geometrically consistent mesh modification a new paradigm of adaptivity is to execute refinement coarsening and smoothing of meshes on manifolds with incomplete information about their geometry and yet preserve position and curvature accuracy we refer to this collectively as geometrically consistent gc mesh modification we discuss the concept of discrete gc show the failure of naive approaches and propose and analyze a simple algorithm that is gc and accuracy preserving

higher order finite element methods and pointwise error estimates for elliptic problems on surfaces we define higher order analogues to the piecewise linear surface finite element method studied in g dziuk finite elements for the beltrami operator on arbitrary surfaces in partial differential equations and calculus of variations springer verlag berlin 1988 pp 142 155 and prove error estimates in both pointwise and l 2 based norms using the laplace beltrami problem on an implicitly defined surface gamma as a model pde we define lagrange finite element methods of arbitrary degree on polynomial approximations to gamma which likewise are of arbitrary degree then we prove a priori error estimates in the l 2 h 1 and corresponding pointwise norms that demonstrate the interaction between the pde error that arises from employing a finite dimensional finite element space and the geometric error that results from approximating gamma we also consider parametric finite element approximations that are defined on gamma and thus induce no geometric error computational examples confirm the sharpness of our error estimates

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

shallow deep and very deep cuts in the analytic center cutting plane method in this paper we show that the cut does not need to go through the query point it can be deep or shallow the primal framework leads to a simple analysis of the potential variation which shows that the inequality needed for convergence of the algorithm is in fact attained at the first iterate of the feasibility step

large scale image geo localization using dominant sets this paper presents a new approach for the challenging problem of geo localization using image matching in a structured database of city wide reference images with known gps coordinates we cast the geo localization as a clustering problem of local image features akin to existing approaches to the problem our framework builds on low level features which allow local matching between images for each local feature in the query image we find its approximate nearest neighbors in the reference set next we cluster the features from reference images using dominant set clustering which affords several advantages over existing approaches first it permits variable number of nodes in the cluster which we use to dynamically select the number of nearest neighbors for each query feature based on its discrimination value second this approach is several orders of magnitude faster than existing approaches thus we obtain multiple clusters different local maximizers and obtain a robust final solution to the problem using multiple weak solutions through constrained dominant set clustering on global image features where we enforce the constraint that the query image must be included in the cluster this second level of clustering also bypasses heuristic approaches to voting and selecting the reference image that matches to the query we evaluate the proposed framework on an existing dataset of 102k street view images as well as a new larger dataset of 300k images and show that it outperforms the state of the art by 20 and 7 percent respectively on the two datasets

an interior point cutting plane method for the convex feasibility problem with second order cone inequalities the convex feasibility problem in general is a problem of finding a point in a convex set that contains a full dimensional ball and is contained in a compact convex set we assume that the outer set is described by second order cone inequalities and propose an analytic center cutting plane technique to solve this problem we discuss primal and dual settings simultaneously two complexity results are reported the complexity of restoration procedure and complexity of the overall algorithm we prove that an approximate analytic center is updated after adding a second order cone cut socc ino 1 newton step and that the analytic center cutting plane method accpm with socc is a fully polynomial algorithm

a posteriori error analysis for the mean curvature flow of graphs we study the equation describing the motion of a nonparametric surface according to its mean curvature flow this is a nonlinear nonuniformly parabolic pde that can be discretized in space via a finite element method we conduct an aposteriori error analysis of the spatial discretization and derive upper bounds on the error in terms of computable estimators based on local residual indicators the reliability of the estimators is illustrated with two numerical simulations one of which treats the case of a singular solution

constraint reduction for linear programs with many inequality constraints consider solving a linear program in standard form where the constraint matrix a is m times n with n gg m gg 1 such problems arise for example as the result of finely discretizing a semi infinite program the cost per iteration of typical primal dual interior point methods on such problems is o m 2n we propose to reduce that cost by replacing the normal equation matrix ad 2a t where d is a diagonal matrix with a reduced version of same dimension a qd q 2a q t where q is an index set including the indices of m most nearly active or most violated dual constraints at the current iterate with m geq m a prescribed integer this can result in a speedup of close to n q at each iteration promising numerical results are reported for constraint reduced versions of a dual feasible affine scaling algorithm and of mehrotra s predictor corrector method s mehrotra it siam j optim 2 1992 pp 575 601 in particular while it could be expected that neglecting a large portion of the constraints especially at early iterations may result in a significant deterioration of the search direction it appears that the total number of iterations typically remains essentially constant as the size of the reduced constraint set is decreased down to some threshold in some cases this threshold is a small fraction of the total set in the case of the affine scaling algorithm global convergence and local quadratic convergence are proved

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

on the von neumann economic growth problem we study the complexity of the von neumann economic growth problem gamma ast max gamma mid exists y ne 0 y ge 0 b gamma ay ge 0 where a and b are given two nonnegative and rational m n matrices and a has no all zero column let the binary data length of a and b be l we develop an interior point algorithm to generate a i such that 2 1 i in om nl minm nt iterations where each iteration solves a system of m n linear equations

a multiple cut analytic center cutting plane method for semidefinite feasibility problems we consider the problem of finding a point in a nonempty bounded convex body gamma in the cone of symmetric positive semidefinite matrices cal s m assume that gamma is defined by a separating oracle which for any given m ti m symmetric matrix hat y either confirms that hat y in gamma or returns several selected cuts i e a number of symmetric matrices ai i 1 p p le p max such that gamma is in the polyhedron y in cal s m mid a i bullet y le a i bullet hat y i 1 ldots p we present a multiple cut analytic center cutting plane algorithm starting from a trivial initial point the algorithm generates a sequence of positive definite matrices which are approximate analytic centers of a shrinking polytope in cal s m the algorithm terminates with a point in gamma within o m 3p max epsilon 2 newton steps to leading order where epsilon is the maximum radius of a ball contained in gamma

discrete gradient flows for shape optimization and applications we present a variational framework for shape optimization problems that establishes clear and explicit connections among the continuous formulation its full discretization and the resulting linear algebraic systems our approach hinges on the following essential features shape differential calculus a semi implicit time discretization and a finite element method for space discretization we use shape differential calculus to express variations of bulk and surface energies with respect to domain changes the semi implicit time discretization allows us to track the domain boundary without an explicit parametrization and has the flexibility to choose different descent directions by varying the scalar product used for the computation of normal velocity we propose a schur complement approach to solve the resulting linear systems efficiently we discuss applications of this framework to image segmentation optimal shape design for pde and surface diffusion along with the choice of suitable scalar products in each case we illustrate the method with several numerical experiments some developing pinch off and topological changes in finite time

parametric fem for geometric biomembranes we consider geometric biomembranes governed by an l 2 gradient flow for bending energy subject to area and volume constraints helfrich model we give a concise derivation of a novel vector formulation based on shape differential calculus and corresponding discretization via parametric fem using quadratic isoparametric elements and a semi implicit euler method we document the performance of the new parametric fem with a number of simulations leading to dumbbell red blood cell and toroidal equilibrium shapes while exhibiting large deformations

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

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

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

a matrix generation approach for eigenvalue optimization we study the extension of a column generation technique to nonpolyhedral models in particular we study the problem of minimizing the maximum eigenvalue of an affine combination of symmetric matrices at each step of the algorithm a restricted master problem in the primal space corresponding to the relaxed dual original problem is formed a query point is obtained as an approximate analytic center of a bounded set that contains the optimal solution of the dual problem the original objective function is evaluated at the query point and depending on its differentiability a column or a matrix is added to the restricted master problem we discuss the issues of recovering feasibility after the restricted master problem is updated by a column or a matrix the computational experience of implementing the algorithm on randomly generated problems are reported and the cpu time of the matrix generation algorithm is compared with that of the primal dual interior point methods on dense and sparse problems using the software sdpt3 our numerical results illustrate that the matrix generation algorithm outperforms primal dual interior point methods on dense problems with no structure and also on a class of sparse problems

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

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

complexity analysis of the analytic center cutting plane method that uses multiple cuts we analyze the complexity of the analytic center cutting plane or column generation algorithm for solving general convex problems defined by a separation oracle the oracle is called at the analytic center of a polytope which contains a solution set and is given by the intersection of the linear inequalities previously generated from the oracle if the center is not in the solution set separating hyperplanes will be placed through the center to shrink the containing polytope while the complexity result has been recently established for the algorithm when one cutting plane is placed in each iteration the result remains open when multiple cuts are added moreover adding multiple cuts actually is a key to practical effectiveness in solving many problems and it presents theoretical difficulties in analyzing cutting plane methods in this paper we show that the analytic center cutting plane algorithm with multiple cuts added in each iteration still is a fully polynomial approximation algorithm

a new algorithm for minimizing convex functions over convex sets let s subseteq mathbb r n be a convex set for which there is an oracle with the following property given any pointz n the oracle returns a yes ifz s whereas ifz s then the oracle returns a no together with a hyperplane that separatesz froms the feasibility problem is the problem of finding a point ins the convex optimization problem is the problem of minimizing a convex function overs we present a new algorithm for the feasibility problem the notion of a volumetric center of a polytope and a related ellipsoid of maximum volume inscribable in the polytope are central to the algorithm our algorithm has a significantly better global convergence rate and time complexity than the ellipsoid algorithm the algorithm for the feasibility problem easily adapts to the convex optimization problem

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

computational parametric willmore flow we propose a new algorithm for the computation of willmore flow this is the l 2 gradient flow for the willmore functional which is the classical bending energy of a surface willmore flow is described by a highly nonlinear system of pdes of fourth order for the parametrization of the surface the spatially discrete numerical scheme is stable and consistent the discretization relies on an adequate calculation of the first variation of the willmore functional together with a derivation of the second variation of the area functional which is well adapted to discretization techniques with finite elements the algorithm uses finite elements on surfaces we give numerical examples and tests for piecewise linear finite elements a convergence proof for the full algorithm remains an open question

image geo localization based on multiplenearest neighbor feature matching usinggeneralized graphs in this paper we present a new framework for geo locating an image utilizing a novel multiple nearest neighbor feature matching method using generalized minimum clique graphs gmcp first we extract local features e g sift from the query image and retrieve a number of nearest neighbors for each query feature from the reference data set next we apply our gmcp based feature matching to select a single nearest neighbor for each query feature such that all matches are globally consistent our approach to feature matching is based on the proposition that the first nearest neighbors are not necessarily the best choices for finding correspondences in image matching therefore the proposed method considers multiple reference nearest neighbors as potential matches and selects the correct ones by enforcing consistency among their global features e g gist using gmcp in this context we argue that using a robust distance function for finding the similarity between the global features is essential for the cases where the query matches multiple reference images with dissimilar global features towards this end we propose a robust distance function based on the gaussian radial basis function g rbf we evaluated the proposed framework on a new data set of 102k street view images the experiments show it outperforms the state of the art by 10 percent

specialized fast algorithms for iqc feasibility and optimization problems the conventional way to treat integral quadratic constraint iqc problems is to transform them into semi definite programs sdps sdps can then be solved using interior point methods which have been proven efficient this approach however is not always the most efficient since it introduces additional decision variables to the sdp and the additional decision variables sometimes largely increase the complexity of the problem in this paper we demonstrate how to solve iqc problems by other alternatives more specifically we consider two cutting plane algorithms we will show that in certain cases these cutting plane algorithms can solve iqc problems much faster than the conventional approach numerical examples as well as some explanations from the point of view of computational complexity are provided to support our point

on the complexity of approximating the maximal inscribed ellipsoid for a polytope we give a new polynomial bound on the complexity of approximating the maximal inscribed ellipsoid for a polytope

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

modeling the shape of the scene a holistic representation of the spatial envelope in this paper we propose a computational model of the recognition of real world scenes that bypasses the segmentation and the processing of individual objects or regions the procedure is based on a very low dimensional representation of the scene that we term the spatial envelope we propose a set of perceptual dimensions naturalness openness roughness expansion ruggedness that represent the dominant spatial structure of a scene then we show that these dimensions may be reliably estimated using spectral and coarsely localized information the model generates a multidimensional space in which scenes sharing membership in semantic categories e g streets highways coasts are projected closed together the performance of the spatial envelope model shows that specific information about object shape or identity is not a requirement for scene categorization and that modeling a holistic representation of the scene informs about its probable semantic category

potential reduction methods in mathematical programming we provide a survey of interior point methods for linear programming and its extensions that are based on reducing a suitable potential function at each iteration we give a fairly complete overview of potential reduction methods for linear programming focusing on the possibility of taking long steps and the properties of the barrier function that are necessary for the analysis we then describe briefly how the methods and results can be extended to certain convex programming problems following the approach of nesterov and todd we conclude with some open problems

a cutting plane method for solving kyp sdps semidefinite programs originating from the kalman yakubovich popov lemma are convex optimization problems and there exist polynomial time algorithms that solve them however the number of variables is often very large making the computational time extremely long algorithms more efficient than general purpose solvers are thus needed to this end structure exploiting algorithms have been proposed based on the dual formulation in this paper a cutting plane algorithm is proposed in a comparison with a general purpose solver and a structure exploiting solver it is shown that the cutting plane based solver can handle optimization problems of much higher dimension

multiple cuts in the analytic center cutting plane method we analyze the multiple cut generation scheme in the analytic center cutting plane method we propose an optimal primal and dual updating direction when the cuts are central the direction is optimal in the sense that it maximizes the product of the new dual slacks and of the new primal variables within the trust regions defined by dikin s primal and dual ellipsoids the new primal and dual directions use the variance covariance matrix of the normals to the new cuts in the metric given by dikin s ellipsoid we prove that the recovery of a new analytic center from the optimal restoration direction can be done in o plog p 1 damped newton steps where p is the number of new cuts added by the oracle which may vary with the iteration the results and the proofs are independent of the specific scaling matrix primal dual or primal dual that is used in the computations the computation of the optimal direction uses newton s method applied to a self concordant function of p variables the convergence result of ye math programming 78 1997 pp 85 104 holds here also the algorithm stops after o frac bar p 2n 2 varepsilon 2 cutting planes have been generated where bar p is the maximum number of cuts generated at any given iteration

the analytic center cutting plane method with semidefinite cuts we analyze an analytic center cutting plane algorithm for convex feasibility problems with semidefinite cuts the problem of interest seeks a feasible point in a bounded convex set which contains a full dimensional ball with varepsilon 0 is a condition number of the field of cuts

parametric approximation of willmore flow and related geometric evolution equations we present various variational approximations of willmore flow in mathbb r d d 2 3 as well as the classic willmore flow we also consider variants that are a volume preserving and b volume and area preserving the latter evolution law is the so called helfrich flow in addition we consider motion by gaus curvature the presented fully discrete schemes are easy to solve as they are linear at each time level and they have good properties with respect to the distribution of mesh points finally we present numerous numerical experiments including simulations for energies appearing in the modeling of biological cell membranes

image search with selective match kernels aggregation across single and multiple images this paper considers a family of metrics to compare images based on their local descriptors it encompasses the vector or locally aggregated descriptors descriptor and matching techniques such as hamming embedding making the bridge between these approaches leads us to propose a match kernel that takes the best of existing techniques by combining an aggregation procedure with a selective match kernel the representation underpinning this kernel is approximated providing a large scale image search both precise and scalable as shown by our experiments on several benchmarks we show that the same aggregation procedure originally applied per image can effectively operate on groups of similar features found across multiple images this method implicitly performs feature set augmentation while enjoying savings in memory requirements at the same time finally the proposed method is shown effective for place recognition outperforming state of the art methods on a large scale landmark recognition benchmark

region based hierarchical image matching this paper presents an approach to region based hierarchical image matching where given two images the goal is to identify the largest part in image 1 and its match in image 2 having the maximum similarity measure defined in terms of geometric and photometric properties of regions e g area boundary shape and color as well as region topology e g recursive embedding of regions to this end each image is represented by a tree of recursively embedded regions obtained by a multiscale segmentation algorithm this allows us to pose image matching as the tree matching problem to overcome imaging noise one to one many to one and many to many node correspondences are allowed the trees are first augmented with new nodes generated by merging adjacent sibling nodes which produces directed acyclic graphs dags then transitive closures of the dags are constructed and the tree matching problem reformulated as finding a bijection between the two transitive closures on dags while preserving the connectivity and ancestor descendant relationships of the original trees the proposed approach is validated on real images showing similar objects captured under different types of noise including differences in lighting conditions scales or viewpoints amidst limited occlusion and clutter

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

an analytic center cutting plane method for semidefinite feasibility problems semidefinite feasibility problems arise in many areas of operations research the abstract form of these problems can be described as finding a point in a nonempty bounded convex body g in the cone of symmetric positive semidefinite matrices assume that g is defined by an oracle which for any givenm xm symmetric positive semidefinite matrix y either confirms that y e g or returns a cut i e a symmetric matrixa such that g is in the half space y a y a y we study an analytic center cutting plane algorithm for this problem at each iteration the algorithm computes an approximate analytic center of a working set defined by the cutting plane system generated in the previous iterations if this approximate analytic center is a solution then the algorithm terminates otherwise the new cutting plane returned by the oracle is added into the system as the number of iterations increases the working set shrinks and the algorithm eventually finds a solution to the problem all iterates generated by the algorithm are positive definite matrices the algorithm has a worst case complexity of o m3 e 2 on the total number of cuts to be used where e is the maximum radius of a ball contained by g