on the computational complexity of piecewise linear homotopy algorithms we show that piecewise linear homotopy algorithms may take a number of steps that grows exponentially with the dimension when solving a system of linear equations whose solution lies close to the starting point our examples are based on an example of murty exhibiting exponential growth for lemke s algorithm for the linear complementarity problem

approximate labelling for simplicial algorithms and two classes of special subsets of the sphere call a subset of rn complete if the origin is in its convex hull we are concerned with the construction of large finite sets of points on the unit sphere whose complete subsets are in some sense easy to describe our motivation comes from simplicial algorithms for approximating zeroes of functions we construct two classes of such subsets of the sphere both of which generalize a subset that corresponds to usual integer labelling

solving discrete zero point problems in this paper we present two theorems on the existence of a discrete zero point of a function from the n dimensional integer lattice n to the n dimensional euclidean space n the theorems differ in their boundary conditions for both theorems we give a proof using a combinatorial lemma and present a constructive proof based on a simplicial algorithm that finds a discrete zero point within a finite number of steps

an efficient simplicial algorithm for computing a zero of a convex union of smooth functions we present an efficient simplicial algorithm for computing a zero of a point to set mapping that is formed by piecing together smooth functions such mappings arise in nonlinear programming and economic equilibrium problems our algorithm under suitable regularity conditions on the problem generates a sequence converging at least q superlinearly to a zero of the mapping asymptotically it operates in a space of reduced dimension analogous to an active set strategy in the optimization setting but it switches active sets automatically results of computational experiments are given

a simplicial approach to the determination of an integer point of a simplex we propose an algorithm to find an integer point of a simplex the algorithm is based on an integer labeling rule and a triangulation of the space starting at an integer point the algorithm leaves it along one of n rays and generates a sequence of adjacent simplices of varying dimension within a finite number of iterations the algorithm either yields an integer point of the simplex or proves that no such point exists

solving discrete systems of nonlinear equations abstract we study the existence problem of a zero point of a function defined on a finite set of elements of the integer lattice z n of the n dimensional euclidean space r n it is assumed that the set is integrally convex which implies that the convex hull of the set can be subdivided in simplices such that every vertex is an element of z n and each simplex of the triangulation lies in an n dimensional cube of size one with respect to this triangulation we assume that the function satisfies some property that replaces continuity under this property and some boundary condition the function has a zero point to prove this we use a simplicial algorithm that terminates with a zero point within a finite number of iterations the standard technique of applying a fixed point theorem to a piecewise linear approximation cannot be applied because the continuity property is too weak to assure that a zero point of the piecewise linear approximation induces a zero point of the function itself we apply the main existence result to prove the existence of a pure cournot nash equilibrium in a cournot oligopoly model we further obtain a discrete analogue of the well known borsuk ulam theorem and a theorem for the existence of a solution for the discrete nonlinear complementarity problem

robust inventory management an optimal control approach we formulate and solve static and dynamic models of inventory management that lie at the intersection of robust optimization and optimal control theory our objective is to minimize cumulative ordering holding and shortage costs over a horizon 0 t where the variable is a nonnegative ordering rate function q t 2 0 t the demand rate function d t is unknown and is only assumed to belong to an uncertainty set d t l2 0 t a 1 t 0td t dt b a d t b t 0 t this set is motivated by the strong law of large numbers for stochastic processes limt 1 t 0td t dt where is the mean drift we analyze a static model where the ordering rate function must be fully specified at time zero and three dynamic variants where re optimizations are allowed during the planning horizon 0 t at prespecified review epochs in the dynamic models at review epoch 0 t the past demand on 0 is observable in the first dynamic model we ignore this information and define a variant of that i

variable dimension algorithms basic theory interpretations and extensions of some existing methods in this paper we establish a basic theory for variable dimension algorithms which were originally developed for computing fixed points by van der laan and talman we introduce a new concept primal dual pair of subdivided manifolds and by utilizing it we propose a basic model which will serve as a foundation for constructing a wide class of variable dimension algorithms our basic model furnishes interpretations to several existing methods lemke s algorithm for the linear complementarity problem its extension to the nonlinear complementarity problem a variable dimension algorithm on conical subdivisions and merrill s algorithm we shall present a method for solving systems of equations as an application of the second method and an efficient implementation of the fourth method to which our interpretation leads us a method for constructing triangulations with an arbitrary refinement factor of mesh size is also proposed

a variable dimension fixed point algorithm and the orientation of simplices a variable dimension algorithm with integer labelling is proposed for solving systems ofn equations inn variables the algorithm is an integer labelling version of the 2 ray algorithm proposed by the author the orientation of lower dimensional simplices is studied and is shown to be preserved along a sequence of adjacent simplices

asymptotically optimal inventory control for assemble to order systems with identical lead times optimizing multiproduct assemble to order ato inventory systems is a long standing difficult problem we consider ato systems with identical component lead times and a general bill of materials we use a related two stage stochastic program sp to set a lower bound on the average inventory cost and develop inventory control policies for the dynamic ato system using this sp we apply the first stage sp optimal solution to specify a base stock replenishment policy and the second stage sp recourse linear program to make allocation decisions we prove that our policies are asymptotically optimal on the diffusion scale so the percentage gap between the average cost from its lower bound diminishes to zero as the lead time grows subject classifications assemble to order inventory management stochastic linear program stochastic control asymptotic optimality diffusion scale area of review stochastic models history received may 2013 revisions received july 2014 january 2015 accepted february 2015

on the parabolic kernel of the schrodinger operator etude des equations paraboliques du type q x t t u x t 0 sur une variete riemannienne generale introduction estimations de gradients inegalites de harnack majorations et minorations des solutions fondamentales equation de la chaleur et noyau de green operateur de schrodinger

computing an integer point of a simplex with an arbitrary starting homotopy like simplicial algorithm abstract an arbitrary starting homotopy like simplicial algorithm is developed for computing an integer point of an n dimensional simplex the algorithm is derived from the use of an integer labeling rule and a triangulation of r n 0 1 and consists of two interchanging phases one phase of the algorithm constitutes a homotopy simplicial algorithm which generates n 1 dimensional simplices in r n 0 1 and the other phase of the algorithm constitutes a pivoting procedure which generates n dimensional simplices in either r n 0 or r n 1 the algorithm varies from one phase to the other when the matrix defining the simplex is in the so called canonical form starting at an arbitrary integer point in r n 0 the algorithm within a finite number of iterations either yields an integer point of the simplex or proves that no such point exists

near optimal dynamic lead time quotation and scheduling under convex concave customer delay costs we consider a make to order system where customers are dynamically quoted lead times and prices customers are homogenous but have general nonlinear disutility for delay because the firm is a monopolist the pricing problem is trivial and the dynamic problem reduces to one of lead time quotation and order sequencing we also consider the static problem of up front capacity installation we use a large capacity asymptotic regime to make the problem tractable we provide recommended policies for convex concave and convex concave lead time cost functions and prove that these policies are asymptotically optimal the policies are both highly intuitive and readily implementable moreover they provide delay guarantees for all served customers they are tested numerically we find that significant benefits can accrue by using the prescribed dynamic policies instead of first come first served type policies

a new simplicial variable dimension algorithm to find equilibria on the product space of unit simplices a new simplicial variable dimension restart algorithm is introduced to solve the nonlinear complementarity problem on the product spaces of unit simplices the triangulation which underlies the algorithm differs from the triangulations ofs used thus far moreover the number of rays along which the algorithm can leave the arbitrarily chosen starting point is much larger more precisely there is a ray leading from the starting point to each vertex ofs in cases is the product ofn one dimensional unit simplices the alogrithm is similar to the octahedral algorithm onrn having 2n rays also the accuracy of an approximate solution in the terminal simplex of the algorithm is in general better than for the other algorithms ons computational results will show that the number of iterations for the new algorithm is much less the examples concern the computation of equilibria in noncooperative games exchange economies and trade models

a vector labeling method for solving discrete zero point and complementarity problems in this paper we establish the existence of a discrete zero point of a function from the n dimensional integer lattice mathbb z n to the n dimensional euclidean space mathbb r n under very general conditions with respect to the behavior of the function the proof is constructive and uses a combinatorial argument based on a simplicial algorithm with vector labeling and lexicographic linear programming pivot steps the algorithm provides an efficient method to find an exact solution we also discuss how to adapt the algorithm for two related problems namely to find a discrete zero point of a function under a general antipodal condition and to find a solution to a discrete nonlinear complementarity problem in both cases the modified algorithm provides a constructive existence proof too we further show that the algorithm for the discrete nonlinear complementarity problem generalizes the well known lemke s method to nonlinear environments an economic application is also presented

a discrete multivariate mean value theorem with applications we establish a discrete multivariate mean value theorem for the class of positive maximum component sign preserving functions a constructive and combinatorial proof is given based upon a simplicial algorithm and vector labeling moreover we apply this theorem to a discrete nonlinear complementarity problem and an economic equilibrium problem with indivisibilities and show the existence of solutions in both problems under certain mild conditions

on the computation of fixed points in the product space of unit simplices and an application to noncooperative n person games in this paper an algorithm based on the principle of simplicial approximation is introduced to compute fixed points of upper semicontinuous point to set mappings from the product space s of unit simplices into itself the algorithm is a modification of an algorithm introduced in an earlier paper the main feature is that it starts with an arbitrary chosen point in s and that the triangulation of s depends on the starting point moreover the algorithm can terminate with a non full dimensional subsimplex yielding a good approximation an application is given for non cooperative n person games where s is the strategy space some computational experiences are given

a unified approach to the implementation of several restart fixed point algorithms and a new variable dimension algorithm we present a unified description of a class of restart fixed point algorithms including merrill s method and several variable dimension algorithms for their implementation on computers based on this description we show that some pivot saving techniques originally developed for the homotopy methods can be applied to the class we also propose a new variable dimension algorithm having 3 n 1 rays along which we can move toward a solution some numerical comparisons of the simplicial restart algorithms merrill s method the 2n method the octahedral method and the new one support that the latter two methods are more efficient than the others

a continuous deformation algorithm on the product space of unit simplices a continuous deformation algorithm is introduced on s 1 where s denotes the product space of unit simplices with arbitrary grid refinement between two subsequent levels the set s 1 is triangulated in such a way that for each m m 1 2 s m is triangulated by the so called v triangulation the algorithm starts by applying a variable dimension algorithm on s until an approximating simplex has been found on level 1 then the algorithm follows a path of approximating simplices in s 1 starting on level 1 until a certain level or a certain accuracy of a solution of the underlying problem has been reached if the algorithm returns to level 1 then we again apply the variable dimension algorithm until a new approximating simplex is found on level 1 etc we allow solutions to lie on the boundary of s 1 in which case the algorithm in general will follow a path on the boundary of s 1

simplicial variable dimension algorithms for solving the nonlinear complementarity problem on a product of unit simplices using a general labelling this paper deals with the nonlinear complementarity problem on the product space of unit simplices s a simplicial variable dimension algorithm developed by van der laan and talman for proper labellings of s is extended to the case of general labellings general labellings allow a more natural description of the complementarity problem on the boundary of s a distinctive feature of the new algorithm is that lower dimensional simplicial movement can occur both on the boundary and in the interior of s in contrast the van der laan and talman algorithm for proper labellings of s allows lower dimensional simplicial movement only in the interior of s computational experiments confirm the usefulness of general labellings for solving nonlinear complementarity problems

a universal bound on the gradient of logarithm of the heat kernel for manifolds with bounded ricci curvature abstract we derive a gradient estimate for the logarithm of the heat kernel on a riemannian manifold with ricci curvature bounded from below the bound is universal in the sense that it depends only on the lower bound of ricci curvature dimension and diameter of the manifold imposing a more restrictive non collapsing condition allows one to sharpen this estimate for the values of time parameter close to zero

efficiency and implementation of simplicial zero point algorithms in this paper we compare the efficiency of several simplicial variable dimension algorithms to do so we first treat the issues of degeneracy and accelerating we present a device for solving degeneracy furthermore we compare several accelerating techniques the technique of iterated quasi newton steps after each major cycle of the simplicial algorithm is implemented in a computer code which is used to compare the efficiency of the n 1 ray 2n ray 2n ray and 3n 1 ray algorithms except for the n 1 ray algorithm the number of function evaluations does not differ very much between the various algorithms it appeared however that the 2n algorithm needs considerably less computation time

optimal control of a high volume assemble to order system we consider an assemble to order system with a high volume of prospective customers arriving per unit time our objective is to maximize expected infinite horizon discounted profit by choosing product prices component production capacities and a dynamic policy for sequencing customer orders for assembly we prove that a myopic discrete review sequencing policy which allocates scarce components among orders for different products to minimize instantaneous physical and financial holding costs is asymptotically optimal furthermore we prove that optimal prices and production capacity nearly balance the supply and demand for components i e it is economically optimal to operate the system in heavy traffic so system performance is characterized by a diffusion approximation the diffusion approximation exhibits state space collapse its dimension equals the number of components rather than the number of components plus the number of products these results complement the existing assemble to order literature which focuses on managing component inventory and assumes fifo sequencing of orders for assembly

simplicial approximation of solutions to the nonlinear complementarity problem with lower and upper bounds ideas of a simplicial variable dimension restart algorithm to approximate zero points onrn developed by the authors and of a linear complementarity problem pivoting algorithm are combined to an algorithm for solving the nonlinear complementarity problem with lower and upper bounds the algorithm can be considered as a modification of the2n ray zero point finding algorithm onrn it appears that for the new algorithm the number of linear programming pivot steps is typically less than for the2n ray algorithm applied to an equivalent zero point problem this is caused by the fact that the algorithm utilizes the complementarity conditions on the variables

on the existence and approximation of zeroes in this paper we consider the problem of finding zeroes of a continuous functionf from a convex compact subsetu of n to n in the first part of the paper it is proved thatf has a computable zero iff cn n satisfies the nonparallel condition for any two antipodal points on bdcn i e if for anyx bdcn f x f x 0 holds therefore we describe a simplicial algorithm to approximate such a zero it is shown that generally the degree of the approximate zero depends on the number of reflection steps made by the algorithm i e the number of times the algorithm switches from a face on bdcn to the face therefore the index of a terminal simplex is defined which equals the local brouwer degree of the function if is full dimensional in the second part of the paper the algorithm is used to generate possibly several approximate zeroes off two sucessive solutions may have both the same or opposite degrees again depending on the number of reflection steps by extendingf u n to a function g from a cube containingu to n the procedure can be applied to any continuous functionf without having any information about the global and local brouwer degrees a priori

hoffman s error bounds and uniform lipschitz continuity of bestlp approximations in a central paper on smoothness of best approximation in 1968 r holmes and b kripke proved among others that on rn endowed with thelp norm 1 p the metric projection onto a given linear subspace is lipschitz continuous where the lipschitz constant depended on the parameterp usinghoffman s error boundsas a principal tool we prove uniform lipschitz continuity of bestlp approximations as a consequence we reprove and prove respectively lipschitz continuity of thestrict best approximation sba p and of thenatural best approximation nba p 1

a new variable dimension algorithm for the fixed point problem we propose a new variable dimension algorithm for solving a system ofn equations inn variables the algorithm solves a sequence of subproblems consisting of the first several equations the dimension of subproblems does not however increase monotonically a convergence condition is also presented

advertising models stochastic games and myopic strategies this paper analyzes dynamic advertising models we present conditions that imply that the payoffs under myopic strategies are linear in the state variable which represents the number of customers for each firm these conditions which determine a subclass of stochastic games are satisfied by several dynamic advertising models we give a criterion to check whether this model contains an equilibrium point within the class of myopic strategies that is also an equilibrium point with regard to all strategies

a lexicographic algebraic theorem and its applications abstract in this paper we prove the following result given any full dimensional simple polytope p x r n a i t x b i i 1 m without redundant constraints and any vector c r n there exists a unique vertex x of p such that the matrix 1 1 1 1 c a i 1 a i 2 a i n 1 exists and is lexicographic positive where x is the solution of the n linear equations a i t t x b i t t 1 n this theorem generalizes a result on the n dimensional cube and can be used to resolve the degeneracy problem for simplicial fixed point algorithms we also discuss several applications for this result

analysis of deterministic control and its improvements for an inventory problem with multiproduct batch differentiation we consider a multiperiod planning problem faced by a biopharmaceutical firm that must coordinate the production and allocation of batches of intermediate products to end products for multiple markets this is a challenging problem to solve optimally so we derive a theoretical bound on the performance of a deterministic control dc in which all random variables are replaced by their expected values and the corresponding deterministic optimization problem is solved this is a variant of an approach that is widely used in practice we show that while dc can perform poorly in some instances simple reoptimizations of dc i e certainty equivalent control cec improve the performance of dc that said the benefit of reoptimization decreases as the magnitude of demand variation increases to address this and also to provide guidance for heuristic design we derive performance bounds for two additional heuristic controls 1 open loop feedback control olfc which reoptimizes a sequence of stochastic pro

a better triangulation for wright s 2n ray algorithm abstract we give a triangulation for wright s 2n ray algorithm to compute solutions of nonlinear equations according to measures of efficiency of triangulations it is better than any other available triangulation for the 2n ray algorithm

a class of simplicial restart fixed point algorithms without an extra dimension in an earlier paper we introduced an algorithm for approximating a fixed point of a mapping on the product space of unit simplices ideas of that paper are used to construct a class of triangulations ofrn more precisely for somek 1 k n and positive integersm1 mk with sumn a triangulation ofrn is obtained by triangulating the cells which are formed by taking the product of given triangulations ofrmj j 1 k the triangulation of each cell will be defined in relation to an arbitrarily chosen pointv inrn being the starting point of the algorithm fork n we obtain thek triangulation originally due to todd each element of the class can be used to find a simplex which approximates a fixed point of a mapping onrn by generating a unique path of adjacent simplices of variable dimension starting with the pointv

perelman s entropy formula for the witten laplacian on riemannian manifolds via bakry emery ricci curvature in this paper we study perelman s mathcal w entropy formula for the heat equation associated with the witten laplacian on complete riemannian manifolds via the bakry emery ricci curvature under the assumption that the m dimensional bakry emery ricci curvature is bounded from below we prove an analogue of perelman s and ni s entropy formula for the mathcal w entropy of the heat kernel of the witten laplacian on complete riemannian manifolds with some natural geometric conditions in particular we prove a monotonicity theorem and a rigidity theorem for the mathcal w entropy on complete riemannian manifolds with non negative m dimensional bakry emery ricci curvature moreover we give a probabilistic interpretation of the mathcal w entropy for the heat equation of the witten laplacian on complete riemannian manifolds and for the ricci flow on compact riemannian manifolds