one dimensional transport equations with discontinuous coefficients we consider one dimensional linear transport equations with bounded but possibly discontinuous coefficient a the cauchy problem is studied from two different points of view in the first case we assume that a is piecewise continuous we give an existence result and a precise description of the solutions on the lines of discontinuity in the second case we assume that a satisfies a one sided lipschitz condition we give existence uniqueness and general stability results for backward lipschitz solutions and forward measure solutions by using a duality method we prove that the flux associated to these measure solutions is a product by some canonical representative of a key words linear transport equations discontinuous coefficients weak stability duality product of a measure by a discontinuous function nonnegative solutions 1991 mathematics subject classification primary 35f10 35b35 34a12 to appear in nonlinear analysis tma departement de mathematiques et applications umr cnrs 8553 ecole normale superieure et cnrs 45 rue d ulm 75230 paris cedex 05 france francois bouchut ens fr mathematiques applications et physique mathematique d orleans umr cnrs 6628 universite d orleans 45067 orleans cedex 2 france james math cnrs fr

an implementation of the dual affine scaling algorithm for minimum cost flow on bipartite uncapacitated networks this paper describes an implementation of the dual affine scaling algorithm for linear programming specialized to solve minimum cost flow problems on bipartite uncapacitated networks this implementation uses a preconditioned conjugate gradient algorithm to solve the system of linear equations that determines the search direction at each iteration of the interior point algorithm two preconditioners are considered a diagonal preconditioner and a preconditioner based on the incidence matrix of an approximate maximum weighted spanning tree of the network under dual nondegeneracy this spanning tree allows for early identification of the optimal solution by applying an epsilon perturbation to the cost vector an optimal extreme point primal solution is produced in the presence of dual degeneracy the implementation is tested by solving several large instances of randomly generated assignment problems comparing solution times with the network simplex code netflo and the relaxation algorithm code relax

solving large scale linear multicommodity flow problems with an active set strategy and proximal accpm in this paper we propose to solve the linear multicommodity flow problem using a partial lagrangian relaxation the relaxation is restricted to the set of arcs that are likely to be saturated at the optimum this set is itself approximated by an active set strategy the partial lagrangian dual is solved with proximal accpm a variant of the analytic center cutting plane method the new approach makes it possible to solve huge problems when few arcs are saturated at the optimum as appears to be the case in many practical problems

simplex and interior point specialized algorithms for solving nonoriented multicommodity flow problems multicommodity network flow models arise in a wide variety of contexts typical among which is the dimensioning of telecommunication networks in this paper we present various approaches based on specialization of the simplex algorithm and interior point methods to solve nonoriented multicommodity flowproblems algorithms are tested with data from the france telecom paris district transmission network first we focus on a specialization for the node arc formulation of the problem a primal simplex and dual affine scaling algorithms exploiting the particular structure of the constraint matrix are presented and compared numerical results are provided for problems up to about 800 000 constraints and 6 000 000 variables however much more powerful approaches based on specialized decomposition methods can be implemented for solving the problem a dantzig wolfe decomposition method is designed and compared with a specialized implementation of the analytic center cutting plane method accpm partitioning techniques are used to exploit the structure of the master programs involved in those methods

a potential reduction algorithm allowing column generation using the dantzig wolfe decomposition technique a potential reduction algorithm allowing column generation for the linear feasibility lf problem is developed the point of departure is a simple containing polytope and its analytic center in each iteration an inequality violated at the current center is selected used to cut the polytope and then used to find the new center for the shrunken polytope the potential value associated with the containing polytope is reduced by a constant at each step and the algorithm is terminated in a polynomial time that depends only on the number and data length of the inequalities generated in the iterative process

large cone nonnegative matrix factorization nonnegative matrix factorization nmf has been greatly popularized by its parts based interpretation and the effective multiplicative updating rule for searching local solutions in this paper we study the problem of how to obtain an attractive local solution for nmf which not only fits the given training data well but also generalizes well on the unseen test data based on the geometric interpretation of nmf we introduce two large cone penalties for nmf and propose large cone nmf lcnmf algorithms compared with nmf lcnmf will obtain bases comprising a larger simplicial cone and therefore has three advantages 1 the empirical reconstruction error of lcnmf could mostly be smaller 2 the generalization ability of the proposed algorithm is much more powerful and 3 the obtained bases of lcnmf have a low overlapping property which enables the bases to be sparse and makes the proposed algorithms very robust experiments on synthetic and real world data sets confirm the efficiency of lcnmf

discriminative transfer subspace learning via low rank and sparse representation in this paper we address the problem of unsupervised domain transfer learning in which no labels are available in the target domain we use a transformation matrix to transfer both the source and target data to a common subspace where each target sample can be represented by a combination of source samples such that the samples from different domains can be well interlaced in this way the discrepancy of the source and target domains is reduced by imposing joint low rank and sparse constraints on the reconstruction coefficient matrix the global and local structures of data can be preserved to enlarge the margins between different classes as much as possible and provide more freedom to diminish the discrepancy a flexible linear classifier projection is obtained by learning a non negative label relaxation matrix that allows the strict binary label matrix to relax into a slack variable matrix our method can avoid a potentially negative transfer by using a sparse matrix to model the noise and thus is more robust to different types of noise we formulate our problem as a constrained low rankness and sparsity minimization problem and solve it by the inexact augmented lagrange multiplier method extensive experiments on various visual domain adaptation tasks show the superiority of the proposed method over the state of the art methods the matlab code of our method will be publicly available at http www yongxu org lunwen html

a cutting plane algorithm for convex programming that uses analytic centers anoracle for a convex sets n accepts as input any pointz in n and ifz s then it returns yes while ifz s then it returns no along with a separating hyperplane we give a new algorithm that finds a feasible point ins in cases where an oracle is available our algorithm uses the analytic center of a polytope as test point and successively modifies the polytope with the separating hyperplanes returned by the oracle the key to establishing convergence is that hyperplanes judged to be unimportant are pruned from the polytope if a ball of radius 2 l is contained ins ands is contained in a cube of side 2l 1 then we can show our algorithm converges after o nl2 iterations and performs a total of o n4l3 tnl2 arithmetic operations wheret is the number of arithmetic operations required for a call to the oracle the bound is independent of the number of hyperplanes generated in the algorithm an important application in which an oracle is available is minimizing a convex function overs

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

coupling of general lagrangian systems this work is devoted to the coupling of two fluid models such a s two euler systems in lagrangian coordinates at a fixed interface we define coupling conditi ons which can be expressed in terms of continuity of some well chosen variables and then solve the coupled riemann problem in the present setting where the interface is characteristic a particula r choice of dependent variables which are transmitted ensures the overall conservativity

choice of measure source terms in interface coupling for a model problem in gas dynamics this paper is devoted to the mathematical and numerical analysis of a coupling procedure for one dimensional euler systems the two systems have different closure laws and are coupled through a thin fixed interface following the work of 5 we propose to couple these systems by a bounded vector valued dirac measure concentrated at the coupling interface which in the applications may have a physical meaning we show that the proposed framework allows to control the coupling conditions and we propose an approximate riemann solver based on a relaxation approach preserving equilibrium solutions of the coupled problem numerical experiments in constrained optimization problems are then presented to assess the performances of the present method 1 introduction the study of large scale and complex problems exhibiting a wide range of physical space and time scales see for instance 62 35 14 usually requires separate solvers adapted to the resolution of specific scales this is the case of many industrial flows let us quote for example the numerical simulation of two phase flows applied to the burning liquid oxygen hydrogen gas in rocket engines 58 this kind of flow contains both separated and dispersed two phase flows due to atomization and evaporation phenomena this requires appropriate models and solvers for separated and dispersed phases that have to be appropriately coupled another example concerns turbomachine flows which can be modeled by the euler equations of gas dynamics with different closure laws between the stages of the turbine where the conditions of temperature and pressure are strongly heterogeneous the coupling of these different systems is thus necessary to give a complete description of the flow inside the whole turbine the method of interface coupling allows to represent the evolution of such flows where different models are separated by fixed interfaces first coupling conditions are specified at the interface to exchange information between the systems the definition of transmission conditions generally results from physical consideration e g the conservation or the continuity of given variables then the transmission conditions are represented at the discrete level the study of interface coupling for nonlinear hyperbolic systems has received attention for several years in 43 the authors study the scalar case from both mathematical and numerical points of view

strong traces for solutions of multidimensional scalar conservation laws in this paper we consider multidimensional scalar conservation laws without bv estimates defined in a subset d we show that with a non degeneracy hypothesis on the flux we can define a strong notion of trace at the boundary of reached by l 1 convergence

a nonlinear analytic center cutting plane method for a class of convex programming problems a cutting plane algorithm for minimizing a convex function subject to constraints defined by a separation oracle is presented the algorithm is based on approximate analytic centers the nonlinearity of the objective function is taken into account yet the feasible region is approximated by a containing polytope this containing polytope is regularly updated by adding a new cut through a test point each test point is an approximate analytic center of the intersection of a containing polytope and a level set of the nonlinear objective function we establish the complexity of the algorithm our complexity estimate is given in terms of the problem dimension the desired accuracy of an approximate solution and other parameters that depend on the geometry of a specific instance of the problem

boundary layers in weak solutions of hyperbolic conservation laws this paper is concerned with the initial boundary value problem for a nonlinear hyperbolic system of conservation laws we study the boundary layers that may arise in approximations of entropy discontinuous solutions we consider both the vanishing viscosity method and finite difference schemes lax friedrichs type schemes and the godunov scheme we demonstrate that different regularization methods generate different boundary layers hence the boundary condition can be formulated only if an approximation scheme is selected first assuming solely uniform l infty bounds on the approximate solutions and so dealing with l infty solutions we derive several entropy inequalities satisfied by the boundary layer in each case under consideration a young measure is introduced to describe the boundary trace when a uniform bound on the total variation is available the boundary young measure reduces to a dirac mass from the above analysis we deduce several formulations for the boundary condition which apply whether the boundary is characteristic or not each formulation is based on a set of admissible boundary values following the terminology of dubois lefloch 15 the local structure of these sets and the well posedness of the corresponding initial boundary value problem are investigated the results are illustrated with convex and nonconvex conservation laws and examples from continuum mechanics

complexity estimates of some cutting plane methods based on the analytic barrier in this paper we establish the efficiency estimates for two cutting plane methods based on the analytic barrier we prove that the rate of convergence of the second method is optimal uniformly in the number of variables we present a modification of the second method in this modified version each test point satisfies an approximate centering condition we also use the standard strategy for updating approximate hessians of the logarithmic barrier function we prove that the rate of convergence of the modified scheme remains optimal and demonstrate that the number of newton steps in the auxiliary minimization processes is bounded by an absolute constant we also show that the approximate hessian strategy significantly improves the total arithmetical complexity of the method

solving combinatorial optimization problems using karmakar s algorithm we describe a cutting plane algorithm for solving combinatorial optimization problems the primal projective standard form variant of karmarkar s algorithm for linear programming is applied to the duals of a sequence of linear programming relaxations of the combinatorial optimization problem

a scaling technique for finding the weighted analytical center of a polytope let a bounded full dimensional polytope be defined by the systemax b wherea is anm n matrix letai denote theith row of the matrixa and define theweighted analytic center of the polytope to be the point that minimizes the strictly convex barrier function i 1mwi ln aitx bi the proper selection of weightswi can make any desired point in the interior of the polytope become the weighted analytic center as a result the weighted analytic center has applications in both linear and general convex programming for simplicity we assume that the weights are positive integers if some of thewi s are much larger than others then newton s method for minimizing the resulting barrier function is very unstable and can be very slow previous methods for finding the weighted analytic center relied upon a rather direct application of newton s method potentially resulting in very slow global convergence we present a method for finding the weighted analytic center that is based on the scaling technique of edmonds and karp and is an enhancement of newton s method the scaling algorithm runs in o sqrt m log w iterations wherem is the number of constraints defining the polytope andw is the largest weight given on any constraint each iteration involves taking a step in the newton direction and its complexity is dominated by the time needed to solve a system of linear equations

the numerical interface coupling of nonlinear hyperbolic systems of conservation laws i the scalar case we study the theoretical and numerical coupling of two general hyperbolic conservation laws the coupling preserves in a weak sense the continuity of the solution at the interface without imposing the overall conservativity of the coupled model in order to analyze the convergence of the coupled numerical scheme we first revisit the approximation of the boundary value problems we then prove the convergence and characterize the limit solution of the coupled schemes in a few simple but significative coupling situations the general coupling problem is analyzed for riemann initial data and illustrated by numerical simulations resume nous nous interessons a une nouvelle forme de couplage de deux systemes hyperboliques de lois de conservation ce couplage assure de facon faible la continuite de la solution a l interface sans imposer la conservativite du modele couple pour etudier la convergence du schema d approximation numerique nous commencons par reprendre les resultats concernant l approximation du probleme aux limites nous demontrons ensuite la convergence du schema couple dans un certain nombre de cas interessants le cas general du couplage est etudie et illustre numeriquement pour une donnee initiale de riemann

a cutting plane method from analytic centers for stochastic programming the stochastic linear programming problem with recourse has a dual block angular structure it can thus be handled by benders decomposition or by kelley s method of cutting planes equivalently the dual problem has a primal block angular structure and can be handled by dantzig wolfe decomposition the two approaches are in fact identical by duality here we shall investigate the use of the method of cutting planes from analytic centers applied to similar formulations the only significant difference form the aforementioned methods is that new cutting planes or columns by duality will be generated not from the optimum of the linear programming relaxation but from the analytic center of the set of localization

initial boundary value problem for conservation laws this paper concerns the initial boundary value problems for some systems of quasilinear hyperbolic conservation laws in the space of bounded measurable functions the main assumption is that the system under study admits a convex entropy extension it is proved that then any twicely differentiable entropy fluxes have traces on the boundary if the bounded solutions are generated by either godunov schemes or by suitable viscous approximations furthermore in the case that the weak interior solutions are generated by godunov schemes any lipschitz continuous entropy fluxes corresponding to convex entropies have traces on the boundary and the traces are bounded above by computable numerical boundary values this in particular gives a trace formula for the flux functions in terms of the numerical boundary data we also investigate the formulation of boundary conditions for systems of hyperbolic conservation laws it is shown that the set of expected boundary values derived from the viscous approximation contains the one derived in terms of the boundary riemann problems and the converse is not true in general the general theory is then applied to some specific examples first several new facts are obtained for convex scalar conservation laws for example we give example which show that godunov schemes produce numerical boundary layers it is shown that any continuous functions of density have traces on the boundary instead of only entropy fluxes we also obtain interior and boundary regularity of the weak solutions for bounded measurable initial and boundary data a generalized oleinik entropy condition is also obtained next we prove the existence of a weak solution to the initial boundary value problem for a family of quadratic system with a uniformly characteristic boundary condition

automatic relevance determination in nonnegative matrix factorization with the beta divergence this paper addresses the estimation of the latent dimensionality in nonnegative matrix factorization nmf with the divergence the divergence is a family of cost functions that includes the squared euclidean distance kullback leibler kl and itakura saito is divergences as special cases learning the model order is important as it is necessary to strike the right balance between data fidelity and overfitting we propose a bayesian model based on automatic relevance determination ard in which the columns of the dictionary matrix and the rows of the activation matrix are tied together through a common scale parameter in their prior a family of majorization minimization mm algorithms is proposed for maximum a posteriori map estimation a subset of scale parameters is driven to a small lower bound in the course of inference with the effect of pruning the corresponding spurious components we demonstrate the efficacy and robustness of our algorithms by performing extensive experiments on synthetic data the swimmer dataset a music decomposition example and a stock price prediction task

integrated low rank based discriminative feature learning for recognition feature learning plays a central role in pattern recognition in recent years many representation based feature learning methods have been proposed and have achieved great success in many applications however these methods perform feature learning and subsequent classification in two separate steps which may not be optimal for recognition tasks in this paper we present a supervised low rank based approach for learning discriminative features by integrating latent low rank representation latlrr with a ridge regression based classifier our approach combines feature learning with classification so that the regulated classification error is minimized in this way the extracted features are more discriminative for the recognition tasks our approach benefits from a recent discovery on the closed form solutions to noiseless latlrr when there is noise a robust principal component analysis pca based denoising step can be added as preprocessing when the scale of a problem is large we utilize a fast randomized algorithm to speed up the computation of robust pca extensive experimental results demonstrate the effectiveness and robustness of our method

on vaidya s volumetric cutting plane method for convex programming we describe a simplified and strengthened version of vaidya s volumetric cutting plane method for finding a point in a convex set x1d49e rn at each step the algorithm has a system of linear inequality constraints which defines a polyhedron x1d4ab x1d49e and an interior point x x1d4ab the algorithm then either drops one constraint or calls an oracle to check if x x1d49e and if not obtain a new constraint that separates x from x1d49e following the addition or deletion of a constraint the algorithm takes a small number of newton steps for the volumetric barrier v progress of the algorithm is measured in terms of changes in v the algorithm is terminated when either it is discovered that x x1d49e or v becomes large enough to demonstrate that the volume of x1d49e must be below some prescribed amount the complexity of the algorithm compares favorably with that of the ellipsoid method especially in terms of the number of calls to the separation oracle compared to vaidya s original analysis we decrease the total number

affine scaling for linear programs with free variables the affine scaling modification of karmarkar s algorithm is extended to solve problems with free variables this extended primal algorithm is used to prove two important results first the geometrically elegant feasibility algorithm proposed by chandru and kochar is the same algorithm as the one obtained by appending a single column of residuals to the constraint matrix second the dual algorithm as first described by adler et al is the same as the extended primal algorithm applied to the dual

decomposition and nondifferentiable optimization with the projective algorithm this paper deals with an application of a variant of karmarkar s projective algorithm for linear programming to the solution of a generic nondifferentiable minimization problem this problem is closely related to the dantzig wolfe decomposition technique used in large scale convex programming the proposed method is based on a column generation technique defining a sequence of primal linear programming maximization problems associated with each problem one defines a weighted potential function which is minimized using a variant of the projective algorithm when a point close to the minimum of the potential function is reached a corresponding point in the dual space is constructed which is close to the analytic center of a polytope containing the solution set of the nondifferentiable optimization problem an admissible cut of the polytope corresponding to a new supporting hyperplane of the epigraph of the function to minimize is then generated at this approximate analytic center in the primal space this new cut translates into a new column for the associated linear programming problem the algorithm has performed well on a set of convex nondifferentiable programming problems

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

a faster strongly polynomial minimum cost flow algorithm in this paper we present a new strongly polynomial time algorithm for the minimum cost flow problem based on a refinement of the edmonds karp scaling technique our algorithm solves the uncapacitated minimum cost flow problem as a sequence of on log n shortest path problems on networks with n nodes and m arcs and runs in on log nm n log n time using a standard transformation this approach yields an om log nm n log n algorithm for the capacitated minimum cost flow problem this algorithm improves the best previous strongly polynomial time algorithm due to z galil and e tardos by a factor of n2 m our algorithm for the capacitated minimum cost flow problem is even more efficient if the number of arcs with finite upper bounds say m is much less than m in this case the running time of the algorithm is om n log nm n log n

inexact cuts in benders decomposition benders decomposition is a well known technique for solving large linear programs with a special structure in particular it is a popular technique for solving multistage stochastic linear programming problems early termination in the subproblems generated during benders decomposition assuming dual feasibility produces valid cuts that are inexact in the sense that they are not as constraining as cuts derived from an exact solution we describe an inexact cut algorithm prove its convergence under easily verifiable assumptions and discuss a corresponding dantzig wolfe decomposition algorithm the paper is concluded with some computational results from applying the algorithm to a class of stochastic programming problems that arise in hydroelectric scheduling

a primal dual decomposition based interior point approach to two stage stochastic linear programming decision making under uncertainty is a challenge faced by many decision makers stochastic programming is a major tool developed to deal with optimization with uncertainties which has found applications in e g finance such as asset liability and bond portfolio management computationally however many models in stochastic programming remain unsolvable because of overwhelming dimensionality for a model to be well solvable its special structure must be explored most of the solution methods are based on decomposing the data in this paper we propose a new decomposition approach for two stage stochastic programming based on a direct application of the path following method combined with the homogeneous self dual technique numerical experiments show that our decomposition algorithm is very efficient for solving stochastic programs in particular we apply our decomposition method to a two period portfolio selection problem using options on a stock index in this model the investor can invest in a money market account a stock index and european options on this index with different maturities we experiment with our model with market prices of options on the s p500

robust recovery of subspace structures by low rank representation in this paper we address the subspace clustering problem given a set of data samples vectors approximately drawn from a union of multiple subspaces our goal is to cluster the samples into their respective subspaces and remove possible outliers as well to this end we propose a novel objective function named low rank representation lrr which seeks the lowest rank representation among all the candidates that can represent the data samples as linear combinations of the bases in a given dictionary it is shown that the convex program associated with lrr solves the subspace clustering problem in the following sense when the data is clean we prove that lrr exactly recovers the true subspace structures when the data are contaminated by outliers we prove that under certain conditions lrr can exactly recover the row space of the original data and detect the outlier as well for data corrupted by arbitrary sparse errors lrr can also approximately recover the row space with theoretical guarantees since the subspace membership is provably determined by the row space these further imply that lrr can perform robust subspace clustering and error correction in an efficient and effective way

an analytic center based column generation algorithm for convex quadratic feasibility problems we consider the analytic center based column generation algorithm for the problem of finding a feasible point in a set defined by a finite number of convex quadratic inequalities at each iteration the algorithm computes an approximate analytic center of the set defined by the intersection of quadratic inequalities generated in the previous iterations if this approximate analytic center is a solution then the algorithm terminates otherwise a quadratic inequality violated at the current center is selected and a new quadratic cut defined by a convex quadratic inequality is placed near the approximate center as the number of cuts increases the set defined by their intersection shrinks and the algorithm eventually finds a solution to the problem previously similar analytic center based column generation algorithms were studied first for the linear feasibility problem and later for the general convex feasibility problem our method differs from these early methods in that we use quadratic cuts in the computation instead of linear cuts moreover our method has a polynomial worst case complexity of o n ln frac 1 varepsilon on the total number of cuts to be used where n is the number of convex quadratic polynomial inequalities in the problem and varepsilon is the radius of the largest ball contained in the feasible set in contrast the early column generation methods using linear cuts can only solve the convex quadratic feasibility problem in pseudopolynomial time

graph regularized nonnegative matrix factorization for data representation matrix factorization techniques have been frequently applied in information retrieval computer vision and pattern recognition among them nonnegative matrix factorization nmf has received considerable attention due to its psychological and physiological interpretation of naturally occurring data whose representation may be parts based in the human brain on the other hand from the geometric perspective the data is usually sampled from a low dimensional manifold embedded in a high dimensional ambient space one then hopes to find a compact representation which uncovers the hidden semantics and simultaneously respects the intrinsic geometric structure in this paper we propose a novel algorithm called graph regularized nonnegative matrix factorization gnmf for this purpose in gnmf an affinity graph is constructed to encode the geometrical information and we seek a matrix factorization which respects the graph structure our empirical study shows encouraging results of the proposed algorithm in comparison to the state of the art algorithms on real world problems

coupling techniques for nonlinear hyperbolic equations iv well balanced schemes for scalar multi dimensional and multi component laws this series of papers is devoted to the formulation and the approximation of coupling problems for nonlinear hyperbolic equations the coupling across an interface in the physical space is formulated in term of an augmented system of partial differential equations in an earlier work this strategy allowed us to develop a regularization method based on a thick interface model in one space variable for coupling scalar equations in the present paper we significantly extend this framework and in addition encompass equations in several space variables this new formulation includes the coupling of several distinct scalar conservation laws and allows for a possible covering in space our main contributions are on one hand the design and analysis of a well balanced finite volume method on general triangulations and on the other hand a proof of convergence of this method toward entropy solutions extending coquel cockburn and lefloch s theory restricted to a single conservation law without coupling the core of our analysis is first the derivation of entropy inequalities as well as a discrete entropy dissipation estimate and second a proof of convergence toward the entropy solution of the coupling problem

on nonconservative algorithms for grid interfaces in computations of fluid flows by domain decomposition methods the necessity of conservation at grid interfaces is now widely claimed in this paper we consider nonconservative algorithms for the interfaces our investigation begins with discussions about typical nonconservative interface treatments for one dimensional calculations the analysis shows that the conservation error of a numerical solution caused by a nonconservative interface matching has an upper bound when the solution itself is bounded furthermore if the numerical solution converges as the mesh size goes to zero it converges to a weak solution of the problem under certain conditions that may be detected numerically also we acquire similar results for two dimensional calculations on grids intersecting with each other in an arbitrary way in order to illustrate the theoretical results we present numerical examples and demonstrate that under those conditions conservation error reduces and accuracy for jumps as well as locations of discontinuities improves as the mesh size decreases

solving nonlinear multicommodity flow problems by the analytic center cutting plane method the paper deals with nonlinear multicommodity flow problems with convex costs a decomposition method is proposed to solve them the approach applies a potential reduction algorithm to solve the master problem approximately and a column generation technique to define a sequence of primal linear programming problems each subproblem consists of finding a minimum cost flow between an origin and a destination node in an uncapacited network it is thus formulated as a shortest path problem and solved with dijkstra s d heap algorithm an implementation is described that takes full advantage of the supersparsity of the network in the linear algebra operations computational results show the efficiency of this approach on well known nondifferentiable problems and also large scale randomly generated problems up to 1000 arcs and 5000 commodities

finding minimum cost circulations by canceling negative cycles a classical algorithm for finding a minimum cost circulation consists of repeatedly finding a residual cycle of negative cost and canceling it by pushing enough flow around the cycle to saturate an arc we show that a judicious choice of cycles for canceling leads to a polynomial bound on the number of iterations in this algorithm this gives a very simple strongly polynomial algorithm that uses no scaling a variant of the algorithm that uses dynamic trees runs in o nm log n min log nc m log n time on a network of n vertices m arcs and arc costs of maximum absolute value c this bound is comparable to those of the fastest previously known algorithms

convergence analysis of alternating direction method of multipliers for a family of nonconvex problems the alternating direction method of multipliers admm is widely used to solve large scale linearly constrained optimization problems convex or nonconvex in many engineering fields however there is a general lack of theoretical understanding of the algorithm when the objective function is nonconvex in this paper we analyze the convergence of the admm for solving certain nonconvex consensus and sharing problems we show that the classical admm converges to the set of stationary solutions provided that the penalty parameter in the augmented lagrangian is chosen to be sufficiently large for the sharing problems we show that the admm is convergent regardless of the number of variable blocks our analysis does not impose any assumptions on the iterates generated by the algorithm and is broadly applicable to many admm variants involving proximal update rules and various flexible block selection rules

a polynomial primal dual dikin type algorithm for linear programming in this paper we present a new primal dual affine scaling method for linear programming the method yields a strictly complementary optimal solution pair and also allows a polynomial time convergence proof the search direction is obtained by using the original idea of dikin namely by minimizing the objective function which is the duality gap in the primal dual case over some suitable ellipsoid this gives rise to completely new primal dual affine scaling directions having no obvious relation with the search directions proposed in the literature so far the new directions guarantee a significant decrease in the duality gap in each iteration and at the same time they drive the iterates to the central path in the analysis of our algorithm we use a barrier function which is the natural primal dual generalization of karmarkar s potential function the iteration bound is onl which is a factor ol better than the iteration bound of an earlier primal dual affine scaling method monteiro adler and resende monteiro r d c i adler m g c resende 1990 a polynomial time primal dual affine scaling algorithm for linear and convex quadratic programming and its power series extension math oper res 15 191 214

a primal partitioning solution for the arc chain formulation of a multicommodity network flow problem we present a new solution approach for the multicommodity network flow problem mcnf based upon both primal partitioning and decomposition techniques which simplifies the computations required by the simplex method the partitioning is performed on an arc chain incidence matrix of the mcnf similar within a change of variables to the constraint matrix of the master problem generated in a dantzig wolfe decomposition to isolate a very sparse near triangular working basis of greatly reduced dimension the majority of the simplex operations performed on the partitioned basis are simply additive and network operations specialized for the nine possible pivot types identified the columns of the arc chain incidence matrix are generated by a dual network simplex method for updating shortest paths when link costs change

a polynomial method of approximate centers for linear programming we present a path following algorithm for the linear programming problem with a surprisingly simple and elegant proof of its polynomial behaviour this is done both for the problem in standard form and for its dual problem we also discuss some implementation strategies

large step path following methods for linear programming part i barrier function method the algorithm proposed in this paper is the classical logarithmic barrier function method with newton raphson steps for the internal minimization of the penalized function polynomial behavior is obtained by stopping each internal cycle when the iterates approach the central trajectory each master iteration updates the penalty parameter by a constant factor and the overall complexity bound depends on this factor o nl newton iterations for an arbitrary constant and o sqrt n l iterations for a constant dependent on sqrt n

nuclear norm based matrix regression with applications to face recognition with occlusion and illumination changes recently regression analysis has become a popular tool for face recognition most existing regression methods use the one dimensional pixel based error model which characterizes the representation error individually pixel by pixel and thus neglects the two dimensional structure of the error image we observe that occlusion and illumination changes generally lead approximately to a low rank error image in order to make use of this low rank structural information this paper presents a two dimensional image matrix based error model namely nuclear norm based matrix regression nmr for face representation and classification nmr uses the minimal nuclear norm of representation error image as a criterion and the alternating direction method of multipliers admm to calculate the regression coefficients we further develop a fast admm algorithm to solve the approximate nmr model and show it has a quadratic rate of convergence we experiment using five popular face image databases the extended yale b ar eurecom multi pie and frgc experimental results demonstrate the performance advantage of nmr over the state of the art regression based methods for face recognition in the presence of occlusion and illumination variations

robust transfer metric learning for image classification metric learning has attracted increasing attention due to its critical role in image analysis and classification conventional metric learning always assumes that the training and test data are sampled from the same or similar distribution however to build an effective distance metric we need abundant supervised knowledge i e side label information which is generally inaccessible in practice because of the expensive labeling cost in this paper we develop a robust transfer metric learning rtml framework to effectively assist the unlabeled target learning by transferring the knowledge from the well labeled source domain specifically rtml exploits knowledge transfer to mitigate the domain shift in two directions i e sample space and feature space in the sample space domain wise and class wise adaption schemes are adopted to bridge the gap of marginal and conditional distribution disparities across two domains in the feature space our metric is built in a marginalized denoising fashion and low rank constraint which make it more robust to tackle noisy data in reality furthermore we design an explicit rank constraint regularizer to replace the rank minimization np hard problem to guide the low rank metric learning experimental results on several standard benchmarks demonstrate the effectiveness of our proposed rtml by comparing it with the state of the art transfer learning and metric learning algorithms

exploiting special structure in a primal dual path following algorithm a primal dual path following algorithm that applies directly to a linear program of the form min ctx ax b hx u x 0 x n is presented this algorithm explicitly handles upper bounds generalized upper bounds variable upper bounds and block diagonal structure we also show how the structure of time staged problems and network flow problems can be exploited especially on a parallel computer finally using our algorithm we obtain a complexity bound of o sqrt n ds2 log nk fortransportation problems withs origins d destinations s

large step path following methods for linear programming part ii potential reduction method the algorithm proposed in this paper is the affine polynomial potential reduction method with new procedures for updating the lower bounds for an optimal solution of the linear programming problem a method is developed for updating the lower bounds by large steps with strict control over the duality gaps associated with each iterate two algorithms are obtained by this approach the first one has complexity o nl iterations and a very simple updating procedure the second one updates the lower bounds at points very near the central trajectory and achieves a complexity of o sqrt n l iterations

dimensionality dependent generalization bounds for k dimensional coding schemes the k dimensional coding schemes refer to a collection of methods that attempt to represent data using a set of representative k dimensional vectors and include nonnegative matrix factorization dictionary learning sparse coding k means clustering and vector quantization as special cases previous generalization bounds for the reconstruction error of the k dimensional coding schemes are mainly dimensionality independent a major advantage of these bounds is that they can be used to analyze the generalization error when data are mapped into an infinite or high dimensional feature space however many applications use finite dimensional data features can we obtain dimensionality dependent generalization bounds for k dimensional coding schemes that are tighter than dimensionality independent bounds when data are in a finite dimensional feature space yes in this letter we address this problem and derive a dimensionality dependent generalization bound for k dimensional coding schemes by bounding the covering number of the loss function class induced by the reconstruction error the bound is of order where m is the dimension of features k is the number of the columns in the linear implementation of coding schemes and n is the size of sample when n is finite and when n is infinite we show that our bound can be tighter than previous results because it avoids inducing the worst case upper bound on k of the loss function the proposed generalization bound is also applied to some specific coding schemes to demonstrate that the dimensionality dependent bound is an indispensable complement to the dimensionality independent generalization bounds

structure constrained low rank representation benefiting from its effectiveness in subspace segmentation low rank representation lrr and its variations have many applications in computer vision and pattern recognition such as motion segmentation image segmentation saliency detection and semisupervised learning it is known that the standard lrr can only work well under the assumption that all the subspaces are independent however this assumption cannot be guaranteed in real world problems this paper addresses this problem and provides an extension of lrr named structure constrained lrr sc lrr to analyze the structure of multiple disjoint subspaces which is more general for real vision data we prove that the relationship of multiple linear disjoint subspaces can be exactly revealed by sc lrr with a predefined weight matrix as a nontrivial byproduct we also illustrate that sc lrr can be applied for semisupervised learning the experimental results on different types of vision problems demonstrate the effectiveness of our proposed method

quadratic regularizations in an interior point method for primal block angular problems one of the most efficient interior point methods for some classes of primal block angular problems solves the normal equations by a combination of cholesky factorizations and preconditioned conjugate gradient for respectively the block and linking constraints its efficiency depends on the spectral radius in 0 1 of a certain matrix in the definition of the preconditioner spectral radius close to 1 degrade the performance of the approach the purpose of this work is twofold first to show that a separable quadratic regularization term in the objective reduces the spectral radius significantly improving the overall performance in some classes of instances second to consider a regularization term which decreases with the barrier function thus with no need for an extra parameter computational experience with some primal block angular problems confirms the efficiency of the regularized approach in particular for some difficult problems the solution time is reduced by a factor of two to ten by the regularization term outperforming state of the art commercial solvers

on the computation of weighted analytic centers and dual ellipsoids with the projective algorithm the primal projective algorithm for linear programs with unknown optimal objective function value is extended to the case where one uses a weighted karmarkar potential function this potential is defined with respect to a strict lower bound to the optimum the minimization of this potential when the lower bound is kept fixed yields a primal and a dual feasible solution the dual solution is the weighted analytic center of a certain dual polytope finally one exhibits a pair of homothetic dual ellipsoids that extends results obtained by sonnevend todd ye freund and anstreicher

algorithm dependent generalization bounds for multi task learning often tasks are collected for multi task learning mtl because they share similar feature structures based on this observation in this paper we present novel algorithm dependent generalization bounds for mtl by exploiting the notion of algorithmic stability we focus on the performance of one particular task and the average performance over multiple tasks by analyzing the generalization ability of a common parameter that is shared in mtl when focusing on one particular task with the help of a mild assumption on the feature structures we interpret the function of the other tasks as a regularizer that produces a specific inductive bias the algorithm for learning the common parameter as well as the predictor is thereby uniformly stable with respect to the domain of the particular task and has a generalization bound with a fast convergence rate of order mathcal o 1 n where n is the sample size of the particular task when focusing on the average performance over multiple tasks we prove that a similar inductive bias exists under certain conditions on the feature structures thus the corresponding algorithm for learning the common parameter is also uniformly stable with respect to the domains of the multiple tasks and its generalization bound is of the order mathcal o 1 t where t is the number of tasks these theoretical analyses naturally show that the similarity of feature structures in mtl will lead to specific regularizations for predicting which enables the learning algorithms to generalize fast and correctly from a few examples

a multicut algorithm for two stage stochastic linear programs outer linearization methods such as van slyke and wets s l shaped method for stochastic linear programs generally apply a single cut on the nonlinear objective at each major iteration the structure of stochastic programs allows for several cuts to be placed at once this paper describes a multicut algorithm to carry out this procedure it presents experimental and theoretical justification for reductions in major iterations

a numerical method for transport equations with discontinuous flux functions application to mathematical modeling of cell dynamics in this work we propose a numerical method to handle discontinuous fluxes arising in transport like equations more precisely we study hyperbolic pdes with flux transmission conditions at interfaces between subdomains where coefficients are discontinuous a dedicated finite volume scheme with a limited high order enhancement is adapted to treat the discontinuities arising at interfaces the validation of the method is done on one and two dimensional toy problems for which exact solutions are available allowing us to do a thorough convergence study we then apply the method to a biological model focusing on complex cell dynamics that initially motivated this study and illustrates the full potentialities of the scheme

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

homogeneous analytic center cutting plane methods for convex problems and variational inequalities in this paper we consider a new analytic center cutting plane method in an extended space we prove the efficiency estimates for the general scheme and show that these results can be used in the analysis of a feasibility problem the variational inequality problem and the problem of constrained minimization our analysis is valid even for problems whose solution belongs to the boundary of the domain

complexity of some cutting plane methods that use analytic centers we consider cutting plane methods for minimizing a convex possibly nondifferentiable function subject to box constraints at each iteration accumulated subgradient cuts define a polytope that localizes the minimum the objective and its subgradient are evaluated at the analytic center of this polytope to produce one or two cuts that improve the localizing set we give complexity estimates for several variants of such methods our analysis is based on the works of goffin luo and ye

solving a time space network formulation for the convoy movement problem we give a formal specification for a strategic network routing problem known as the convoy movement problem cmp and establish that the corresponding feasibility problem is np complete we then introduce an integer programming ip model based on the concept of a time space network and apply a lagrangian relaxation to this model we discuss how the dual function may be evaluated using a modified version of dijkstra s algorithm suitable to very large implicitly defined graphs and show how heuristic solutions to the primal problem may be obtained we present results for a number of instances of the cmp most of which are based on real world problems the number of convoys in these instances varies between 15 25 and their movement time requires up to several thousand time units in networks ranging in size from a few dozen to several thousand vertices and edges the most difficult instance tested involves 17 long convoys each taking four times the average link travel time to pass through a point in the network this instance is solved within 3 3 of optimality in less than 3 5 hours of computing time on a dell precision 420 dual processor computer every other test instance is solved within 2 of the optimal value in less than 20 minutes of computing time

numerical viscosity and convergence of finite volume methods for conservation laws with boundary conditions the authors study the convergence of finite volume schemes for general multidimensional conservation laws with boundary conditions a unique result for a measure valued solution which generalizes those of diperna for unbounded domains and szepessy for bounded domains is proved it gives us sufficient conditions to get convergence by studying carefully the entropy production for one dimensional e schemes we are able to prove convergence of finite volume e schemes under general assumption

learning fast low rank projection for image classification rooted in a basic hypothesis that a data matrix is strictly drawn from some independent subspaces the low rank representation lrr model and its variations have been successfully applied in various image classification tasks however this hypothesis is very strict to the lrr model as it cannot always be guaranteed in real images moreover the hypothesis also prevents the sub dictionaries of different subspaces from collaboratively representing an image fortunately in supervised image classification low rank signal can be extracted from the independent label subspaces ils instead of the independent image subspaces iis therefore this paper proposes a projective low rank representation plr model by directly training a projective function to approximate the lrr derived from the labels to the best of our knowledge plr is the first attempt to use the ils hypothesis to relax the rigorous iis hypothesis in the lrr models we further prove a low rank effect that the representations learned by plr have high intraclass similarities and large interclass differences which are beneficial to the classification tasks the effectiveness of our proposed approach is validated by the experimental results on three databases

a family of polynomial affine scaling algorithms for positive semidefinite linear complementarity problems in this paper the new polynomial affine scaling algorithm of jansen roos and terlaky for linear programming lp is extended to positive semidefinite psd linear complementarity problems the algorithm is immediately further generalized to allow higher order scaling these algorithms are also new for the lp case the analysis is based on ling s proof for the lp case hence it allows an arbitrary interior feasible pair to start with with the scaling of jansen roos and terlaky the complexity of the algorithm is cal o left frac n rho 2 1 rho 2 ln frac x 0 ts 0 e right where rho 2 is a uniform bound for the ratio of the smallest and largest coordinate of the iterates in the primal dual space we also show that monteiro adler and resende s polynomial complexity result for the classical primal dual affine scaling algorithm can easily be derived from our analysis in addition our result is valid for arbitrary not necessarily centered initial points finally some computational results are presented which indicate the influence of the order of the scaling on the numerical performance

boundary conditions for nonlinear hyperbolic systems of conservation laws we propose two formulations of the boundary conditions for nonlinear hyperbolic systems of conservation laws a first approach is based on the vanishing viscosity method and a second one is related to the riemann problem the equivalence between these two conditions is studied the latter formulation is extended to treat numerically physically relevant boundary conditions monodimensional experiments are presented