of mixed integer linear and continuous convex subproblems the key advance we present is to strengthen the polyhedral approximations by constructing them in a higher dimensional space in order to automate this extended formulation we rely on the algebraic modeling technique of disciplined convex programming dcp and for generality and ease of implementation we use conic representations of the convex constraints although our framework requires a manual translation of existing models into dcp form after performing this transformation on the minlplib2 benchmark library we were able to solve a number of unsolved instances and on many other instances achieve superior performance compared with state of the art solvers like bonmin scip and artelys knitro

a critical review of discrete filled function methods in solving nonlinear discrete optimization problems many real life problems can be modeled as nonlinear discrete optimization problems such problems often have multiple local minima and thus require global optimization methods due to high complexity of these problems heuristic based global optimization techniques are usually required when solving large scale discrete optimization or mixed discrete optimization problems one of the more recent global optimization tools is known as the discrete filled function method nine variations of the discrete filled function method in literature are identified and a review on theoretical properties of each method is given some of the most promising filled functions are tested on various benchmark problems numerical results are given for comparison

global optimization of mixed integer nonlinear programs a theoretical and computational study this work addresses the development of an efficient solution strategy for obtaining global optima of continuous integer and mixed integer nonlinear programs towards this end we develop novel relaxation schemes range reduction tests and branching strategies which we incorporate into the prototypical branch and bound algorithm in the theoretical algorithmic part of the paper we begin by developing novel strategies for constructing linear relaxations of mixed integer nonlinear programs and prove that these relaxations enjoy quadratic convergence properties we then use lagrangian linear programming duality to develop a unifying theory of domain reduction strategies as a consequence of which we derive many range reduction strategies currently used in nonlinear programming and integer linear programming this theory leads to new range reduction schemes including a learning heuristic that improves initial branching decisions by relaying data across siblings in a branch and bound tree finally we incorporate these relaxation and reduction strategies in a branch and bound algorithm that incorporates branching strategies that guarantee finiteness for certain classes of continuous global optimization problems in the computational part of the paper we describe our implementation discussing wherever appropriate the use of suitable data structures and associated algorithms we present computational experience with benchmark separable concave quadratic programs fractional 0 1 programs and mixed integer nonlinear programs from applications in synthesis of chemical processes engineering design just in time manufacturing and molecular design

improvement of error free splitting for accurate matrix multiplication recently new algorithms for accurate matrix multiplication have been developed by the authors a characteristic of the algorithms is a high dependency on level 3 blas routines which are highly optimized for several architectures an error free splitting for floating point matrices is a key technique in the algorithms in this paper an improvement of the error free splitting is focused on it is shown by numerical examples that the accuracy of computed results of matrix products can be improved by the modified error free splitting compared to that by the previous algorithms

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

estimating the number of stable configurations for the generalized thomson problem given a natural number n one may ask what configuration of n points on the two sphere minimizes the discrete generalized coulomb energy if one applies a gradient based numerical optimization to this problem one encounters many configurations that are stable but not globally minimal this led the authors of this manuscript to the question how many stable configurations are there in this manuscript we report methods for identifying and counting observed stable configurations and estimating the actual number of stable configurations these estimates indicate that for n approaching two hundred there are at least tens of thousands of stable configurations

accurate and efficient floating point summation we present and analyze several simple algorithms for accurately computing the sum of n floating point numbers using a wider accumulator let f and f be the number of significant bits in the summands and the accumulator respectively then assuming gradual underflow no overflow and round to nearest arithmetic up to approximately 2f f numbers can be added accurately by simply summing the terms in decreasing order of exponents yielding a sum correct to within about 1 5 units in the last place ulps we apply this result to the floating point formats in the ieee floating point standard for example a dot product of single precision vectors of length at most 33 computed using double precision and sorting is guaranteed correct to nearly 1 5 ulps if double extended precision is used the vector length can be as large as 65 537 we also investigate how the cost of sorting can be reduced or eliminated while retaining accuracy

ellipsoid bounds for convex quadratic integer programming solving convex quadratic integer minimization problems by a branch and bound algorithm requires tight lower bounds on the optimal objective value to obtain such dual bounds we follow the approach of c buchheim a caprara and a lodi math program 135 2012 pp 369 395 and underestimate the objective function by another convex quadratic function that can be minimized over the integers by simply rounding its continuous minimizer in geometric terms we approximate the sublevel sets of the original objective function by auxiliary ellipsoids having the strong rounding property introduced by r hu bner and a scho bel european j oper res 237 2014 pp 404 410 we first consider axisparallel ellipsoids corresponding to separable convex quadratic functions and show how to efficiently compute an axisparallel ellipsoid yielding the tightest lower bound both in the situation where the location of the continuous minimizer is given and in the situation where it varies as it happens in a branch and bound approach in the latter case we compute both worst case and average case optimal ellipsoids moreover we consider the class of quasi round ellipsoids from hu bner and scho bel in this case we do not know whether optimal auxiliary ellipsoids can be computed efficiently however we present a heuristic for computing an appropriate quasi round ellipsoid that still yields valid dual bounds it turns out that using this approach within a branch and bound scheme outperforms the approaches based on axisparallel ellipsoid in terms of running times we finally combine both approaches by introducing the concept of quasi axisparallel ellipsoids and present a corresponding extension of the heuristic bound computation