on particular solution of ordinary differential equations with constant coefficients in this paper a new method is presented for obtaining the particular solution of ordinary differential equations with constant coefficients an explicit formula of the particular solution is derived from the use of an upper triangular toeplitz matrix the validity of the approach is illustrated by some examples

iterative bregman projections for regularized transportation problems this article details a general numerical framework to approximate so lutions to linear programs related to optimal transport the general idea is to introduce an entropic regularization of the initial linear program this regularized problem corresponds to a kullback leibler bregman di vergence projection of a vector representing some initial joint distribu tion on the polytope of constraints we show that for many problems related to optimal transport the set of linear constraints can be split in an intersection of a few simple constraints for which the projections can be computed in closed form this allows us to make use of iterative bregman projections when there are only equality constraints or more generally bregman dykstra iterations when inequality constraints are in volved we illustrate the usefulness of this approach to several variational problems related to optimal transport barycenters for the optimal trans port metric tomographic reconstruction multi marginal optimal trans port and in particular its application to brenier s relaxed solutions of in compressible euler equations partial un balanced optimal transport and optimal transport with capacity constraints

piecewise linear programming the compact cplp algorithm a compact algorithm is presented for solving the convex piecewise linear programming problem formulated by means of a separable convex piecewise linear objective function to be minimized and a set of linear constraints this algorithm consists of a finite sequence of cycles derived from the simplex method characteritic of linear programming and the line search characteristic of nonlinear programming both the required storage and amount of calculation are reduced with respect to the usual approach based on a linear programming formulation with an expanded tableau the tableau dimensions arem n 1 wherem is the number of constraints andn the number of the original structural variables and they do not increase with the number of breakpoints of the piecewise linear terms constituting the objective function

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

optimal maps for the multidimensional monge kantorovich problem let 1 n be borel probability measures on d denote by 1 n the set of all n tuples t t1 tn such that ti d d i 1 n are borel measurable and satisfy 1 i v for all borel v d the multidimensional monge kantorovich problem investigated in this paper consists of finding s s1 sn 1 n minimizing over the set 1 n we study the case where the i s have finite second moments and vanish on d 1 rectifiable sets we prove existence and uniqueness of optimal maps s when we impose that s1 x x and give an explicit form of the maps si the result is obtained by variational methods and to the best of our knowledge is the first available in the literature in this generality as a consequence we obtain uniqueness and characterization of an optimal measure for the multidimensional kantorovich problem 1998 john wiley sons inc

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

on lipschitz optimization based on gray box piecewise linearization we address the problem of minimizing objectives from the class of piecewise differentiable functions whose nonsmoothness can be encapsulated in the absolute value function they possess local piecewise linear approximations with a discrepancy that can be bounded by a quadratic proximal term this overestimating local model is continuous but generally nonconvex it can be generated in its abs normal form by a minor extension of standard algorithmic differentiation tools here we demonstrate how the local model can be minimized by a bundle type method which benefits from the availability of additional gray box information via the abs normal form in the convex case our algorithm realizes the consistent steepest descent trajectory for which finite convergence was established earlier specifically covering counterexamples where steepest descent with exact line search famously fails the analysis of the abs normal representation and the design of the optimization algorithm are geared toward the general case whereas the convergence proof so far covers only the convex case

on a generalization of cyclic monotonicity and distances among random vectors abstract we generalize an investigation of distances among random vectors begun by olkin and pukelsheim and continued by olkin and rachev

a decomposition based pricing procedure for large scale linear programs an application to the linear multicommodity flow problem we propose and test a new pricing procedure for solving large scale structured linear programs the procedure interactively solves a relaxed subproblem to identify potential entering basic columns the subproblem is chosen to exploit special structure rendering it easy to solve the effect of the procedure is the reduction of the number of pivots needed to solve the problem our approach is motivated by the column generation approach of dantzig wolfe decomposition we test our procedure on two sets of multicommodity flow problems one group of test problems arises in routing telecommunications traffic and the second group is a set of logistics problem which have been widely used to test multicommodity flow algorithms

an interior point method for block angular optimization an interior point method for block angular optimization is developed and the convergence properties of the method are described a major motivation for such a method is that most of the computation is easily parallelized computational results are presented for a class of large scale linear programming models these models are multicommodity flow problems that arise from an air force military airlift command application and generate problems as large as 100 000 rows and 300 000 columns

a specialized network simplex algorithm for the constrained maximum flow problem abstract the constrained maximum flow problem is to send the maximum flow from a source to a sink in a directed capacitated network where each arc has a cost and the total cost of the flow cannot exceed a budget this problem is similar to some variants of classical problems such as the constrained shortest path problem constrained transportation problem or constrained assignment problem all of which have important applications in practice the constrained maximum flow problem itself has important applications such as in logistics telecommunications and computer networks in this research we present an efficient specialized network simplex algorithm that significantly outperforms the two widely used lp solvers cplex and lp solve we report cpu times of an average of 27 times faster than cplex with its dual simplex algorithm the closest competitor of our algorithm

an implementation of linear and nonlinear multicommodity network flows abstract this work presents a new code for solving the multicommodity network flow problem with a linear or nonlinear objective function considering additional linear side constraints that link arcs of the same or different commodities for the multicommodity network flow problem through primal partitioning the code implements a specialization of murtagh and saunders strategy of dividing the set of variables into basic nonbasic and superbasic several tests are reported using random problems obtained from different network generators and real problems arising from the fields of long and short term hydrothermal scheduling of electricity generation and traffic assignment with sizes of up to 150000 variables and 45 000 constraints the performance of the code developed is compared to that of alternative methodologies for solving the same problems a general purpose linear and nonlinear constrained optimization code a specialised linear multicommodity network flow code and a primal dual interior point code

the gas transmission problem solved by an extension of the simplex algorithm the problem of distributing gas through a network of pipelines is formulated as a cost minimization subject to nonlinear flow pressure relations material balances and pressure bounds the solution method is based on piecewise linear approximations of the nonlinear flow pressure relations the approximated problem is solved by an extension of the simplex method the solution method is tested on real world data and compared with alternative solution methods

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

free boundaries in optimal transport and monge ampere obstacle problems given compactly supported 0 f g 2 l 1 r n the problem of trans porting a fraction m min kfkl1 kgkl1 of the mass of f onto g as cheaply as possible is considered where cost per unit mass transported is given by a cost function c typically quadratic c x y x y 2 2 this question is shown to be equivalent to a double obstacle problem for the monge ampere equation for which sucient conditions are given to guarantee uniqueness of the solution such as f vanishing on sptg in the quadratic case the part of f to be transported increases monotonically with m and if sptf and sptg are separated by a hyperplane h then this part will separated from the balance of f by a semiconcave lipschitz graph over the hyperplane if f f and g g are bounded away from zero and infinity on separated strictly convex domains r n for the quadratic cost this graph is shown to be a c 1 loc hypersurface in whose normal coincides with the direction transported the optimal map between f and g is shown to be holder continuous up to this free bound ary and to those parts of the fixed boundary which map to locally convex parts of the path connected target region

geometric methods for spectral analysis this paper explores a geometric framework for modeling nonstationary but slowly varying time series based on the assumption that short windowed power spectra capture their spectral character and that energy transference in the frequency domain has a physical significance the framework relies on certain notions of transportation distance and their respective geodesics to model possible nonparametric changes in the power spectral density with respect to time we discuss the relevance of this framework to applications in spectral tracking spectral averaging and speech morphing

dual optimal inequalities for stabilized column generation column generation is one of the most successful approaches for solving large scale linear programming problems however degeneracy difficulties and long tail effects are known to occur as problems become larger in recent years several stabilization techniques of the dual variables have proven to be effective we study the use of two types of dual optimal inequalities to accelerate and stabilize the whole convergence process added to the dual formulation these constraints are satisfied by all or a subset of the dual optimal solutions therefore the optimal objective function value of the augmented dual problem is identical to the original one adding constraints to the dual problem leads to adding columns to the primal problem and feasibility of the solution may be lost we propose two methods for recovering primal feasibility and optimality depending on the type of inequalities that are used our computational experiments on the binary and the classical cutting stock problems and more specifically on the so called triplet instances show that the use of relevant dual information has a tremendous effect on the reduction of the number of column generation iterations

a simplex algorithm for piecewise linear programming 11 finiteness feasibility and degeneracy the simplex method for linear programming can be extended to permit the minimization of any convex separable piecewise linear objective subject to linear constraints part i of this paper has developed a general and direct simplex algorithm for piecewise linear programming under convenient assumptions that guarantee a finite number of basic solutions existence of basic feasible solutions and nondegeneracy of all such solutions part ii now shows how these assumptions can be weakened so that they pose no obstacle to effective use of the piecewise linear simplex algorithm the theory of piecewise linear programming is thereby extended and numerous features of linear programming are generalized or are seen in a new light an analysis of the algorithm s computational requirements and a survey of applications will be presented in part iii

a simplex algorithm for piecewise linear fractional programming problems generalizations of the well known simplex method for linear programming are available to solve the piecewise linear programming problem and the linear fractional programming problem in this paper we consider a further generalization of the simplex method to solve piecewise linear fractional programming problems unifying the simplex method for linear programs piecewise linear programs and the linear fractional programs computational results are presented to obtain further insights into the behavior of the algorithm on random test problems

recursive operational method to solve fractional linear systems abstract a method for solving fractional linear systems is presented the method is based on an algebraic version of the usual generalized convolution it does not need any transform and only requires that the fractional derivative is defined on the real line the algorithm proceeds in a recursive way in the sense that it allows to add or remove pseudo pole zero contributions without having to remake all the computations some examples illustrate the behavior of the algorithm

a class of multi marginal c cyclically monotone sets with explicit c splitting potentials abstract multi marginal optimal transport plans are concentrated on c splitting sets it is known that similar to the two marginal case c splitting sets are c cyclically monotone within a suitable framework the converse implication was very recently established by griessler however for an arbitrary cost c given a multi marginal c cyclically monotone set the question whether there exists an analogous explicit construction to the one from the two marginal case of c splitting potentials is still open when the margins are one dimensional and the cost belongs to a certain class carlier proved that the two marginal projections of a c splitting set are monotone for arbitrary products of sets equipped with cost functions which are sums of two marginal costs we show that the two marginal monotonicity condition is a sufficient condition which does give rise to an explicit construction of c splitting potentials our condition is in principle easier to verify than the one of multi marginal c cyclic monotonicity various examples illustrate our results we show that in general our condition is sufficient however it is not necessary on the other hand we conclude that when the margins are one dimensional equipped with classical cost functions our condition is a characterization of c splitting sets and extends classical convex analysis

a fixed point approach to barycenters in wasserstein space abstract let p 2 a c be the set of borel probabilities on r d with finite second moment and absolutely continuous with respect to lebesgue measure we consider the problem of finding the barycenter or frechet mean of a finite set of probabilities 1 k p 2 a c with respect to the l 2 wasserstein metric for this task we introduce an operator on p 2 a c related to the optimal transport maps pushing forward any p 2 a c to 1 k under very general conditions we prove that the barycenter must be a fixed point for this operator and introduce an iterative procedure which consistently approximates the barycenter the procedure allows effective computation of barycenters in any location scatter family including the gaussian case in such cases the barycenter must belong to the family thus it is characterized by its mean and covariance matrix while its mean is just the weighted mean of the means of the probabilities the covariance matrix is characterized in terms of their covariance matrices 1 k through a nonlinear matrix equation the performance of the iterative procedure in this case is illustrated through numerical simulations which show fast convergence towards the barycenter

optimal expansion of capacitated transshipment networks abstract in this paper we address the problem of allocating a given budget to increase the capacities of arcs in a transshipment network to minimize the cost of flow in the network the capacity expansion costs of arcs are assumed to be piecewise linear convex functions we use properties of the optimum solution to convert this problem into a parametric network flow problem the concept of optimum basis structure is used which allows us to consider piecewise linear convex functions without introducing additional arcs the resulting algorithm yields an optimum solution of the capacity expansion problem for all budget levels less than or equal to the given budget for integer data the algorithm performs almost all computations in integers detailed computational results are also presented

wasserstein barycenters over riemannian manifolds we study barycenters in the space of probability measures on a riemannian manifold equipped with the wasserstein metric under reasonable assumptions we establish absolute continuity of the barycenter of general measures p p m p p m on wasserstein space extending on one hand results in the euclidean case for barycenters between finitely many measures of agueh and carlier 1 to the riemannian setting and on the other hand results in the riemannian case of cordero erausquin mccann schmuckenschlager 12 for barycenters between two measures to the multi marginal setting our work also extends these results to the case where is not finitely supported as applications we prove versions of jensen s inequality on wasserstein space and a generalized brunn minkowski inequality for a random measurable set on a riemannian manifold

a simplex algorithm for piecewise linear programming iii computational analysis and application the first two parts of this paper have developed a simplex algorithm for minimizing convex separable piecewise linear functions subject to linear constraints this concluding part argues that a direct piecewiselinear simplex implementation has inherent advantages over an indirect approach that relies on transformation to a linear program the advantages are shown to be implicit in relationships between the linear and piecewise linear algorithms and to be independent of many details of implementation two sets of computational results serve to illustarate these arguments the piecewise linear simplex algorithm is observed to run 2 6 times faster than a comparable linear algorithm not including any additional expense that might be incurred in setting up the equivalent linear program further support for the practical value of a good piecewise linear programming algorithm is provided by a survey of many varied applications

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

degeneracy in the presence of roundoff errors abstract a technique is described for resolving degeneracy in the simplex method for linear programming it is shown that this technique enables a guarantee of termination to be given not only for exact arithmetic but also for inexact arithmetic when roundoff errors are present the method is recursive and each level of recursion is obtained by localizing and dualizing the problem at a lower level the existence of a cost function at any level is exploited to provide the guarantee a number of interesting theoretical properties of the method are given data structures are described which enable the method to be implemented with very little overhead beyond what is normally required for the simplex method there is no difficulty in using any of the currently popular techniques for representing and updating matrix factors the technique is extended to handle degeneracy in l1 linear programming problems including linear l1 approximation and it is shown that all the above properties can be preserved some comparative numerical experiments are described

algorithms for network piecewise linear programs a comparative study abstract piecewise linear programming plp is an important area of mathematical programming and concerns the minimisation of a convex separable piecewise linear objective function subject to linear constraints in this paper a subarea of plp called network piecewise linear programming nplp is explored the paper presents four specialised algorithms for nplp strongly feasible primal simplex dual method out of kilter and strongly polynomial cost scaling and their relative efficiency is studied a statistically designed experiment is used to perform a computational comparison of the algorithms the response variable observed in the experiment is the cpu time to solve randomly generated network piecewise linear problems classified according to problem class transportation transshipment and circulation problem size extent of capacitation and number of breakpoints per arc results and conclusions on performance of the algorithms are reported

a simple approach to the particular solution of constant coefficient ordinary differential equations an eigenfunction approach to compute the particular solution of constant coefficient ordinary differential equations is presented it is shown that the exponentials are the eigenfunctions of such equations solutions corresponding to products of powers and exponentials are obtained the singular case is studied and a fast algorithm for its implementation is presented

piecewise linear multicriteria programs the continuous case and its discontinuous generalization in this paper we study piecewise linear multicriteria programs that is multicriteria programs with either a continuous or discontinuous piecewise linear objective function and a polyhedron set constraint we obtain an algebraic representation of a semi closed polyhedron and apply it to show that the image of a semi closed polyhedron under a continuous linear function is always one semi closed polyhedron we establish that the weak pareto solution point set of a piecewise linear multicriteria program is the union of finitely many semi closed polyhedra we propose an algorithm for finding the pareto point set of a continuous piecewise linear bi criteria program and generalize it to the discontinuous case we apply our algorithm to solve the discontinuous bi criteria portfolio selection problem with an l risk measure and transaction costs and show that this algorithm can be improved by using an ideal point strategy

amplitude and phase variation of point processes we develop a canonical framework for the study of the problem of registration of multiple point processes subjected to warping known as the problem of separation of amplitude and phase variation the amplitude variation of a real random function y x x is an element of 0 1 corresponds to its random oscillations in the y axis typically encapsulated by its co variation around a mean level in contrast its phase variation refers to fluctuations in the x axis often caused by random time changes we formalise similar notions for a point process and nonparametrically separate them based on realisations of i i d copies pi i of the phase varying point process a key element in our approach is to demonstrate that when the classical phase variation assumptions of functional data analysis fda are applied to the point process case they become equivalent to conditions interpretable through the prism of the theory of optimal transportation of measure we demonstrate that these induce a natural wasserstein geometry tailored to the warping problem including a formal notion of bias expressing over registration within this framework we construct nonparametric estimators that tend to avoid over registration in finite samples we show that they consistently estimate the warp maps consistently estimate the structural mean and consistently register the warped point processes even in a sparse sampling regime we also establish convergence rates and derivev root n consistency and a central limit theorem in the cox process case under dense sampling showing rate optimality of our structural mean estimator in that case

classroom note on particular solutions of linear difference equations with constant coefficients this note contains a recursive alternative to the method of undetermined coefficients for a particular solution of a linear difference equation with constant coefficients versatility of the same has been illustrated by solving some equations

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

multi marginal optimal transport on riemannian manifolds we study a multi marginal optimal transportation problem on a riemannian manifold with cost function given by the average distance squared from multiple points to their barycenter under a standard regularity condition on the first marginal we prove that the optimal measure is unique and concentrated on the graph of a function over the first variable thus inducing a monge solution this result generalizes mccann s polar factorization theorem on manifolds from two to several marginals in the same sense that a well known result of gangbo and swiech generalizes brenier s polar factorization theorem on bbb r n

smoothing of transport plans with fixed marginals and rigorous semiclassical limit of the hohenberg kohn functional we prove rigorously that the exact n electron hohenberg kohn density functional converges in the strongly interacting limit to the strictly correlated electrons sce functional and that the absolute value squared of the associated constrained search wavefunction tends weakly in the sense of probability measures to a minimizer of the multi marginal optimal transport problem with coulomb cost associated to the sce functional this extends our previous work for n 2 cotar etal in commun pure appl math 66 548 599 2013 the correct limit problem has been derived in the physics literature by seidl phys rev a 60 4387 4395 1999 and seidl gorigiorgi and savin phys rev a 75 042511 1 12 2007 in these papers the lack of a rigorous proofwas pointed out we also give amathematical counterexample to this type of result by replacing the constraint of given one body density an infinite dimensional quadratic expression in the wavefunction by an infinite dimensional quadratic expression in the wavefunction and its gradient connections with the lawrentiev phenomenon in the calculus of variations are indicated

the distance between two random vectors with given dispersion matrices for two p dimensional random vectors x and y with dispersion matrices 11 and 22 respectively we determine that covariance matrix 0 of x and y that minimizes the l2 distance between x and y there is a dual to this problem that is of interest in another context

optimal transportation with infinitely many marginals abstract we formulate and study an optimal transportation problem with infinitely many marginals this is a natural extension of the multi marginal problem studied by gangbo and wi ch 1998 15 we prove results on the existence uniqueness and characterization of the optimizer which are natural extensions of the results in gangbo and wi ch 1998 15 the proof relies on a relationship between this problem and the problem of finding barycenters in the wasserstein space a connection first observed for finitely many marginals by agueh and carlier 2011 1

explanation of variability and removal of confounding factors from data through optimal transport a methodology based on the theory of optimal transport is developed to attribute variability in data sets to known and unknown factors and to remove such attributable components of the variability from the data denoting by x the quantities of interest and by z the explanatory factors the procedure transforms x into filtered variables y through a z dependent map so that the conditional probability distributions x z are pushed forward into a target distribution y independent of z among all maps and target distributions that achieve this goal the procedure selects the one that minimally distorts the original data the barycenter of the x z connections are found to unsupervised learning and to fundamental problems in statistics such as conditional density estimation and sampling particularly simple instances of the methodology are shown to be equivalent to k means and principal component analysis an application is shown to a time series of ground temperature hourly data across the united states 2017 wiley periodicals inc

on the particular solution of constant coefficient fractional differential equations the paper presents an approach that comes directly from the formulation for integer order differential equations it generalises previous results it presents a matricial algorithm for dealing with the singular case the eigenfunction approach to compute the particular solution of constant coefficient ordinary differential equations is extended to the fractional case it is shown that the exponentials are also the eigenfunctions of such equations solutions corresponding to products of powers and exponentials are presented the singular case is studied and a matricial algorithm for its implementation is presented

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

on solving the sum of ratios problem this paper addresses the development of efficient global search methods for fractional programming problems such problems are in general nonconvex with numerous local extremums and belong to a class of global optimization problems first we reduce a rather general fractional programming problem with d c functions to solving an equation with a vector parameter that satisfies some nonnegativity assumption this theorem allows the justified use of the generalized dinkelbachs approach for solving fractional programming problems with a d c goal function based on solving of some d c minimization problem we developed a global search algorithm for fractional programming problems which was tested on a set of low dimensional test problems taken from the literature as well as on randomly generated problems with up to 200 variables or 200 terms in the sum

optimal objective function approximation for separable convex quadratic programming we present an optimal piecewise linear approximation method for the objective function of separable convex quadratic programs the method provides guidelines on how many grid points to use and how to position them for a piecewise linear approximation if the error induced by the approximation is to be bounded a priori

large scale portfolio optimization this paper describes a practical algorithm for large scale mean variance portfolio optimization the emphasis is on developing an efficient computational approach applicable to the broad range of portfolio models employed by the investment community what distinguishes these from the usual quadratic program is i the form of the covariance matrix arising from the use of factor and scenario models of return and ii the inclusion of transactions limits and costs a third aspect is the question of whether the problem should be solved parametrically in the risk reward trade off parameter or separately for several discrete values of we show how the parametric algorithm can be made extremely efficient by sparsifying the covariance matrix with the introduction of a few additional variables and constraints and by treating the transaction cost schedule as an essentially nonlinear nondifferentiable function then we show how these two seemingly unrelated approaches can be combined to yield good approximate solutions when minimum trading size restrictions buy or sell at least a certain amount or not at all are added in combination these approaches make possible the parametric solution of problems on a scale not heretofore possible on computers where cpu time and storage are the constraining factors

polar factorization and monotone rearrangement of vector valued functions given a probability space x p and a bounded domain r in rd equipped with the lebesgue measure 1 i normalized so that 10 i i it is shown under additional technical assumptions on x and q that for every vector valued function u e lp x p rd there is a unique polar factorization u v s where is a convex function defined on r and s is a measure preserving mapping from x p into q i i provided that u is nondegenerate in the sense that p u e 0 for each lebesgue negligible subset e of rd through this result the concepts of polar factorization of real matrices helmholtz decomposition of vector fields and nondecreasing rearrangements of real valued functions are unified the monge amgre equation is involved in the polar factorization and the proof relies on the study of an appropriate monge kantorovich problem

duality theorems for marginal problems given topological spaces x1 xn with product space x probability measures i on xi together with a real function h on x define a marginal problem as well as a dual problem using an extended version of choquet s theorem on capacities an analogue of the classical duality theorem of linear programming is established imposing only weak conditions on the topology of the spaces xi and the measurability resp boundedness of the function h applications concern among others measures with given support stochastic order and general marginal problems

discontinuous piecewise linear optimization a theoretical framework and a practical algorithm are presented to solve discontinuous piecewise linear optimization problems dealing with functions for which theridges are known a penalty approach allows one to consider such problems subject to a wide range of constraints involving piecewise linear functions although the theory is expounded in detail in the special case of discontinuous piecewiselinear functions it is straightforwardly extendable using standard nonlinear programming techniques tononlinear discontinuous piecewise differentiable functions the descent algorithm which is elaborated uses active set and projected gradient approaches it is a generalization of the ideas used by conn to deal with nonsmoothness in thel1 exact penalty function and it is based on the notion ofdecomposition of a function into a smooth and a nonsmooth part the constrained case is reduced to the unconstrained minimization of a piecewise linear l1 exact penalty function we also discuss how the algorithm is modified when it encounters degenerate points preliminary numerical results are presented the algorithm is applied to discontinuous optimization problems from models in industrial engineering 1998 the mathematical programming society inc published by elsevier science b v

the geometry of optimal transportation introduction 113 1 summary of main results 120 2 background on optimal measures 126 part i strictly convex costs 133 3 existence and uniqueness of optimal maps 133 4 characterization of the optimal map 137 part ii costs which are strictly concave as a function of d i s t a n c e 141 5 the role of optimal maps 141 6 uniqueness of optimal solutions 144 part iii appendices 148 a legendre transforms and conjugate costs 148 b examples of c concave potentials 152 c regularity of c concave potentials 154 references 159