on modified factorizations for large scale linearly constrained optimization we consider the algebraic issues concerning the solution of general large scale linearly constrained nonlinear optimization problems particular attention is given to suitable methods for solving the linear systems that occur at each iteration of such methods the main issue addressed is how to ensure that a quadratic model of the objective function is positive definite in the null space of the constraints while neither adversely affecting the convergence of newton s method nor incurring a significant computational overhead numerical evidence to support the theoretical developments is provided

automatic mesh generation over intersecting surfaces a fully automatic general scheme is proposed to determine and analyse the intersections between two groups of surfaces composed of triangular facets by carefully maintaining the spatial position of the lines of intersection in regenerating mesh locally around surface intersections pieces of surfaces can be arbitrarily combined and merged a great variety of objects can be easily created by selectively putting together different surface parts derived from surface intersections the new algorithm is best applied in conjunction with an existing surface mesh generator to enhance its general capability in dealing with objects built from intersecting surfaces in fact it is a powerful surface mesh manipulator and through the repeated use of the process complex structures can be rapidly and accurately constructed

nondifferentiable minimax fractional programming under generalized univexity in this paper we are concerned with a nondifferentiable minimax fractional programming problem we derive a kuhn tucker type sufficient optimality condition for an optimal solution to the problem and establish week strong and converse duality theorems for the problem and its three different forms of dual problems the results in this paper extend a few known results in the literature

on second order duality for nondifferentiable minimax fractional programming two types of second order dual models for a nondifferentiable minimax fractional programming problem are formulated and proved weak strong strict converse duality theorems using bonvexity assumptions special cases are also discussed to show that this work extends some known results of the literature

duality for a class of nondifferentiable mathematical programming problems abstract two dual problems are proposed for the minimax problem minimize maxy y x y subject to g x 0 a duality theorem is established for each dual problem it is revealed that these problems are intimately related to a class of nondifferentiable programming problems

computing projections for the karmarkar algorithm abstract alternatives are considered for computing the projection of a vector onto the nullspace of a matrix as is required to compute the step direction for the karmarkar projective algorithm among the possibilities considered are forming and factoring the normal matrix and working with a larger but sparser extended matrix the extended matrices are symmetric but indefinite a modification to the harwell ma27 set of subroutines to make them more efficient for indefinite matrices is presented computational results are given to support the conclusion that computing these projections using one of the extended matrices offers significant computational advantages over the other alternatives with which it is compared

on signed incomplete cholesky factorization preconditioners for saddle point systems limited memory incomplete cholesky factorizations can provide robust preconditioners for sparse symmetric positive definite linear systems in this paper the focus is on extending the approach to sparse symmetric indefinite systems in saddle point form a limited memory signed incomplete cholesky factorization of the form ldl t is proposed where the diagonal matrix d has entries pm 1 the main advantage of this approach is its simplicity as it avoids the use of numerical pivoting instead a global shift strategy involving two shifts one for the 1 1 block and one for the 2 2 block of the saddle point matrix is used to prevent breakdown and to improve performance the matrix is optionally prescaled and preordered using a standard sparse matrix ordering scheme that is then postprocessed to give a constrained ordering that reduces the likelihood of breakdown and need for shifts the use of intermediate memory memory used in the construction of the incomplete factorization but subsequently d

splitting dense columns in sparse linear systems abstract we consider systems of equations of the form aa t x b where a is a sparse matrix having a small number of columns which are much denser than the other columns these dense columns in a cause aat to be very or even completely dense which greatly limits the effectiveness of sparse matrix techniques for directly solving the above system of equations in the literature on interior point methods for linear programming the usual technique for dealing with this problem is to split a into a sparse part s and a dense part d a s d and to solve systems involving aat in terms of the solution of systems involving sst using either conjugate gradient techniques or the sherman morrison woodbury formula this approach has the difficulty that sst is often rank deficient even when aat has full rank in this paper we propose an alternative method which avoids the rank deficiency problem and at the same time allows for the effective use of sparse matrix techniques the resulting algorithm is both efficient and robust

on the augmented system approach to sparse least squares problems we study the augmented system approach for the solution of sparse linear least squares problems it is well known that this method has better numerical properties than the method based on the normal equations we use recent work by arioli et al 1988 to introduce error bounds and estimates for the components of the solution of the augmented system in particular we find that using iterative refinement we obtain a very robust algorithm and our estimates of the error are accurate and cheap to compute the final error and all our error estimates are much better than the classical or skeel s error analysis 1979 indicates moreover we prove that our error estimates are independent of the row scaling of the augmented system and we analyze the influence of the bjorck scaling 1967 on these estimates we illustrate this with runs both on large scale practical problems and contrived examples comparing the numerical behaviour of the augmented systems approach with a code using the normal equations these experiments show that while the augmented system approach with iterative refinement can sometimes be less efficient than the normal equations approach it is comparable or better when the least squares matrix has a full row and is in any case much more stable and robust

on implementing mehrotra s predictor corrector interior point method for linear programming mehrotra tech report 90 03 department of industrial engineering and management sciences northwestern university evanston il 1990 recently described a predictor corrector variant of the primal dual interior point algorithm for linear programming this paper describes a full implementation of this algorithm with extensions for solving problems with free variables and problems with bounds on primal variables computational results on the netlib test set are given to show that this new method almost always improves the performance of the primal dual algorithm and that the improvement increases dramatically as the size and complexity of the problem increases a numerical instability in using schur complements to remove dense columns is identified and a numerical remedy is given

an implementation of karmarkar s algorithm for linear programming this paper describes the implementation of power series dual affine scaling variants of karmarkar s algorithm for linear programming based on a continuous version of karmarkar s algorithm two variants resulting from first and second order approximations of the continuous trajectory are implemented and tested linear programs are expressed in an inequality form which allows for the inexact computation of the algorithm s direction of improvement resulting in a significant computational advantage implementation issues particular to this family of algorithms such as treatment of dense columns are discussed the code is tested on several standard linear programming problems and compares favorably with the simplex codeminos 4 0

multiobjective control problem with generalized invexity abstract a multiobjective control problem is introduced the definition of invexity for continuous functions is extended to invexity pseudoinvexity and quasi invexity duality results are established for wolfe as well as mond wier type duals

discrete transfinite mappings for the description and meshing of three dimensional surfaces using interactive computer graphics a system for describing three dimensional surfaces in a form suitable for finite element analysis is described the system makes extensive use of real time interactive computer graphics techniques for both input and display discrete transfinite mappings are used as the mathematical basis for the surface representation the mathematical basis and the reasons for choosing this form of representation are discussed explicit forms of the mappings based on lagrange polynomial interpolation functions are presented finally the interactive graphics procedures for defining finite element meshes are described

the augmented system variant of ipms in two stage stochastic linear programming computation abstract the application of interior point methods ipm to solve the deterministic equivalent of two stage stochastic linear programming problems is a known and natural idea experiments have proved that among the interior point methods the augmented system approach gives the best performance on these problems however most of their implementations encounter numerical difficulties in certain cases which can result in loss of efficiency we present a new approach for the decomposition of the augmented system which automatically exploits the special behavior of the problems we show that the suggested approach can be implemented in a fast and numerically robust way by solving a number of large scale two stage stochastic linear programming problems the comparison of our solver with fo1aug which is considered as a state of the art augmented system implementation of interior point methods is also given

on minimax fractional optimality conditions with invexity abstract under different forms of invexity conditions sufficient kuhn tucker conditions and three dual models are presented for the minimax fractional programming

intersecting and trimming parametric meshes on finite element shells luiz cristovao g coelho marcelo gattass department of computer science puc rio rio de janeiro brazilluiz henrique de figueiredolncc laborato rio nacional de computac a o cient i ca p etro polis brazilabstractwe present an algorithm for intersecting nite element meshes de ned on parametric surface patches the inter section curves are modeled precisely and both meshes are adjusted to the newly formed borders without unwantedreparametrizations the algorithm is part of an interactive shell modeling program that has been used in the designof large offshore oil structures to achieve good interactive response we represent meshes with a topological datastructure that stores its entities in spatial indexing trees instead of linear lists these trees speed up the intersectioncomputations required to determine points of the trimming curves moreover when combined with the topologicalinformation they allow remeshing using only local queries

convergence of a class of inexact interior point algorithms for linear programs we present a convergence analysis for a class of inexact infeasible interior point methods for solving linear programs the main feature of inexact methods is that the linear systems defining the search direction at each interior point iteration need not be solved to high accuracy more precisely we allow that these linear systems are only solved to a moderate accuracy in the residual but no assumptions are made on the accuracy of the search direction in the search space in particular our analysis does not require that feasibility is maintained even if the initial iterate happened to be a feasible solution of the linear program we also present a few numerical examples to illustrate the effect of using inexact search directions on the number of interior point iterations

formulating two stage stochastic programs for interior point methods this paper describes an approach for modeling two stage stochastic programs that yields a form suitable for interior point algorithms a staircase constraint structure is created by replacing first stage variables with sparse split variables in conjunction with side constraints dense columns are thereby eliminated the resulting model is larger than traditional stochastic programs but computational savings are substantial over a tenfold improvement for the problems tested a series of experiments with stochastic networks drawn from financial planning demonstrates the attained efficiencies comparisons with minos and the dual block angular stochastic programming model are provided as benchmarks the split variable approach is applicable to general two stage stochastic programs and other dual block angular models

effects of finite precision arithmetic on interior point methods for nonlinear programming we show that the effects of finite precision arithmetic in forming and solving the linear system that arises at each iteration of primal dual interior point algorithms for nonlinear programming are benign provided that the iterates satisfy centrality and feasibility conditions of the type usually associated with path following methods when we replace the standard assumption that the active constraint gradients are independent by the weaker mangasarian fromovitz constraint qualification rapid convergence usually is attainable even when cancellation and roundoff errors occur during the calculations in deriving our main results we prove a key technical result about the size of the exact primal dual step this result can be used to modify existing analysis of primal dual interior point methods for convex programming making it possible to extend the superlinear local convergence results to the nonconvex case

a general parametric analysis approach and its implication to sensitivity analysis in interior point methods adler and monteiro 1992 developed a parametric analysis approach that is naturally related to the geometry of the linear program this approach is based on the availability of primal and dual optimal solutions satisfying strong complementarity in this paper we develop an alternative geometric approach for parametric analysis which does not require the strong complementarity condition this parametric analysis approach is used to develop range and marginal analysis techniques which are suitable for interior point methods two approaches are developed namely the lu factorization approach and the affine scaling approach

symmetric indefinite systems for interior point methods we present a unified framework for solving linear and convex quadratic programs via interior point methods at each iteration this method solves an indefinite system whose matrix is left begin array 20 c d 2 a t a 0 end array right instead of reducing to obtain the usualad 2 a t system this methodology affords two advantages 1 it avoids the fill created by explicitly forming the productad 2 a t whena has dense columns and 2 it can easily be used to solve nonseparable quadratic programs since it requires only thatd be symmetric we also present a procedure for converting nonseparable quadratic programs to separable ones which yields computational savings when the matrix of quadratic coefficients is dense

an algorithm for the intersection of quadrilateral surfaces by tracing of neighbours the use of discrete data to represent engineering structures as derivatives from intersecting components requires algorithms to perform boolean operations between groups of quadrilateral and triangular surfaces in the intersection process an accurate and efficient method for the determination of intersection lines is a crucial step for large scale and complex surface intersections an algorithm based on tracing the neighbours of intersecting quadrilaterals is proposed to determine the intersection lines a background grid is employed to limit the scope of searching for candidate quadrilaterals that may intersect this will drastically cut down the time of geometrical check for intersections between quadrilaterals making the surface intersection and mesh generation a quasi linear process with respect to the number of elements involved given the node numbers at the vertices of the candidate quadrilaterals the neighbour relationship is then established in the determination of intersection each quadrilateral is divided into two triangles and four fundamental cases are identified and treated systematically to enhance robustness and reliability tracing the neighbours for the determination of intersection lines not only greatly increases the efficiency of the process it also improves the reliability as branching and degenerated cases can all be dealt with in a consistent manner on the intersecting surfaces concerned examples on a great variety of surface and mesh characteristics are given to demonstrate the effectiveness and robustness of the algorithm

on duality theorems for a nondifferentiable minimax fractional programming the optimality conditions of lai et al j math anal appl 230 1999 311 can be used to construct two kinds of parameter free dual models of nondifferentiable minimax fractional programming problems which involve pseudo quasi convex functions in this paper the weak duality strong duality and strict converse duality theorems are established for the two dual models

a product form cholesky factorization method for handling dense columns in interior point methods for linear programming cholesky factorization has become the method of choice for solving the symmetric system of linear equations arising in interior point methods ipms for linear programming lp its main advantages being numerical stability and efficiency for sparse systems however in the presence of dense columns there is dramatic loss of efficiency a typical approach to remedy this problem is to apply the sherman morrison woodbury smw update formula to the dense part this approach while being very efficient is not numerically stable here we propose using product form cholesky factorization to handle dense columns the proposed approach is essentially as stable as the original cholesky factorization and nearly as efficient as the smw approach we demonstrate these properties both theoretically and computationally a key part of our theoretical analysis is the proof that the elements of the cholesky factors of the matrices that arise in ipms for lp are uniformly bounded as the duality gap converges to zero

on sufficiency of the kuhn tucker conditions if all the functions fo x f x f x are convex on c then these conditions are also sufficient various classes of functions have been defined for the purpose of weakening this limitation of convexity in mathematical programming mangasarian 11 has speculated that pseudo convexity of x and quasiconvexity off x are the weakest conditions that can be imposed so that the above conditions are sufficient for optimality it will be shown that there are other wide classes of functions for which the conditions are sufficient 545

computational geometry an introduction from the reviews this book offers a coherent treatment at the graduate textbook level of the field that has come to be known in the last decade or so as computational geometry the book is well organized and lucidly written a timely contribution by two founders of the field it clearly demonstrates that computational geometry in the plane is now a fairly well understood branch of computer science and mathematics it also points the way to the solution of the more challenging problems in dimensions higher than two

second order duality in nondifferentiable minmax programming involving type i functions two types of second order dual models are formulated for a nondifferentiable minmax programming problem and usual duality results are established involving generalized type i functions

a qmr based interior point algorithm for solving linear programs a new approach for the implementation of interior point methods for solving linear programs is proposed its main feature is the iterative solution of the symmetric but highly indefinite 2 2 block systems of linear equations that arise within the interior point algorithm these linear systems are solved by a symmetric variant of the quasi minimal residual qmr algorithm which is an iterative solver for general linear systems the symmetric qmr algorithm can be combined with indefinite preconditioners which is crucial for the efficient solution of highly indefinite linear systems yet it still fully exploits the symmetry of the linear systems to be solved to support the use of the symmetric qmr iteration a novel stable reduction of the original unsymmetric 3 3 block systems to symmetric 2 2 block systems is introduced and a measure for a low relative accuracy for the solution of these linear systems within the interior point algorithm is proposed some indefinite preconditioners are discussed finally we report results of a few preliminary numerical experiments to illustrate the features of the new approach

necessary and sufficient conditions for minimax fractional programming we establish the necessary and sufficient optimality conditions for a class of nondifferentiable minimax fractional programming problems solving generalized convex functions subsequently we apply the optimality conditions to formulate one parametric dual problem and we prove weak duality strong duality and strict converse duality theorems

optimality and duality for generalized fractional programming involving nonsmooth pseudoinvex functions using a parametric approach we establish necessary and sufficient conditions and derive duality theorems for a class of nonsmooth generalized minimax fractional programming problems containing pseudoinvex functions

solving symmetric indefinite systems in an interior point method for linear programming we describe an implementation of a primal dual path following method for linear programming that solves symmetric indefinite augmented systems directly by bunch parlett factorization rather than reducing these systems to the positive definite normal equations that are solved by cholesky factorization in many existing implementations the augmented system approach is seen to avoid difficulties of numerical instability and inefficiency associated with free variables and with dense columns in the normal equations approach solving the indefinite systems does incur an extra overhead whose median is about 40 in our tests but the augmented system approach proves to be faster for a minority of cases in which the normal equations have relatively dense cholesky factors a detailed analysis shows that the augmented system factorization is reliable over a fairly large range of the parameter settings that control the tradeoff between sparsity and numerical stability

on a homogeneous algorithm for the monotone complementarity problem we present a generalization of a homogeneous self dual linear programming lp algorithm to solving the monotone complementarity problem mcp the algorithm does not need to use any big m parameter or two phase method and it generates either a solution converging towards feasibility and comple mentarity simultaneously or a certificate proving infeasibility moreover if the mcp is polynomially solvable with an interior feasible starting point then it can be polynomially solved without using or knowing such information at all to our knowledge this is the first interior point and infeasible starting algorithm for solving the mcp that possesses these desired features preliminary computational results are presented

convergence conditions and krylov subspace based corrections for primal dual interior point method we present convergence conditions for a generic primal dual interior point algorithm with multiple corrector directions the corrector directions can be generated by any approach the search direction is obtained by combining predictor and corrector directions through a small linear program we also propose a new approach to generate corrector directions this approach generates directions using information from an appropriately defined krylov subspace we propose efficient implementation strategies for our approach that follow the analysis of this paper numerical experiments illustrating the features of the proposed approach and its practical usefulness are reported

generalized α invexity and nondifferentiable minimax fractional programming in this paper we study a nondifferentiable minimax fractional programming problem under the assumptions of a invex function in this paper we utilize the concept of a invexity m a noor on generalized preinvex functions and monotonicities j inequalities pure appl math 5 2004 1 9 and pseudo a invexity s k mishra m a noor on vector variational like inequality problems j math anal appl 311 2005 69 75 we also introduce the concept of strict pseudo a invex and quasi a invex functions we derive karush kuhn tucker type sufficient optimality conditions and establish weak strong and converse duality theorems for the problem and its three different dual problems the results in this paper extend several known results in the literature

interface handling for three dimensional higher order xfem computations in fluid structure interaction three dimensional higher order extended finite element method xfem computations still pose challenging computational geometry problems especially for moving interfaces this paper provides a method for the localization of a higher order interface finite element fe mesh in an underlying three dimensional higher order fe mesh additionally it demonstrates how a subtetrahedralization of an intersected element can be obtained which preserves the possibly curved interface and allows therefore exact numerical integration the proposed interface algorithm collects initially a set of possibly intersecting elements by comparing their extended axis aligned bounding boxes the intersection method is applied to a highly reduced number of intersection candidates the resulting linearized interface is used as input for an elementwise constrained delaunay tetrahedralization which computes an appropriate subdivision for each intersected element the curved interface is recovered from the linearized interface in the last step the output comprises triangular integration cells representing the interface and tetrahedral integration cells for each intersected element application of the interface algorithm currently concentrates on fluid structure interaction problems on low order and higher order fe meshes which may be composed of any arbitrary element types such as hexahedra tetrahedra wedges etc nevertheless other xfem problems with explicitly given interfaces or discontinuities may be tackled in addition multiple structures and interfaces per intersected element can be handled without any additional difficulties several parallelization strategies exist depending on the desired domain decomposition approach numerical test cases including various geometrical exceptions demonstrate the accuracy robustness and efficiency of the interface handling

generalized α univexity and duality for nondifferentiable minimax fractional programming abstract the aim of the present paper is to study a nondifferentiable minimax fractional programming problem under the assumptions of univex and related functions in this paper we introduce the concepts of univex pseudo univex strict pseudo univex and quasi univex functions respectively by unifying the notions of invex and univex functions we derive karush kuhn tucker type sufficient optimality conditions and establish weak strong and converse duality theorems for the problem and its three different dual problems the results in this paper extend several known results in the literature

three types dual model for minimax fractional programming abstract we establish necessary and sufficient optimality conditions for minimax fractional programming involving nonsmooth generalized t convex functions three types dual model of 1 are considered and certain duality results have been derived in the framework of generalized t convex functions