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

isomorphic rings of monomial representations abstract let d g and d g be the rings of monomial representations of finite groups g and g of odd order we prove that the isomorphy d g d g implies the isomorphy b g b g of the respective burnside rings moreover we prove that the isomorphy d g d g with finite z groups g and g i e groups which have only cyclic sylow subgroups implies the isomorphy g g in particular a group g of squarefree order is uniquely determined up to isomorphy by the ring d g

the cholesky factorization in interior point methods the paper concerns the cholesky factorization of symmetric positive definite matrices arising in interior point methods our investigation is based on a property of the cholesky factorization which interprets small diagonal values during factorization as degeneracy in the scaled optimization problem a practical scaling independent technique based on the above property is developed for the modified cholesky factorization of interior point methods this technique increases the robustness of cholesky factorizations performed during interior point iterations when the optimization problem is degenerate our investigations show also the limitations of interior point methods with the recent implementation technology and floating point arithmetic standard we present numerical results on degenerate linear programming problems of netlib

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

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

new super integrable hierarchies associated with osp 2 2 and spo 2 2 and their applications abstract based on the lie superalgebras osp 2 2 and spo 2 2 we generate some new super soliton hierarchies these hierarchies can reduce to the akns hierarchy and the super akns hierarchy and by using the supertrace identity on lie superalgebras we construct the super hamiltonian structures of these new super soliton hierarchies then as the applications of spo 2 2 osp 2 2 and spo 2 2 sl 1 2 we obtain the relations of different hierarchies associated with different lie superalgebras

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

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

a new class of preconditioners for large scale linear systems from interior point methods for linear programming a new class of preconditioners for the iterative solution of the linear systems arising from interior point methods is proposed for many of these methods the linear systems are symmetric and indefinite the system can be reduced to a system of normal equations which is positive definite we show that all preconditioners for the normal equations system have an equivalent for the augmented system while the opposite is not true the new class of preconditioners works better near a solution of the linear programming problem when the matrices are highly ill conditioned the preconditioned system can be reduced to a positive definite one the techniques developed for a competitive implementation are rather sophisticated since the subset of columns is not known a priori the new preconditioner applied to the conjugate gradient method compares favorably with the cholesky factorization approach on large scale problems whose cholesky factorization contains too many nonzero entries

a trace identity and its applications to the theory of discrete integrable systems a general theory for studying discrete integrable systems is developed based on a trace identity that the author previously proposed a scheme for generating hierarchies of discrete integrable systems is presented under broad assumptions the resulting hierarchies are shown to consist of liouville integrable hamiltonian systems

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 subalgebra of lie algebra a2 and its associated two types of loop algebras as well as hamiltonian structures of integrable hierarchy in this paper a subalgebra a2 of the lie algebra a2 is constructed which gives a corresponding loop algebra a 2 by properly choosing the gradation of the basis elements it follows that an isospectral problem is established and a new liouville integrable hamiltonian hierarchy is obtained by making use of a matrix transformation a subalgebra a2 of the lie algebra a1 is presented which possesses the same communicative operations of basis elements as those in a2 again we expand the lie algebra a1 into a high dimensional loop algebra g and a type of expanding integrable system of the hierarchy obtained above is worked out furthermore hamiltonian structures of hierarchy are presented by use of the quadratic form identity

the trace identity a powerful tool for constructing the hamiltonian structure of integrable systems a new approach to hamiltonian structures of integrable systems is proposed by making use of a trace identity for a variety of isospectral problems that can be unified to one model x u it is shown that both the related hierarchy of evolution equations and the hamiltonian structure can be obtained from the same solution of the equation vx u v

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

constraint softening in model predictive control with off line optimized admissible sets for systems with additive and multiplicative disturbances abstract an approach to the softening of constraints is explored for a class of mpc algorithms that employ off line computed constraint admissible sets for simplified on line computations the proposed approach relies on the use of exact penalty functions to ensure that the solution coincides with the actual optimal solution if the original mpc problem is feasible and that there are constraint violations at minimum possible levels if the original problem is infeasible the approach is implemented for a class of linear systems with additive and multiplicative disturbances using a dynamic policy based mpc algorithm results specific to the cases of non stochastic and stochastic disturbances are explored and assessed with simulation examples

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

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 the isomorphism problem for the ring of monomial representations of a finite group in this paper we are concerned with the problem of finding properties of a finite group g in the ring d g of monomial representations of g we determine the conductors of the primitive idempotents of q zd g where c is a primitive g th root of unity and prove a structure theorem for the torsion units of d g using these results we show that an abelian group g is uniquely determined by the ring d g we state an explicit formula for the primitive idempotents of z p zd g where z p is a localization of z we get further results for nilpotent and p nilpotent groups and we obtain properties of sylow subgroups of g from d g

multiplicative induction and units for the ring of monomial representations abstract let g be a finite group and let a be a finite abelian g group for each subgroup h of g h a denotes the ring of monomial representations of h with coefficients in a which is a generalization of the burnside ring h of h we research the multiplicative induction map h a g a derived from the tensor induction map h g and also research the unit group of g a the results are explained in terms of the first cohomology groups h 1 k a for k g we see that tensor induction for 1 cocycles plays a crucial role in a description of multiplicative induction the unit group of g a is identified as a finitely generated abelian group we especially study the group of torsion units of g a and study the unit group of g as well

a supertrace identity and its applications to superintegrable systems a supertrace identity on lie superalgebras is established it provides a tool for constructing super hamiltonian structures of zero curvature equations associated with lie superalgebras applications in the case of the lie superalgebra b 0 1 present super hamiltonian structures of a super akns soliton hierarchy and a super dirac soliton hierarchy

fast cholesky factorization for interior point methods of linear programming abstract every iteration of an interior point method of large scale linear programming requires computing at least one orthogonal projection in practice cholesky decomposition seems to be the most efficient and sufficiently stable method we studied the column oriented or left looking sparse variant of the cholesky decomposition which is a very popular method in large scale optimization we show some techniques such as using supernodes and loop unrolling for improving the speed of computation we show numerical results on a wide variety of large scale real life linear programming problems

an approach to generate superextensions of integrable systems an algorithm to generate integrable systems is extended to the super case some new examples of superextensions of integrable systems are illustrated we also generalize the trace identity due to tu to the super case and use it to establish hamiltonian structures of superextensions of integrable systems under consideration

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

computing block angular karmarkar projections with applications to stochastic programming we present a variant of karmarkar s algorithm for block angular structured linear programs such as stochastic linear programs by computing the projection efficiently we give a worst case bound on the order of the running time that can be an order of magnitude better than that of karmarkar s standard algorithm further implications for approximations and very large scale problems are given

semidirect sums of lie algebras and discrete integrable couplings a relation between semidirect sums of lie algebras and integrable couplings of lattice equations is established and a practicable way to construct integrable couplings is further proposed an application of the resulting general theory to the generalized toda spectral problem yields two classes of integrable couplings for the generalized toda hierarchy of lattice equations the construction of integrable couplings using semidirect sums of lie algebras provides a good source of information on complete classification of integrable lattice equations

global representation rings abstract in this paper we combine the concepts of burnside ring and character ring in what we call the global representation ring of a finite group we compute all ring homomorphisms from this ring to the complex numbers we compute its spectrum its connected components and determine the primitive idempotents

the species and idempotents of the green algebra of a finite group with a cyclic sylow subgroup we study the green algebra a kg generated by the isomorphism classes of indecomposable kg modules where k is a field with chark p 0 and g a finite group with a cyclic sylow p subgroup more precisely we discuss the homomorphisms a kg c called species and state a formula for the idempotents of a kg