fast sweeping algorithms for a class of hamilton jacobi equations we derive a godunov type numerical flux for the class of strictly convex homogeneous hamiltonians that includes h p q sqrt ap 2 bq 2 2cpq c 2 ab we combine our godunov numerical fluxes with simple gauss seidel type iterations for solving the corresponding hamilton jacobi hj equations the resulting algorithm is fast since it does not require a sorting strategy as found e g in the fast marching method in addition it providesa way to compute solutions to a class of hj equations for which the conventional fast marching method is not applicable our experiments indicate convergence after a few iterations even in rather difficult cases

spectra of digraphs in this primarily expository paper we survey classical and some more recent results on the spectra of digraphs equivalently the spectra of 0 1 matrices with emphasis on the spectral radius

adjustable robust solutions of uncertain linear programs we consider linear programs with uncertain parameters lying in some prescribed uncertainty set where part of the variables must be determined before the realization of the uncertain parameters non adjustable variables while the other part are variables that can be chosen after the realization adjustable variables we extend the robust optimization methodology 1 3 6 9 13 14 to this situation by introducing the adjustable robust counterpart arc associated with an lp of the above structure often the arc is significantly less conservative than the usual robust counterpart rc however in most cases the arc is computationally intractable np hard this difficulty is addressed by restricting the adjustable variables to be affine functions of the uncertain data the ensuing affinely adjustable robust counterpart aarc problem is then shown to be in certain important cases equivalent to a tractable optimization problem typically an lp or a semidefinite problem and in other cases having a tight approximation which is tractable the aarc approach is illustrated by applying it to a multi stage inventory management problem

optimality of the triangular lattice for a particle system with wasserstein interaction we prove strong crystallization results in two dimensions for an energy that arises in the theory of block copolymers the energy is defined on sets of points and their weights or equivalently on the set of atomic measures it consists of two terms the first term is the sum of the square root of the weights and the second is the quadratic optimal transport cost between the atomic measure and the lebesgue measure

second order cone relaxations for binary quadratic polynomial programs several types of relaxations for binary quadratic polynomial programs can be obtained using linear second order cone or semidefinite techniques in this paper we propose a general framework to construct conic relaxations for binary quadratic polynomial programs based on polynomial programming using our framework we re derive previous relaxation schemes and provide new ones in particular we present three relaxations for binary quadratic polynomial programs the first two relaxations based on second order cone and semidefinite programming represent a significant improvement over previous practical relaxations for several classes of nonconvex binary quadratic polynomial problems from a practical point of view due to the computational cost semidefinite based relaxations for binary quadratic polynomial problems can be used only to solve small to midsize instances to improve the computational efficiency for solving such problems we propose a third relaxation based purely on second order cone programming computational tests on different classes of nonconvex binary quadratic polynomial problems including quadratic knapsack problems show that the second order cone based relaxation outperforms the semidefinite based relaxations that are proposed in the literature in terms of computational efficiency and it is comparable in terms of bounds

bounds for levinger s function of nonnegative almost skew symmetric matrices the analysis of the perron eigenspace of a nonnegative matrix a whose symmetric part has rank one is continued improved bounds for the perron root of levinger s transformation 1 a a t 0 1 and its derivative are obtained the relative geometry of the corresponding left and right perron vectors is examined the results are applied to tournament matrices to obtain a comparison result for their spectral radii

distributionally robust optimization under moment uncertainty with application to data driven problems stochastic programming can effectively describe many decision making problems in uncertain environments unfortunately such programs are often computationally demanding to solve in addition their solution can be misleading when there is ambiguity in the choice of a distribution for the random parameters in this paper we propose a model that describes uncertainty in both the distribution form discrete gaussian exponential etc and moments mean and covariance matrix we demonstrate that for a wide range of cost functions the associated distributionally robust or min max stochastic program can be solved efficiently furthermore by deriving a new confidence region for the mean and the covariance matrix of a random vector we provide probabilistic arguments for using our model in problems that rely heavily on historical data these arguments are confirmed in a practical example of portfolio selection where our framework leads to better performing policies on the true distribution underlying the daily returns of financial assets

an upper bound on the perron value of an almost regular tournament matrix we provide an asymptotic upper bound on the perron value of an almost regular tournament matrix improving upon an existing bound due to friedland our approach employs techniques from contrained optimization

an explicit equivalent positive semidefinite program for nonlinear 0 1 programs we consider the general nonlinear optimization problem in 0 1 variables and provide an explicit equivalent positive semidefinite program in 2n 1 variables the optimal values of both problems are identical from every optimal solution of the former one easily finds an optimal solution of the latter and conversely from every solution of the latter one may construct an optimal solution of the former for illustration the equivalent positive semidefinite program is explicated for quadratic 0 1 programs and max cut in mathbb r 3 for unconstrained 0 1 programs a special representation in terms of a weighted sum of squares is provided

a hierarchy of near optimal policies for multistage adaptive optimization in this paper we propose a new tractable framework for dealing with linear dynamical systems affected by uncertainty applicable to multistage robust optimization and stochastic programming we introduce a hierarchy of near optimal polynomial disturbance feedback control policies and show how these can be computed by solving a single semidefinite programming problem the approach yields a hierarchy parameterized by a single variable the degree of the polynomial policies which controls the trade off between the optimality gap and the computational requirements we evaluate our framework in the context of three classical applications two in inventory management and one in robust regulation of an active suspension system in which very strong numerical performance is exhibited at relatively modest computational expense

least squares quantization in pcm it has long been realized that in pulse code modulation pcm with a given ensemble of signals to handle the quantum values should be spaced more closely in the voltage regions where the signal amplitude is more likely to fall it has been shown by panter and dite that in the limit as the number of quanta becomes infinite the asymptotic fractional density of quanta per unit voltage should vary as the one third power of the probability density per unit voltage of signal amplitudes in this paper the corresponding result for any finite number of quanta is derived that is necessary conditions are found that the quanta and associated quantization intervals of an optimum finite quantization scheme must satisfy the optimization criterion used is that the average quantization noise power be a minimum it is shown that the result obtained here goes over into the panter and dite result as the number of quanta become large the optimum quautization schemes for 2 b quanta b 1 2 cdots 7 are given numerically for gaussian and for laplacian distribution of signal amplitudes

on almost regular tournament matrices abstract spectral and determinantal properties of a special class m n of 2n 2n almost regular tournament matrices are studied in particular the maximum perron value of the matrices in this class is determined and shown to be achieved by the brualdi li matrix which has been conjectured to have the largest perron value among all tournament matrices of even order we also establish some determinantal inequalities for matrices in m n and describe the structure of their associated walk spaces

supermodularity and affine policies in dynamic robust optimization this paper considers a particular class of dynamic robust optimization problems where a large number of decisions must be made in the first stage which consequently fix the constraints and cost structure underlying a one dimensional linear dynamical system we seek to bridge two classical paradigms for solving such problems namely 1 dynamic programming dp and 2 policies parameterized in model uncertainties also known as decision rules obtained by solving tractable convex optimization problems we show that if the uncertainty sets are integer sublattices of the unit hypercube the dp value functions are convex and supermodular in the uncertain parameters and a certain technical condition is satisfied then decision rules that are affine in the uncertain parameters are optimal we also derive conditions under which such rules can be obtained by optimizing simple i e linear objective functions over the uncertainty sets our results suggest new modeling paradigms for dynamic robust optimiz

the price of robustness a robust approach to solving linear optimization problems with uncertain data was proposed in the early 1970s and has recently been extensively studied and extended under this approach we are willing to accept a suboptimal solution for the nominal values of the data in order to ensure that the solution remains feasible and near optimal when the data changes a concern with such an approach is that it might be too conservative in this paper we propose an approach that attempts to make this trade off more attractive that is we investigate ways to decrease what we call the price of robustness in particular we flexibly adjust the level of conservatism of the robust solutions in terms of probabilistic bounds of constraint violations an attractive aspect of our method is that the new robust formulation is also a linear optimization problem thus we naturally extend our methods to discrete optimization problems in a tractable way we report numerical results for a portfolio optimization problem a knapsack problem and a problem from the net lib library

adaptive multiscale redistribution for vector quantization vector quantization is a classical problem that appears in many fields unfortunately the quantization problem is generally nonconvex and therefore affords many local minima the main problem is finding an initial approximation which is close to a good local minimum once such an approximation is found the lloyd max method may be used to reach a local minimum near it in recent years much improvement has been made with respect to reducing the computational costs of quantization algorithms whereas the task of finding better initial approximations received somewhat less attention we present a novel multiscale iterative scheme for the quantization problem the scheme is based on redistributing the representation levels among aggregates of decision regions at changing scales the rule governing the redistribution relies on the so called point density function and on the number of representation levels in each aggregate our method focuses on achieving better local minima than those achieved by other contemporary methods such as lbg when quantizing signals with sparse and patchy histograms as may occur in color images for example the improvement in distortion relative to lbg may be arbitrarily large

geometric self assembly of rigid shapes self assembly of shapes from spheres to nonsmooth and possibly nonconvex shapes is pervasive throughout the sciences these arrangements arise in biology for animal flocking and herding in condensed matter physics with molecular and nano self assembly and in control theory for coordinated motion problems while idealizing these often nonconvex objects as points or spheres aids in analysis the effects of shape curvature and convexity are often dramatic especially for short range interactions in this paper we develop a general purpose model for arranging rigid shapes in euclidean domains and on flat tori the shapes are arranged optimally with respect to minimization of a geometric voronoi based cost function which generalizes the notion of a centroidal voronoi tessellation from point sources to general rigid shapes building upon our previous work in l j larsson r choksi and j c nave siam j sci comput 36 2014 pp a792 a827 we present an efficient and fast algorithm for the minimization of this nonlocal albeit finite dimensional variational problem the algorithm applies in any space dimension and can be used to generate self assemblies of any collection of nonconvex piecewise smooth shapes we also provide a result which supports the intuition that self assembled shapes should be centered in and aligned with their voronoi regions

the value of stochastic modeling in two stage stochastic programs with cost uncertainty although stochastic programming is probably the most effective framework for handling decision problems that involve uncertain variables it is always a costly task to formulate the stochastic model that accurately embodies our knowledge of these variables in practice this might require one to collect a large amount of observations to consult with experts of the specialized field of practice or to make simplifying assumptions about the underlying system when none of these options seem feasible a common heuristic has been to simply seek the solution of a version of the problem where each uncertain variable takes on its expected value otherwise known as the solution of the mean value problem in this paper we show that when 1 the stochastic program takes the form of a two stage mixed integer stochastic linear programs and 2 the uncertainty is limited to the objective function the solution of the mean value problem is in fact robust with respect to the selection of a stochastic model we also propose tractable methods that will bound the actual value of stochastic modeling i e how much improvement can be achieved by investing more efforts in the resolution of the stochastic model our framework is applied to an airline fleet composition problem in the three cases that are considered our results indicate that resolving the stochastic model can not lead to more than a 7 improvement of expected profits thus providing arguments against the need to develop these more sophisticated models

the optimal centroidal voronoi tessellations and the gersho s conjecture in the three dimensional space optimal centroidal voronoi tessellations have important applications in many different areas such as vector quantization data and image processing clustering analysis and resource management in the three dimensional euclidean space they are also useful to the mesh generation and optimization in this paper we conduct extensive numerical simulations to investigate the asymptotic structures of optimal centroidal voronoi tessellations for a given domain such a problem is intimately related to the famous gersho s conjecture for which a full proof is still not available we provide abundant evidence to substantiate the claim of the conjecture the body centered cubic lattice or par6 based centroidal voronoi tessellation has the lowest cost or energy per unit volume and is the most likely congruent cell predicted by the three dimensional gersho conjecture more importantly we probe the various properties of this optimal configuration including its dual triangulations which bear significant consequences in applications to three dimensional high quality meshing

optimal allocation of surgery blocks to operating rooms under uncertainty the allocation of surgeries to operating rooms ors is a challenging combinatorial optimization problem there is also significant uncertainty in the duration of surgical procedures which further complicates assignment decisions in this paper we present stochastic optimization models for the assignment of surgeries to ors on a given day of surgery the objective includes a fixed cost of opening ors and a variable cost of overtime relative to a fixed length of day we describe two types of models the first is a two stage stochastic linear program with binary decisions in the first stage and simple recourse in the second stage the second is its robust counterpart in which the objective is to minimize the maximum cost associated with an uncertainty set for surgery durations we describe the mathematical models bounds on the optimal solution and solution methodologies including an easy to implement heuristic numerical experiments based on real data from a large health care provider are used to contrast the results for the two models and illustrate the potential for impact in practice based on our numerical experimentation we find that a fast and easy to implement heuristic works fairly well on average across many instances we also find that the robust method performs approximately as well as the heuristic is much faster than solving the stochastic recourse model and has the benefit of limiting the worst case outcome of the recourse problem

robust truss topology design via semidefinite programming we present and motivate a new model of the truss topology design problem where the rigidity of the resulting truss with respect both to given loading scenarios and small occasional loads is optimized it is shown that the resulting optimization problem is a semidefinite program we derive and analyze several equivalent reformulations of the problem and present illustrative numerical examples

robust optimization of sums of piecewise linear functions with application to inventory problems robust optimization is a methodology that has gained a lot of attention in the recent years this is mainly due to the simplicity of the modeling process and ease of resolution even for large scale models unfortunately the second property is usually lost when the cost function that needs to be robustified is not concave or linear with respect to the perturbing parameters in this paper we study robust optimization of sums of piecewise linear functions over polyhedral uncertainty set given that these problems are known to be intractable we propose a new scheme for constructing conservative approximations based on the relaxation of an embedded mixed integer linear program and relate this scheme to methods that are based on exploiting affine decision rules our new scheme gives rise to two tractable models that respectively take the shape of a linear program and a semidefinite program with the latter having the potential to provide solutions of better quality than the former at the price of heavie

cones of matrices and set functions and 0 1 optimization it has been recognized recently that to represent a polyhedron as the projection of a higher dimensional but simpler polyhedron is a powerful tool in polyhedral combinatorics a general method is developed to construct higher dimensional polyhedra or in some cases convex sets whose projection approximates the convex hull of 0 1 valued solutions of a system of linear inequalities an important feature of these approximations is that one can optimize any linear objective function over them in polynomial time in the special case of the vertex packing polytope a sequence of systems of inequalities is obtained such that the first system already includes clique odd hole odd antihole wheel and orthogonality constraints in particular for perfect and many other graphs this first system gives the vertex packing polytope for various classes of graphs including t perfect graphs it follows that the stable set polytope is the projection of a polytope with a polynomial number of facets an extension o

robust portfolio selection problems in this paper we show how to formulate and solve robust portfolio selection problems the objective of these robust formulations is to systematically combat the sensitivity of the optimal portfolio to statistical and modeling errors in the estimates of the relevant market parameters we introduce uncertainty structures for the market parameters and show that the robust portfolio selection problems corresponding to these uncertainty structures can be reformulated as second order cone programs and therefore the computational effort required to solve them is comparable to that required for solving convex quadratic programs moreover we show that these uncertainty structures correspond to confidence regions associated with the statistical procedures employed to estimate the market parameters finally we demonstrate a simple recipe for efficiently computing robust portfolios given raw market data and a desired level of confidence

robust convex optimization we study convex optimization problems for which the data is not specified exactly and it is only known to belong to a given uncertainty set u yet the constraints must hold for all possible values of the data from u the ensuing optimization problem is called robust optimization in this paper we lay the foundation of robust convex optimization in the main part of the paper we show that if u is an ellipsoidal uncertainty set then for some of the most important generic convex optimization problems linear programming quadratically constrained programming semidefinite programming and others the corresponding robust convex program is either exactly or approximately a tractable problem which lends itself to efficientalgorithms such as polynomial time interior point methods

a note on the sequence of brualdi li matrices abstract we consider the sequence 2n n 1 2 2n where 2n is the perron value of the brualdi li matrix of order 2n we prove that the sequence converges find its limit and show that asymptotically the sequence is monotonically decreasing in particular this addresses some problems raised by friedland and katz

bounds for the perron root using max eigenvalues abstract using the techniques of max algebra a new proof of al pin s lower and upper bounds for the perron root of a nonnegative matrix is given the bounds depend on the row sums of the matrix and its directed graph if the matrix has zero main diagonal entries then these bounds may improve the classical row sum bounds this is illustrated by a generalized tournament matrix

a fast sweeping method for eikonal equations in this paper a fast sweeping method for computing the numerical solution of eikonal equations on a rectangular grid is presented the method is an iterative method which uses upwind difference for discretization and uses gauss seidel iterations with alternating sweeping ordering to solve the discretized system the crucial idea is that each sweeping ordering follows a family of characteristics of the corresponding eikonal equation in a certain direction simultaneously the method has an optimal complexity of o n for n grid points and is extremely simple to implement in any number of dimensions monotonicity and stability properties of the fast sweeping algorithm are proven convergence and error estimates of the algorithm for computing the distance function is studied in detail it is shown that 2 n gauss seidel iterations is enough for the distance function in n dimensions an estimation of the number of iterations for general eikonal equations is also studied numerical examples are used to verify the analysis

on the limited memory bfgs method for large scale optimization we study the numerical performance of a limited memory quasi newton method for large scale optimization which we call the l bfgs method we compare its performance with that of the method developed by buckley and lenir 1985 which combines cycles of bfgs steps and conjugate direction steps our numerical tests indicate that the l bfgs method is faster than the method of buckley and lenir and is better able to use additional storage to accelerate convergence we show that the l bfgs method can be greatly accelerated by means of a simple scaling we then compare the l bfgs method with the partitioned quasi newton method of griewank and toint 1982a the results show that for some problems the partitioned quasi newton method is clearly superior to the l bfgs method however we find that for other problems the l bfgs method is very competitive due to its low iteration cost we also study the convergence properties of the l bfgs method and prove global convergence on uniformly convex problems

uncertain linear programs extended affinely adjustable robust counterparts in this paper we introduce the extended affinely adjustable robust counterpart to modeling and solving multistage uncertain linear programs with fixed recourse our approach first reparameterizes the primitive uncertainties and then applies the affinely adjustable robust counterpart proposed in the literature in which recourse decisions are restricted to be linear in terms of the primitive uncertainties we propose a special case of the extended affinely adjustable robust counterpart the splitting based extended affinely adjustable robust counterpart and illustrate both theoretically and computationally that the potential of the affinely adjustable robust counterpart method is well beyond the one presented in the literature similar to the affinely adjustable robust counterpart our approach ends up with deterministic optimization formulations that are tractable and scalable to multistage problems

solving large scale sparse semidefinite programs for combinatorial optimization we present a dual scaling interior point algorithm and show how it exploits the structure and sparsity of some large scale problems we solve the positive semidefinite relaxation of combinatorial and quadratic optimization problems subject to boolean constraints we report the first computational results of interior point algorithms for approximating maximum cut semidefinite programs with dimension up to 3 000

robust counterparts of inequalities containing sums of maxima of linear functions abstract this paper addresses the robust counterparts of optimization problems containing sums of maxima of linear functions these problems include many practical problems e g problems with sums of absolute values and arise when taking the robust counterpart of a linear inequality that is affine in the decision variables affine in a parameter with box uncertainty and affine in a parameter with general uncertainty in the literature often the reformulation is used that is exact when there is no uncertainty however in robust optimization this reformulation gives an inferior solution and provides a pessimistic view we observe that in many papers this conservatism is not mentioned some papers have recognized this problem but existing solutions are either conservative or their performance for different uncertainty regions is not known a comparison between them is not available and they are restricted to specific problems we describe techniques for general problems and compare them with numerical examples in inventory management regression and brachytherapy based on these examples we give recommendations for reducing the conservatism

the hexagon theorem we wish to int s z f z 2 s the unit square over all choices of f z which take just n values for large n it is shown that f z approaches that function which on each face of a hexagonal mesh has the value of the centroid of that face this problem in the design of quantizers for a pulse code modulation pcm system arose in which pairs of adjacent samples are quantized in an effort to capitalize on the correlation between samples and on the geometry of the plane

uniform convergence of a nonlinear energy based multilevel quantization scheme a popular vector quantization scheme can be constructed by centroidal voronoi tessellations cvts which also have many other applications in diverse areas of science and engineering the development of efficient algorithms for their construction is a key to the successful applications of cvts in practice this paper studies the details of a new optimization based multilevel algorithm for the numerical computation of cvts the rigorous proof of its uniform convergence in one space dimension and the results of computational simulations are provided they substantiate recent claims on the significant speedup demonstrated by the new scheme in comparison with traditional methods

normal matrices and their principal submatrices of co order one abstract let a be a normal matrix v be any of its indices a v be the matrix obtained from a by deleting the v th row and column and be an eigenvalue of a v in our paper we construct the eigenspace of a associated with from that of a v in particular it is shown that if there is a unique jordan block of size strictly greater than one in the part of the jordan form of a v corresponding to then the geometric multiplicity of decreases by one under the transition from a v to a in other words the typical change of the spectral properties holds for the results obtained are applied to circulant matrices moreover in appendix to our paper we consider almost regular tournament matrices as principal submatrices of co order one of regular tournament matrices in particular it is observed that the brualdi li tournament matrix b 2 n of order 2n is permutationally similar to a principal submatrix of co order one of the circulant matrix of order 2n 1 with the first row 0 1 1 n 0 0 n as a consequence of this fact the weak brualdi li conjecture is formulated for principal submatrices of co order one of the adjacency matrices of cayley tournaments

brief robust maximum likelihood estimation in the linear model this paper addresses the problem of maximum likelihood parameter estimation in linear models affected by gaussian noise whose mean and covariance matrix are uncertain the proposed estimate maximizes a lower bound on the worst case with respect to the uncertainty likelihood of the measured sample and is computed solving a semidefinite optimization problem sdp the problem of linear robust estimation is also studied in the paper and the statistical and optimality properties of the resulting linear estimator are discussed

convergence of the lloyd algorithm for computing centroidal voronoi tessellations centroidal voronoi tessellations cvts are voronoi tessellations of a bounded geometric domain such that the generating points of the tessellations are also the centroids mass centers of the corresponding voronoi regions with respect to a given density function centroidal voronoi tessellations may also be defined in more abstract and more general settings due to the natural optimization properties enjoyed by cvts they have many applications in diverse fields the lloyd algorithm is one of the most popular iterative schemes for computing the cvts but its theoretical analysis is far from complete in this paper some new analytical results on the local and global convergence of the lloyd algorithm are presented these results are derived through careful utilization of the optimization properties shared by cvts numerical experiments are also provided to substantiate the theoretical analysis

a note on perron vectors for almost regular tournament matrices abstract let t be an almost regular tournament matrix of order n with right perron vector w we show that if the i th row sum of t is n 2 2 and the j th row sum is n 2 then w i w j thus in the round robin competition corresponding to t the ranking schemes of kendall and wei and of ramanujacharyula agree with the ranking generated by the row sums of t

new upper bounds on sphere packings i we continue the study of the linear programming bounds for sphere packing introduced by cohn and elkies we use theta series to give another proof of the principal theorem and present some related results and conjectures this article is in the arxiv as arxiv math mg 0110010

perron vector bounds for a tournament matrix with applications to a conjecture of brualdi and li abstract for a primitive generalized tournament matrix we present upper and lower bounds on an entry in its perron vector in terms of the corresponding row sum of the matrix the bounds are then used to help prove that if n is even and sufficiently large any tournament matrix of order n which maximizes the perron value must be almost regular throughout we use both analytic and combinatorial techniques

an iterative algorithm for computing measures of generalized voronoi regions we present and analyze a fast algorithm for directly computing measures of generalized voronoi regions associated with generators of arbitrary codimension the algorithm consists of solving one eikonal equation to construct a kernel based operator whose iteration accumulates mass along the closest generator in particular the algorithm does not require the computation of the voronoi diagram or the gradient of the solution to the eikonal equation the algorithm is shown to be first order and converge very quickly by discretizing the distance to the generators on the grid and not the generators themselves very accurate geometric information is used even for coarse grids several examples are presented including the computation of the population influence associated with the los angeles county highway system the method is also one of the key ingredients for the fast computation of centroidal voronoi tessellations cvts of general rigid objects e g rigid curves and surfaces in higher dimensions w

solution of the conjecture of brualdi and li abstract we establish the conjecture of brualdi and li on the maximal perron root of a tournament matrix of even order

a multigrid approach to the scalar quantization problem a multigrid framework for the one dimensional scalar fixed rate quantization problem is presented the framework is based on the lloyd max iterative process which is a central building block in many quantization algorithms this process iteratively improves a given initial solution and generally converges to a local minimum of the quantization distortion contrary to the classical lloyd max process the convergence of the multigrid algorithm is practically independent of the number of representation levels sought using this approach a local minimum is reached at the cost of just a few lloyd max iterations the complexity of the proposed method is o n operations for the continuous case and 3n o n for the discrete case where n is the number of representation levels sought and n is the size of the discrete probability density function in addition to its independent attributes this work is a precursor to the more important vector quantization problem for which a multiscale framework is also being developed

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

centroidal voronoi tessellations applications and algorithms a centroidal voronoi tessellation is a voronoi tessellation whose generating points are the centroids centers of mass of the corresponding voronoi regions we give some applications of such tessellations to problems in image compression quadrature finite difference methods distribution of resources cellular biology statistics and the territorial behavior of animals we discuss methods for computing these tessellations provide some analyses concerning both the tessellations and the methods for their determination and finally present the results of some numerical experiments