very large scale linear programming a case study in combining interior point and simplex methods experience with solving a 12 753 313 variable linear program is described this problem is the linear programming relaxation of a set partitioning problem arising from an airline crew scheduling application a scheme is described that requires successive solutions of small subproblems yielding a procedure that has little growth in solution time in terms of the number of variables experience using the simplex method as implemented in cplex an interior point method as implemented in obi and a hybrid interior point simplex approach is reported the resulting procedure illustrates the power of an interior point simplex combination for solving very large scale linear programs

mean square stability of second order runge kutta methods for stochastic differential equations in a previous paper we proposed the stochastic generalization of classical second order two stage explicit runge kutta rk methods the obtained stochastic schemes have second order in the weak sense in this paper the numerical stability of these rk schemes is studied the study focuses on stability with respect to the second moment ms stability figures of the stability domains of the numerical schemes are shown numerical examples that confirm the theoretical results are also presented

complementarity framework for non linear dynamic analysis of skeletal structures with softening plastic hinges in this paper we describe an algorithm for the incremental state update of elasto plastic systems with softening the algorithm uses a complementary pivoting technique and is based on casting the incremental state update as a complementarity problem in developing the algorithm we take advantage of the special features of solid and structural mechanics problems to achieve good computational performance and hence the ability to compute numerical solutions to practical size problems for example the notion of a tangent stiffness matrix arises numerical examples using models of skeletal structures are presented to demonstrate the practicability of the algorithm the numerical examples also raise some interesting questions about multiplicity of the solutions copyright 2010 john wiley sons ltd

linearly constrained global optimization a general solution algorithm with applications this paper presents an efficient enumerative approach to solve general linearly constrained optimization problems this class of optimization problems includes fractional nonlinear network models quadratic convex and non convex programs the unified approach is accomplished by converting the constrained optimization problem to an unconstrained optimization problem through a parametric representation of its feasible region the proposed solution algorithm consists of three phases in phase 1 it finds all interior critical points in phase 2 the parametric representation of the feasible region is constructed to identify any critical points on the edges and faces of the feasible region this is done by a modified version of an algorithm for finding the v representation of the polyhedron then in phase 3 the global optimal value of the objective function is found by evaluating the objective function at the critical points as well as at the vertices for an illustration of the algorithm and a comparison with the existing methods small size numerical examples are presented

large scale linear programming geometry working bases and factorizations this paper is concerned with linear programming problems in which many of the constraints are handled implicitly by requiring that the vector of decision variables lie in a polyhedronx it is shown that the simplex method can be implemented using a working basis whose size is the number of explicit constraints as long as the local structure ofx around the current point is known various ways of describing this local structure lead to known implementations whenx is defined by generalized or variable upper bounds or flow conservation constraints in the general case a decomposition principle can be used to generate this local structure we also show how to update factorizations of the working basis

mean square stability of second order runge kutta methods for multi dimensional linear stochastic differential systems in this paper the mean square stability of second order runge kutta schemes for multi dimensional linear stochastic differential systems is studied motivated by the work of tocino mean square stability of second order runge kutta methods for stochastic differential equations j comput appl math 175 2005 355 367 and saito and mitsui mean square stability of numerical schemes for stochastic differential systems in international conference on scientific computation and differential equations july 29 august 3 2001 vancouver british columbia canada we investigate the mean square stability of second order runge kutta schemes for multi dimensional linear stochastic differential systems with one multiplicative noise stability criteria are established and numerical examples that confirm the theoretical results are also presented

maintaining lu factors of a general sparse matrix abstract the authors describe a set of procedures for computing and updating an lu factorization of a sparse matrix a where a may be square possibly singular or rectangular the procedures include a markowitz factorization and a bartels golub update similar to those of reid 1976 1982 the updates provided are addition deletion or replacement of a row or column of a and rank one modification previously column replacement has been the only update available various design features of the implementation lusol are described and computational comparisons are made with the la05 and ma28 packages of reid 1976 and duff 1977 keywords sparse matrix lu factors matrix factorization matrix updates rank one modification fortran software lower and upper triangular

weak second order conditions for stochastic runge kutta methods a general procedure to construct weak methods for the numerical solution of stochastic differential systems is presented as in the deterministic case the procedure consists of comparing the stochastic expansion of the approximation with the corresponding taylor scheme in this way the authors obtain the order conditions that a stochastic runge kutta method must satisfy to have weak order two explicit examples of generalizations of the classical family of second order two stage explicit runge kutta methods are shown also numerical examples are presented

an efficient simplex type algorithm for sparse and dense linear programs abstract we present a big m method for solving general linear programming problems with a recently developed exterior point simplex algorithm epsa we also provide intuitive motivations to epsa we describe major steps in implementing epsa for large scale linear programming our implementation was carried out under the matlab environment this algorithm seems to be more efficient than the classical primal simplex algorithm psa employing dantzig s rule preliminary computational studies on randomly generated sparse and dense small and medium linear programs support this belief in particular epsa is up to 10 times faster than psa on medium size linear programs this translates into corresponding savings in cpu time although the computational effort required in each step of epsa requires more time compared to an iteration step of psa the improvement of epsa comes from the fact that it requires adequately less iterations than psa moreover as the problem size increases and the problem density decreases epsa gets relatively faster

inertia controlling methods for general quadratic programming active set quadratic programming qp methods use a working set to define the search direction and multiplier estimates in the method proposed by fletcher in 1971 and in several subsequent mathematically equivalent methods the working set is chosen to control the inertia of the reduced hessian which is never permitted to have more than one nonpositive eigenvalue such methods will be called inertia controlling this paper presents an overview of a generic inertia controlling qp method including the equations satisfied by the search direction when the reduced hessian is positive definite singular and indefinite recurrence relations are derived that define the search direction and lagrange multiplier vector through equations related to the karush kuhn tucker system discussion is included of connections with inertia controlling methods that maintain an explicit factorization of the reduced hessian matrix

a simplex algorithm for piecewise linear programming i derivation and proof the simplex method for linear programming can be extended to permit the minimization of any convex separable piecewise linear objective subject to linear constraints this three part paper develops and analyzes a general computationally practical simplex algorithm for piecewiselinear programming

a basis deficiency allowing variation of the simplex method for linear programming h one of the most important and fundamental concepts in the simplex methodology basis is restricted to being a square matrix of the order exactly equal to the number of rows of the coefficient matrix such inflexibility might have been the source of too many zero steps taken by the simplex method in solving real world linear programming problems which are usually highly degen erate to dodge this difficulty we first generalize the basis to allow the deficient case characterized as one that has columns fewer than rows of the coefficient matrix variations of the primal and dual simplex procedures are then made and used to form a two phase method based on such a basis the number of whose columns varies dynamically in the solution process generally speaking the more degenerate a problem to be handled is the fewer columns the basis will have so thii renders the possibility of efficiently solving highly degenerate problems we also provide a valuable insight into the distinctive and favorable behavior of the proposed method by reporting our computational experiments 1998 elsevier science ltd all rights reserved

an efficient approach to updating simplex multipliers in the simplex algorithm the simplex algorithm computes the simplex multipliers by solving a system or two triangular systems at each iteration this note offers an efficient approach to updating the simplex multipliers in conjunction with the bartels golub and forrest tomlin updates for lu factors of the basis it only solves one triangular system the approach was implemented within and tested against minos 5 51 on 129 problems from netlib kennington and bpmpd computational results show that the new approach improves simplex implementations

a phase 1 approach for the generalized simplex algorithm abstract a new simplex variant allowing basis deficiency has recently been proposed to attack the degeneracy 1 as a generalization of the simplex algorithm it uses a phase 1 procedure solving an auxiliary problem with piecewise linear sums of infeasibilities as its objective in this paper we develop another phase 1 approach that only introduces a single artificial variable unlike the former which needs a crash procedure to supply an initial basis the proposed phase 1 is able to get itself started from scratch with an artificial basis having a single column computational results with a set of standard test problems from netlib are also reported

convergence and stability of the split step θ method for stochastic differential equations in this paper we construct a new split step method for solving stochastic differential equations namely the split step q method under lipschitz and linear growth conditions we establish a mean square convergence theory of split step q approximate solutions moreover the mean square stability of the method for a linear test equation with real parameters is considered and the real mean square stability region is plotted finally numerical results are presented to demonstrate the efficiency of the split step q method

weak second order explicit stabilized methods for stiff stochastic differential equations we introduce a new family of explicit integrators for stiff ito stochastic differential equa tions sdes of weak order two these numerical methods belong to the class of one step stabilized methods with extended stability domains and do not suffer from the stepsize re duction faced by standard explicit methods the family is based on the standard second or der orthogonal runge kutta chebyshev methods rock2 for deterministic problems the convergence and the mean square and asymptotic stability properties of the methods are analyzed numerical experiments including applications to nonlinear sdes and parabolic stochastic partial differential equations are presented and confirm the theoretical results

on wolfe s method for resolving degeneracy in linearly constrained optimization wolfe j soc indust appl math 11 1963 pp 205 211 describes a novel and very useful method for resolving degeneracy in the simplex method for linear programming lp the simplicity and reliability of the method makes it an excellent choice in this author s opinion the method is described here in the context of an active set method asm format for lp the method solves recursively generated subproblems that are smaller than the original which contributes to efficiency data structures are described that enable the recursive aspect of the method to be implemented very easily with minimal storage overhead the method is extended to solve a general form of linearly constrained optimization problem that includes quadratic programming qp and allows simple bounds on the variables and both equation and inequality general linear constraints issues of round off error are discussed and heuristics are described that have proved to be very reliable in practice the method is further extended to qp pr

pirock a swiss knife partitioned implicit explicit orthogonal runge kutta chebyshev integrator for stiff diffusion advection reaction problems with or without noise a partitioned implicit explicit orthogonal runge kutta method pirock is proposed for the time integration of diffusion advection reaction problems with possibly severely stiff reaction terms and stiff stochastic terms the diffusion terms are solved by the explicit second order orthogonal chebyshev method rock2 while the stiff reaction terms solved implicitly and the advection and noise terms solved explicitly are integrated in the algorithm as finishing procedures it is shown that the various coupling between diffusion reaction advection and noise can be stabilized in the pirock method the method implemented in a single black box code that is fully adaptive provides error estimators for the various terms present in the problem and requires from the user solely the right hand side of the differential equation numerical experiments and comparisons with existing chebyshev methods imex methods and partitioned methods show the efficiency and flexibility of our new algorithm

abs algorithms for linear equations and optimization in this paper we review basic properties and the main achievements obtained by the class of abs methods developed since 1981 to solve linear and nonlinear algebraic equations and optimization problems

stability analysis of numerical schemes for stochastic differential equations stochastic differential equations sdes represent physical phenomena dominated by stochastic processes as for deterministic ordinary differential equations odes various numerical schemes are proposed for sdes in this paper we study the stability of numerical schemes for scalar sdes with respect to the mean square norm which we call ms stability we will show some figures of the ms stability domain or regions for some numerical schemes and present numerical results which confirm it this notion is an extension of absolute stability in numerical methods for odes

second order balanced stochastic runge kutta methods with multi dimensional studies abstract in this paper we have considered two classes of second order balanced stochastic runge kutta methods to multidimensional ito stochastic differential equations the control functions in the proposed methods are used to improve and enhance the convergence and stability properties of the method the strong convergence of the second order balanced stochastic runge kutta methods is analyzed the preservation of sign of initial value to the second order balanced stochastic runge kutta methods for a linear scalar test equations is discussed under suitable conditions moreover the mean square stability of the proposed methods is investigated using the properties of kronecker product subsequently the stability region of the balanced stochastic runge kutta methods for s 4 stages is compared with the other existing stochastic runge kutta methods from this comparison we understood the balanced stochastic runge kutta methods are more stable than the stochastic runge kutta methods finally the mean square stability for different step sizes h is illustrated as well convergence results compared with other current methods are tabulated

higher order semi implicit taylor schemes for ito stochastic differential equations the paper considers the derivation of families of semi implicit schemes of weak order n 3 0 general case and n 4 0 additive noise case for the numerical solution of ito stochastic differential equations the degree of implicitness of the schemes depends on the selection of n parameters which vary between 0 and 1 and the families contain as particular cases the 3 0 and 4 0 weak order explicit taylor schemes since the implementation of the multiple integrals that appear in these theoretical schemes is difficult for the applications they are replaced by simpler random variables obtaining simplified schemes in this way for the multidimensional case with one dimensional noise we present an infinite family of semi implicit simplified schemes of weak order 3 0 and for the multidimensional case with additive one dimensional noise we give an infinite family of semi implicit simplified schemes of weak order 4 0 the mean square stability of the 3 0 family is analyzed concluding that as in the deterministic case the stability behavior improves when the degree of implicitness grows numerical experiments confirming the theoretical results are shown

a dual projective simplex method for linear programming abstract the method proposed in this paper is a dual version of the projective simplex method developed by the author providing a stable alternative setting for the dual simplex method by handling a sequence of linear least squares problems using orthogonalization the method is capable of handling a basis with columns fewer than rows of the coefficient matrix and amenable to problems with n m large relative to m a wide range of problems with which the projective simplex method performs unsatisfactorily in general based on a plausible characterization of an optimal solution a dual crash heuristic is described to produce an initial good basis computational results obtained with a set of standard test problems from netlib are very encouraging

selected topics in column generation dantzig wolfe decomposition and column generation devised for linear programs is a success story in large scale integer programming we outline and relate the approaches and survey mainly recent contributions not yet found in textbooks we emphasize the growing understanding of the dual point of view which has brought considerable progress to the column generation theory and practice it stimulated careful initializations sophisticated solution techniques for the restricted master problem and subproblem as well as better overall performance thus the dual perspective is an ever recurring concept in our selected topics

a projective simplex method for linear programming linear programming problems with quite square coefficient matrix form a wide range of problems that are not amenable to existing algorithms the method proposed in this paper attacks such problems from the dual side alternatively arranging computations of the simplex method using the qr factorization in each iteration its tableau version handles an n m n 1 tableau rather than the m 1 n 1 conventional tableau where m and n are the numbers of rows and columns of the coefficient matrix in contrast to the simplex method where two m m systems are solved per iteration the new approach solves a single s s s n m system only it allows nonbasis deficiency and hence could reduce computational work dramatically a favorable complexity analysis is given for one of its implementations against its conventional counterpart in addition a new crash heuristic having a clear geometrical meaning towards an optimal basis is developed to provide good input we also report numerical results obtained from our trials to give an insight into the method s behavior

mean square and asymptotic stability of the stochastic theta method stability analysis of numerical methods for ordinary differential equations odes is motivated by the question for what choices of stepsize does the numerical method reproduce the characteristics of the test equation we study a linear test equation with a multiplicative noise term and consider mean square and asymptotic stability of a stochastic version of the theta method we extend some mean square stability results in saito and mitsui siam j numer anal 33 1996 pp 2254 2267 in particular we show that an extension of the deterministic a stability property holds we also plot mean square stability regions for the case where the test equation has real parameters for asymptotic stability we show that the issue reduces to finding the expected value of a parametrized random variable we combine analytical and numerical techniques to get insights into the stability properties for a variant of the method that has been proposed in the literature we obtain precise analytic expressions for the asymptotic stability region this allows us to prove a number of results the technique introduced is widely applicable and we use it to show that a fully implicit method suggested by kloeden and platen numerical solution of stochastic differential equations springer verlag 1992 has an asymptotic stability extension of the deterministic a stability property we also use the approach to explain some numerical results reported in milstein platen and schurz siam j numer anal 35 1998 pp 1010 1019

a projective simplex algorithm using lu decomposition abstract recently we proposed a so called projective simplex method which is amenable to linear programming problems with quite square coefficient matrix since it is based on qr decomposition however the method is not a suitable choice for large and sparse problems unless n m is far less than m where m and n are the numbers of rows and columns of the coefficient matrix respectively to dodge this flaw in this paper we propose a method using lu decomposition in contrast to the simplex method in which an m 1 n 1 tableau is used its tableau version handles an n m n 1 tableau in each iteration its revised version solves a single n m n m system only compared with the two m m systems solved in the revised simplex method a complexity analysis establishes its superiority over an implementation of the simplex method in the case of the coefficient matrix being not too flat of particular interest might be the introduction of deficient nonbasis via exploiting dual degeneracy to reduce computational work further an lu decomposition based crash heuristic is furnished to provide good input computational results are also reported to give an insight into its interesting and distinctive behavior

a sparsity exploiting variant of the bartels golub decomposition for linear programming bases we describe a sparsity exploiting variant of the bartels golub decomposition for linear programming bases it includes interchanges that whenever this is possible avoid the use of any eliminations with consequent fill ins when revising the factorization at an iteration test results on some medium scale problems are presented and comparisons made with the algorithm of forrest and tomlin

numerical solutions of stochastic differential equations implementation and stability issues stochastic differential equations sdes arise from physical systems where the parameters describing the system can only be estimated or are subject to noise there has been much work done recently on developing numerical methods for solving sdes this paper will focus on stability issues and variable stepsize implementation techniques for numerically solving sdes effectively

upper bound limit analysis using discontinuous velocity fields abstract a new method for computing rigorous upper bounds under plane strain conditions is described it is based on a linear three noded triangular element which has six unknown nodal velocities and a fixed number of unknown plastic multiplier rates and uses the kinematic theorem to define a kinematically admissible velocity field as the solution of a linear programming problem unlike existing formulations which permit only a limited number of velocity discontinuities whose directions of shearing must be specified a priori the new formulation permits velocity discontinuities at all edges shared by adjacent triangles and the directions of shearing are found automatically the variation of the velocity jump along each discontinuity is described by an additional set of four unknowns all of the unknowns are subject to the constraints imposed by an associated flow rule and the velocity boundary conditions the objective function corresponds to the dissipated power or some related load parameter of interest and is minimised to yield the desired upper bound since plastic deformation may occur not only in the discontinuities but also throughout the triangular elements as well the method is capable of modelling complex velocity fields accurately and typically produces tight upper bounds on the true limit load the formulation is applicable to materials whose strength is cohesive frictional purely cohesive and uniform or purely cohesive and linearly varying and thus quite general the new procedure is very efficient and always requires fewer elements than existing methods to obtain useful upper bound solutions moreover because of the extra degrees of freedom introduced by the discontinuities the linear elements no longer need to be arranged in a special pattern to model incompressible behaviour accurately

solving staircase linear programs by the simplex method 2 pricing this and a companion paper consider how current implementations of the simplex method may be adapted to better solve linear programs that have a staged or staircase structure the preceding paper considered inversion routines that factorize the basis and solve linear systems the present paper examines pricing routines that compute reduced costs for nonbasic variables and that select a variable to enter the basis at each iteration both papers describe extensive although preliminary computer experiments and can point to some quite promising results for pricing in particular staircase computation strategies appear to offer modest but consistent savings staircase selection strategies properly chosen may offer substantial savings in number of iterations time per iteration or both

new crash procedures for large systems of linear constraints many algorithms for solving linearly constrained optimization problems maintain sets of basic variables the calculation of the initial basis is of great importance as it determines to a large extent the amount of computation that will then be required to solve the problem in this paper we suggest a number of simple methods for obtaining an initial basis and perform tests to indicate how they perform on a variety of real life problems

newton methods for large scale linear inequality constrained minimization newton methods of the linesearch type for large scale minimization subject to linear inequality constraints are discussed the purpose of the paper is twofold i to give an active set type method with the ability to delete multiple constraints simultaneously and ii to give a relatively short general convergence proof for such a method it is also discussed how multiple constraints can be added simultaneously the approach is an extension of a previous work by the same authors for equality constrained problems it is shown how the search directions can be computed without the need to compute the reduced hessian of the objective function the convergence analysis states that every limit point of a sequence of iterates satisfies the second order necessary optimality conditions

a primal deficient basis simplex algorithm for linear programming abstract the standard basis which plays a fundamental role in simplex methodology was recently extended to include a deficient case with fewer columns than rows by taking advantage of primal degeneracy computational results have been favorable with dense implementations in this paper we propose a primal simplex algorithm using sparse lu factors of deficient bases amenable to real world linear programming problems which are often degenerate or even highly degenerate the algorithm would solve them with potentially improved stability compared to the simplex algorithm the proposed algorithm has been implemented and tested on a set of 50 netlib test problems as well as a set of 15 much larger real world problems including 8 kennington and 5 bpmpd problems it significantly outperformed minos 5 3 in terms of both iteration counts and run time in particular these results reveal that there is no inevitable correlation between an algorithm s inefficiency and degeneracy contradicting common belief

runge kutta methods for numerical solution of stochastic differential equations the way to obtain deterministic runge kutta methods from taylor approximations is generalized for stochastic differential equations now by means of stochastic truncated expansions about a point for sufficiently smooth functions of an ito process a class of explicit runge kutta schemes of second order in the weak sense for systems of stochastic differential equations with multiplicative noise is developed also two runge kutta schemes of third order have been obtained for scalar equations with constant diffusion coefficients numerical examples that compare the proposed schemes to standard ones are presented

postprocessed integrators for the high order integration of ergodic sdes the concept of effective order is a popular methodology in the deterministic literature for the construction of efficient and accurate integrators for differential equations over long times the idea is to enhance the accuracy of a numerical method by using an appropriate change of variables called the processor we show that this technique can be extended to the stochastic context for the construction of new high order integrators for the sampling of the invariant measure of ergodic systems the approach is illustrated with modifications of the stochastic theta method applied to brownian dynamics where postprocessors achieving order two are introduced numerical experiments including stiff ergodic systems illustrate the efficiency and versatility of the approach

a steepest edge active set algorithm for solving sparse linear programming problems a steepest edge active set algorithm is described which is suitable for solving linear programming problems where the constraint matrix is sparse and has more rows than columns the algorithm uses a steepest edge criterion for selecting the search direction at each iteration and recurrence relations are derived which enable it to execute efficiently the canonical form for the active set method is convenient for many applications and may be exploited to devise a simple crash procedure which is employed prior to phase one a complete two phase algorithm which incorporates the crash procedure is outlined only one artificial variable is needed to determine if the linear programming problem has a feasible solution in phase one some computational results are given to illustrate the effectiveness of the algorithm for a range of sparse linear programming problems comparisons between the steepest edge criterion and the traditional dantzig criterion suggest that the former usually requires fewer iterations and often leads to substantial savings for large problems

statistical analysis of path losses for sectorized wireless networks in modern mobile communication networks such as 3g and 4g networks sectorized antennas have been widely used to divide each cell into multiple sectors in order to improve coverage spectrum efficiency and quality of service large scale path loss from a transmitting antenna to a receiving antenna should include 1 propagation attenuation that depends on transmission distance 2 shadowing that depends on surrounding environment and 3 antenna loss that depends on a sectorized antenna pattern and transmission angle an in depth analysis of statistical characteristics of large scale path losses involving with these three factors is crucial for the design operation evaluation and optimization of modern sectorized wireless networks in this paper a sectorized antenna pattern is for the first time considered in the derivation of a closed form expression of a probability density function pdf of large scale path losses specifically we first discover that the normalized pdf of propagation attenuation plus shadowing which can be approximated by the gaussian mixture model gmm with all system parameters is fully determined by our newly defined metric 10 ln 10 beta sigma s namely the attenuation exponent beta to standard deviation of shadowing sigma s ratio asr the convolution of gmm and antenna loss statistics is elaborately transformed to a series of differential equations a closed form pdf of large scale path losses with sectorized antenna pattern can be obtained by solving these differential equations to reduce the computational complexity we further prove that the exciting sources of these differential equations can be tightly approximated by weighted gaussian functions and thus the final solutions i e pdf of path losses can be derived in the form of gaussian and dawson functions our analytical results are verified by extensive numerical computation and monte carlo simulation results e g the impact of asr on the shape of pdf of propagation attenuation plus shadowing compared with traditional gaussian fitting approach our newly derived pdf of large scale path losses with sectorized antenna patterns is at least two orders of magnitude more accurate in terms of kullback leibler divergence under typical propagation attenuation and shadowing conditions

pivot versus interior point methods pros and cons linear optimization lo is the fundamental problem of mathematical optimization it admits an enormous number of applications in economics engineering science and many other fields the three most significant classes of algorithms for solving lo problems are pivot ellipsoid and interior point methods because ellipsoid methods are not efficient in practice we will concentrate on the computationally successful simplex and primal dual interior point methods only and summarize the pros and cons of these algorithm classes

a sparse counterpart of reichel and gragg s package qrup we describe how to maintain the triangular factor of a sparse qr factorization when columns are added and deleted and q cannot be stored for sparsity reasons the updating procedures could be thought of as a sparse counterpart of reichel and gragg s package qrup they allow us to solve a sequence of sparse linear least squares subproblems in which each matrix b k is an independent subset of the columns of a fixed matrix a and b k 1 is obtained by adding or deleting one column like coleman and hulbert t coleman l hulbert a direct active set algorithm for large sparse quadratic programs with simple bounds math program 45 1989 373 406 we adapt the sparse direct methodology of bjorck and oreborn of the late 1980s but without forming a ta which may be only positive semidefinite our matlab 5 implementation works with a suitable row and column numbering within a static triangular sparsity pattern that is computed in advance by symbolic factorization of a ta and preserved with placeholders

presolving in linear programming most modern linear programming solvers analyze the lp problem before submitting it to optimization some examples are the solvers whizard tomlin and welch 1983 ob1 lustig et al 1994 osl forrest and tomlin 1992 sciconic 1990 and cplex bixby 1994 the purpose of the presolve phase is to reduce the problem size and to discover whether the problem is unbounded or infeasible in this paper we present a comprehensive survey of presolve methods moreover we discuss the restoration procedure in detail i e the procedure that undoes the presolve computational results on the netlib problems gay 1985 are reported to illustrate the efficiency of the presolve methods

solving embedded generalized network problems abstract a complete unified description is given of the design and implementation of a fast and efficient program which solves linear programming problems with network substructures the very efficient generalized network procedures which have recently been developed are extended in this paper to produce an algorithm called e mnet which can solve generalized network problems with additional constraints and additional complicating variables the only requirement on the network substructure is that it contain at most two nonzero entries in each column this requirement is the most general possible for network substructures many large linear programming problems can be placed in this form by a simple rearrangement of its rows and columns preliminary computational experience indicates e mnft is about five times faster than m inos the algorithm presented should provide a very efficient and fast solution for many linear programming problems involving production scheduling distribution facility location and personnel assignment these problems tend to contain large embedded network substructures

a revised dual projective pivot algorithm for linear programming we revise the dual projective pivot algorithm using sparse rectangular lu factors in each iteration the proposed algorithm solves at most three triangular systems while simplex algorithms solve four instead of the classical basis it uses a so called pseudobasis a rectangular matrix having fewer columns than rows thereby solving smaller linear systems with a potentially improved stability compared to simplex algorithms most importantly it generates good search directions at a low cost we report encouraging computational results on a set of 50 netlib standard test problems as well as a set of 15 much larger real world problems a code named rdppa 1 10 based on the proposed algorithm outperformed minos 5 51 significantly in terms of both iterations and run time in particular it appears that a high proportion of degenerate iterations need not imply many total iterations contradicting the common belief

the fleet assignment problem solving a large scale integer program given a flight schedule and set of aircraft the fleet assignment problem is to determine which type of aircraft should fly each flight segment this paper describes a basic daily domestic fleet assignment problem and then presents chronologically the steps taken to solve it efficiently our model of the fleet assignment problem is a large multi commodity flow problem with side constraints defined on a time expanded network these problems are often severely degenerate which leads to poor performance of standard linear programming techniques also the large number of integer variables can make finding optimal integer solutions difficult and time consuming the methods used to attack this problem include an interior point algorithm dual steepest edge simplex cost perturbation model aggregation branching on set partitioning constraints and prioritizing the order of branching the computational results show that the algorithm finds solutions with a maximum optimality gap of 0 02 and is more than two orders of magnitude faster than using default options of a standard lp based branch and bound code

on the solution of highly degenerate linear programs the solution of scheduling problems often gives rise to highly degenerate linear programmes which cause significant computational difficulties for the revised simplex method wolfe s highly effective ad hoc method for overcoming the cycling or stalling problems associated with degeneracy is described here it is given a geometric interpretation in terms of finding a direction of recession for a reduced problem which is fundamental to a full understanding of the procedure an example of an aircrew scheduling problem is used to illustrate the effectiveness of the method

an efficient linear model and optimisation algorithm for multi site combined heat and power production abstract combined heat and power chp production is an important energy production technology which can help to improve the efficiency of energy production and to reduce the emission of co2 cost efficient operation of a chp system can be planned using an optimisation model based on hourly load forecasts a long term planning model decomposes into hourly models which can be formulated as linear programming lp problems such problems can be solved efficiently using the specialized power simplex algorithm however power simplex can only manage one heat and one power balance since heat cannot be transported over long distances power simplex applies only for local chp planning in this paper we formulate the hourly multi site chp planning problem with multiple heat balances as an lp model with a special structure we then develop the extended power simplex eps algorithm for solving such models efficiently even though the problem can be quite large as the number of demand sites increases eps demonstrates very good efficiency in test runs with realistic models eps is from 29 to 85 times faster than an efficient sparse simplex code using the product form of inverse pfi furthermore the relative efficiency of eps improves as the problem size grows

approximating polyhedra with sparse inequalities in this paper we study how well one can approximate arbitrary polytopes using sparse inequalities our motivation comes from the use of sparse cutting planes in mixed integer programing mip solvers since they help in solving the linear programs encountered during branch bound more efficiently however how well can we approximate the integer hull by just using sparse cutting planes in order to understand this question better given a polyope p p e g the integer hull of a mip let p k pk be its best approximation using cuts with at most k non zero coefficients we consider text d p p k max x in p k left min y in p vert x y vert right d p pk maxx pkminy p x y as a measure of the quality of sparse cuts in our first result we present general upper bounds on text d p p k d p pk which depend on the number of vertices in the polytope our bounds imply that if p p has polynomially many vertices using half sparsity already approximates it very well second we present a lower bound on text d p p k d p pk for random polytopes that show that the upper bounds are quite tight third we show that for a class of hard packing ips sparse cutting planes do not approximate the integer hull well that is d p p k d p pk is large for such instances unless k is very close to n finally we show that using sparse cutting planes in extended formulations is at least as good as using them in the original polyhedron and give an example where the former is actually much better

solving staircase linear programs by the simplex method 1 inversion this and a companion paper consider how current implementations of the simplex method may be adapted to better solve linear programs that have a staged or staircase structure the present paper looks at inversion routines within the simplex method particularly those for sparse triangular factorization of a basis by gaussian elimination and for solution of triangular linear systems the succeeding paper examines pricing routines both papers describe extensive though preliminary computational experience and can point to some quite promising results

the simplest examples where the simplex method cycles and conditions where expand fails to prevent cycling this paper introduces a class of linear programming examples that cause the simplex method to cycle and that are the simplest possible examples showing this behaviour the structure of examples from this class repeats after two iterations cycling is shown to occur for both the most negative reduced cost and steepest edge column selection criteria in addition it is shown that the expand anti cycling procedure of gill et al is not guaranteed to prevent cycling

a practical anti cycling procedure for linearly constrained optimization a procedure is described for preventing cycling in active set methods for linearly constrained optimization including the simplex method the key ideas are a limited acceptance of infeasibilities in all variables and maintenance of a working feasibility tolerance that increases over a long sequence of iterations the additional work per iteration is nominal and stalling cannot occur with exact arithmetic the method appears to be reliable based on computational results for the first 53 linear programming problems in thenetlib set

stability of weak numerical schemes for stochastic differential equations abstract this paper considers numerical stability and convergence of weak schemes solving stochastic differential equations a relatively strong notion of stability for a special type of test equations is proposed these are stochastic differential equations with multiplicative noise for different explicit and implicit schemes the regions of stability are also examined

downlink average rate and sinr distribution in cellular networks the statistical characteristics of the signal to interference plus noise ratio sinr are closely related to many performance metrics of cellular networks in this paper the downlink average rate and sinr distribution are studied for orthogonal frequency division multiple access ofdma based cellular networks subject to the distance dependent path loss and shadow fading sf with the analytical approximate mean and moment generating function mgf of the sinr in the logarithmic domain the closed form approximation for the lower and upper bounds of the average rate is obtained then the distribution of the sinr in the logarithmic domain is proposed to be approximated as the normal inverse gaussian nig distribution whose parameters are computed explicitly through moment matching also the closed form expression for the cumulative distribution function cdf of the sinr based on the nig approximation is derived simulation results not only verify the tightness of the bounds but also show that the nig approximation is up to one order of magnitude more accurate than the pearson type iv approximation and at least one order of magnitude more accurate than the lognormal approximation when the sf correlation coefficient is small or the standard deviations of the sf are large or different

approximate solution of the trust region problem by minimization over two dimensional subspaces the trust region problem minimization of a quadratic function subject to a spherical trust region constraint occurs in many optimization algorithms in a previous paper the authors introduced an inexpensive approximate solution technique for this problem that involves the solution of a two dimensional trust region problem they showed that using this approximation in an unconstrained optimization algorithm leads to the same theoretical global and local convergence properties as are obtained using the exact solution to the trust region problem this paper reports computational results showing that the two dimensional minimization approach gives nearly optimal reductions in then dimension quadratic model over a wide range of test cases we also show that there is very little difference in efficiency and reliability between using the approximate or exact trust region step in solving standard test problems for unconstrained optimization these results may encourage the application of similar approximate trust region techniques in other contexts

steepest edge simplex algorithms for linear programming we present several new steepest edge simplex algorithms for solving linear programming problems including variants of both the primal and the dual simplex method these algorithms differ depending upon the space in which the problem is viewed as residing and include variants in which this space varies dynamically we present computational results comparing steepest edge simplex algorithms and approximate versions of them against simplex algorithms that use standard pivoting rules on truly large scale realworld linear programs with as many as tens of thousands of rows and columns these results demonstrate unambiguously the superiority of steepest edge pivot selection criteria to other pivot selection criteria in the simplex method

on the integer properties of scheduling set partitioning models abstract many scheduling problems arising in the transportation industry can be posed as massive set partitioning zero one integer programmes for reasons of computational complexity it is generally unrealistic to attempt to solve the model in this form using conventional integer linear programming by the imposition of additional structure derived from the real world problem but not already implicit in the mathematical model it is possible to significantly reduce the effects of computational complexity and provide an effective method of obtaining good feasible solutions in this paper recent results in graph theory concerning natural integer properties of set partitioning integer programmes are discussed these results motivate the development of further implicit constraints which simultaneously reduce the dimensionality and increase the proportion of integer basic feasible solutions of the set partitioning linear programme

a multiparametric quadratic programming algorithm with polyhedral computations based on nonnegative least squares model predictive control mpc is one of the most successful techniques adopted in industry to control multivariable systems under constraints on input and output variables to circumvent the main drawback of mpc i e the need to solve a quadratic program qp on line to compute the control action explicit mpc was proposed in the past to precompute the control law off line using multiparametric qp mpqp the resulting form of the mpc law is piecewise affine which is extremely easy to code can be computed online by simple arithmetic operations and requires a maximum number of iterations that can be exactly determined a priori on the other hand the offline computations to solve the mpqp problem require detecting emptiness full dimensionality and minimal hyperplane representations of polyhedra and other computational geometric operations while most of the existing methods solve such operations via linear programming the approach proposed in this paper relies on a nonnegative least squares nnls solver that is very simple to code and fast to execute in addition the new approach exploits qp duality to identify and construct critical regions and to handle degeneracy issues

numerical aspects in developing lp softwares lpako and lpabo we have developed two public domain linear programming programs for several years lpako and lpabo which can solve large scale sparse lp problems stably and fast in this paper several important numerical aspects which were considered in developing lpako and lpabo are presented common issues are scaling tolerances and presolving for the lpako lu factorization and pivoting rule are important aspects in case of lpabo cholesky factorization ordering and dense column handling are important in the end of this paper several issues to be considered in the future development are proposed

numerical treatment of stochastic differential equations we define general runge kutta approximations for the solution of stochastic differential equations sde these approximations are proved to converge in quadratic mean to the solution of an sde with a corrected drift the explicit form of the correction term is given concerning the order of convergence we show that in general it is impossible for the quadratic mean of the one step error to be of an order greater than o h 3 this order is attained e g by the stochastic analogue of heun s method in the n dimensional case the highest order of convergence is in general only o h 2 attained by euler s method the order o h 3 can only be reached if nabla x sigma r sigma s nabla x sigma s sigma r for the diffusion matrix

numerical solution of stochastic differential equations with constant diffusion coefficients we present runge kutta methods of high accuracy for stochastic differential equations with constant diffusion coefficients we analyze l sub 2 convergence of these methods and present convergence proofs for scalar equations a second order method is derived and for systems a method of order one and one half is derived we further consider a variance reduction technique based on hermite expansions for evaluating expectations of functions of sample solutions numerical examples in two dimensions are presented

making progress during a stall in the simplex algorithm abstract all the standard methods for avoiding cycling in the simplex algorithm use row selection rules for resolving degeneracy the method proposed in this paper does not instead the gass saaty parametric method applied to the objective cost form is used to choose the incoming column the pivot row choice among blocking rows i e those tied for pivot is arbitrary a simple anticycling device is used which avoids dual degeneracy of the parametrized objective with probability one tests were run on nine highly degenerate practical test problems ranging in size from small to large using minos software the standard simplex method required 31 195 iterations of which 20 504 or 66 were blocked pivots using the parametric scheme required 13 812 iterations of which 67 were blocked the reduction in the number of iterations for this set of highly degenerate test problems was 56 the cpu time reduction was 48

a new degeneracy method and steepest edge based conditioning for lp a new recursive method for resolving degeneracy in simplex like methods for linear programming lp is described the method provides a guarantee of termination even in the presence of round off errors and is readily implemented in contrast to a previous method of the author this method works throughout in the primal space one consequence is that the steepest edge criterion can be used on all iterations and at all levels of recursion it is also shown that the associated steepest edge coefficients provide information from which the expected condition of the current lp basis can be calculated cheaply this provides a more accurate indication of the actual condition of a system than is obtained from norm based condition numbers this idea also enables the condition of null space matrices to be estimated

primal and dual active set methods for convex quadratic programming computational methods are proposed for solving a convex quadratic program qp active set methods are defined for a particular primal and dual formulation of a qp with general equality constraints and simple lower bounds on the variables in the first part of the paper two methods are proposed one primal and one dual these methods generate a sequence of iterates that are feasible with respect to the equality constraints associated with the optimality conditions of the primal dual form the primal method maintains feasibility of the primal inequalities while driving the infeasibilities of the dual inequalities to zero the dual method maintains feasibility of the dual inequalities while moving to satisfy the primal inequalities in each of these methods the search directions satisfy a kkt system of equations formed from hessian and constraint components associated with an appropriate column basis the composition of the basis is specified by an active set strategy that guarantees the nonsingularity of each set of kkt equations each of the proposed methods is a conventional active set method in the sense that an initial primal or dual feasible point is required in the second part of the paper it is shown how the quadratic program may be solved as a coupled pair of primal and dual quadratic programs created from the original by simultaneously shifting the simple bound constraints and adding a penalty term to the objective function any conventional column basis may be made optimal for such a primal dual pair of shifted penalized problems the shifts are then updated using the solution of either the primal or the dual shifted problem an obvious application of this approach is to solve a shifted dual qp to define an initial feasible point for the primal or vice versa the computational performance of each of the proposed methods is evaluated on a set of convex problems from the cutest test collection

asymptotic mean square stability of weak second order balanced stochastic runge kutta methods for multi dimensional ito stochastic differential systems abstract in this paper the linear asymptotic mean square stability of the weak second order stochastic runge kutta methods for multi dimensional ito stochastic differential equations due to rosler 2009 and tang and xiao 2017 are obtained further we have developed the stochastic runge kutta methods for multi dimensional ito stochastic differential equations to the balanced stochastic runge kutta methods by using the control functions we carry out a linear stability analysis of the class of stochastic runge kutta methods for the linear test equations with multiplicative noise thereby providing an explicit structure of stability matrices some comparisons and illustrations shows that there is an improvement in the stability and error analysis of these new balanced stochastic runge kutta methods comparatively to stochastic runge kutta methods and thus conforming the obtained theoretical results

a linear programming decomposition focusing on the span of the nondegenerate columns abstract the improved primal simplex ips was recently developed by elhalaloui et al to take advantage of degeneracy when solving linear programs with the primal simplex it implements a dynamic constraint reduction based on the compatible columns i e those that belong to the span of a given subset of basic columns including the nondegenerate ones the identification of the compatible variables may however be computationally costly and a large number of linear programs are solved to enlarge the subset of basic variables in this article we first show how the positive edge criterion of raymond et al can be included in ips for a fast identification of the compatible variables our algorithm then proceeds through a series of reduction and augmentation phases until optimality is reached in a reduction phase we identify compatible variables and focus on them to make quick progress toward optimality during an augmentation phase we compute one greatest normalized improving direction and select a subset of variables that should be considered in the reduced problem compared with ips the linear program that is solved to find this direction involves the data of the original constraint matrix this new algorithm is tested over mittelmann s benchmark for linear programming and on instances arising from industrial applications the results show that the new algorithm outperforms the primal simplex of cplex on most highly degenerate instances in which a sufficient number of nonbasic variables are compatible in contrast ips has difficulties on the eleven largest mittelmann instances

generating fenchel cutting planes for knapsack polyhedra the author recently proposed a class of cutting planes for integer programs called fenchel cuts which distinguish themselves from more conventional cuts in that they are generated by directly seeking to solve the separation problem rather than by using explicit knowledge of the polyhedral structure of the integer program an algorithm for generating fenchel cuts is presented and described in detail for the separation problem associated with knapsack polyhedra computational results are presented for a collection of real world integer programs to demonstrate the effectiveness of the cutting planes

a computationally stable solution algorithm for linear programs an active area of research for linear programming is to construct initial simplex tableau that is close to the optimal solution and to improve its pivoting selection strategies efficiently in this paper we present a new approach to the problem of initialization and pivoting rules the algorithm is free from any artificial variables and artificial constraints known as the big m methods by supplying the simplex method without using any large numbers therefore the result is computationally stable and provides a better initial basis that reduces the number of pivoting iterations efficiency a three phase simplex type solution algorithm is developed for solving general linear programs in phase 1 greater than constraints are relaxed and the problem is solved starting at the origin if it is unbounded then the objective function is modified in order to have a bound feasible solution then in phase 2 the greater than constraints are introduced in a dual simplex framework following this in the case the original objective function was modified phase 3 restores optimality for the original objective function the algorithm is numerically stable and works with the original decision variables our initial experimental study indicates that the proposed algorithm is more efficient than both the primal and the dual simplex in terms of the number of iterations

the many facets of linear programming we examine the history of linear programming from computational geometric and complexity points of view looking at simplex ellipsoid interior point and other methods

staircase matrices and systems difference equations spline approximations and multiperiod linear programs all give rise to linear equation systems that have a characteristic staircase structure a staircase system s variables can be partitioned into a natural sequence of periods in such a way that every equation involves at most the variables of two adjacent periods this paper surveys the numerous properties of staircase systems and of the staircase matrices that comprise their coefficients particular attention is given to reduced and sparse staircase forms and to the varieties of gaussian elimination for solving staircase systems

sparse psd approximation of the psd cone while semidefinite programming sdp problems are polynomially solvable in theory it is often difficult to solve large sdp instances in practice one technique to address this issue is to relax the global positive semidefiniteness psd constraint and only enforce psd ness on smaller k times k principal submatrices we call this the sparse sdp relaxation surprisingly it has been observed empirically that in some cases this approach appears to produce bounds that are close to the optimal objective function value of the original sdp in this paper we formally attempt to compare the strength of the sparse sdp relaxation vis a vis the original sdp from a theoretical perspective in order to simplify the question we arrive at a data independent version of it where we compare the sizes of sdp cone and the k psd closure which is the cone of matrices where psd ness is enforced on all k times k principal submatrices in particular we investigate the question of how far a matrix of unit frobenius norm in the k psd closure can be from the sdp cone we provide two incomparable upper bounds on this farthest distance as a function of k and n we also provide matching lower bounds which show that the upper bounds are tight within a constant in different regimes of k and n other than linear algebra techniques we extensively use probabilistic methods to arrive at these bounds one of the lower bounds is obtained by observing a connection between matrices in the k psd closure and matrices satisfying the restricted isometry property