error inhomogeneity in wavelet based compression despite the popularity of wavelet based image compression its shortcoming of having error inhomogeneity ei namely the error that is different for even and odd pixel location has not been previously analyzed and formally addressed the difference can be substantial up to 3 4 db peak signal to noise ratio psnr for some images and compression ratios in this paper we show that the ei is caused by asymmetrical filtering of quantization errors after the upsampling step in wavelet synthesis process nonuniformity and correlation of quantization errors can also contribute to this ei albeit to a smaller degree in addition to explaining the source of ei the model we developed in this paper also allows predicting its amount for a given wavelet furthermore we show how to redesign wavelet filters to reduce this ei at a cost of a small reduction in the overall psnr performance for applications that are sensitive to psnr degradation we also show how to design wavelet filters that can gradually tradeoff psnr performance for reduced ei

boundary behavior of interior point algorithms in linear programming this paper studies the boundary behavior of some interior point algorithms for linear programming the algorithms considered are karmarkar s projective rescaling algorithm the linear rescaling algorithm which was proposed as a variation on karmarkar s algorithm and the logarithmic barrier technique the study includes both the continuous trajectories of the vector fields induced by these algorithms and also the discrete orbits it is shown that although the algorithms are defined on the interior of the feasible polyhedron they actually determine differentiable vector fields on the closed polyhedron conditions are given under which a vector field gives rise to trajectories that each visit the neighborhoods of all the vertices of the klee minty cube the linear rescaling algorithm satisfies these conditions thus limits of such trajectories obtained when a starting point is pushed to the boundary may have an exponential number of breakpoints it is shown that limits of projective rescaling trajectories may have a linear number of such breakpoints however projective rescaling trajectories may visit the neighborhoods of linearly many vertices the behavior of the linear rescaling algorithm near vertices is analyzed it is proved that all the trajectories have a unique asymptotic direction of convergence to the optimum

how good are interior point methods klee minty cubes tighten iteration complexity bounds by refining a variant of the klee minty example that forces the central path to visit all the vertices of the klee minty n cube we exhibit a nearly worst case example for path following interior point methods namely while the theoretical iteration complexity upper bound is o 2 n n frac 5 2 we prove that solving this n dimensional linear optimization problem requires at least 2 n 1 iterations

superlinear convergence of the affine scaling algorithm in this paper we show that a variant of the long step affine scaling algorithm with variable stepsizes is two step superlinearly convergent when applied to general linear programming lp problems superlinear convergence of the sequence of dual estimates is also established for homogeneous lp problems having the origin as the unique optimal solution we also show that 2 3 is a sharp upper bound on the fixed stepsize that provably guarantees that the sequence of primal iterates converge to the optimal solution along a unique direction of approach since the point to which the sequence of dual estimates converge depend on the direction of approach of the sequence of primal iterates this result gives a plausible but not accurate theoretical explanation for why 2 3 is a sharp upper bound on the fixed stepsize that guarantees the convergence of the dual estimates

on the number of iterations of karmarkar s algorithm for linear programming karmarkar s algorithm for linear programming was published in 1984 and it is highly important to both theory and practice on the practical side some of its variants have been found to be far more efficient than the simplex method on a wide range of very large calculations while its polynomial time properties are fundamental to research on complexity these properties depend on the fact that each iteration reduces a potential function by an amount that is bounded away from zero the bound being independent of all the coefficients that occur it follows that under mild conditions on the initial vector of variables the number of iterations that are needed to achieve a prescribed accuracy in the final value of the linear objective function is at most a multiple ofn wheren is the number of inequality constraints by considering a simple example that allowsn to be arbitrarily large we deduce analytically that the magnitude of this complexity bound is correct specifically we prove that the solution of the example by karmarkar s original algorithm can require aboutn 20 iterations further we find that the algorithm makes changes to the variables that are closely related to the steps of the simplex method

a survey of search directions in interior point methods for linear programming a basic characteristic of an interior point algorithm for linear programming is the search direction many papers on interior point algorithms only give an implicit description of the search direction in this report we derive explicit expressions for the search directions used in many well known algorithms comparing these explicit expressions gives a good insight into the similarities and differences between the various algorithms moreover we give a survey of projected gradient and newton directions for all potential and barrier functions this is done both for the affine and projective variants

an implementation of karmarkar s algorithm for linear programming this paper describes the implementation of power series dual affine scaling variants of karmarkar s algorithm for linear programming based on a continuous version of karmarkar s algorithm two variants resulting from first and second order approximations of the continuous trajectory are implemented and tested linear programs are expressed in an inequality form which allows for the inexact computation of the algorithm s direction of improvement resulting in a significant computational advantage implementation issues particular to this family of algorithms such as treatment of dense columns are discussed the code is tested on several standard linear programming problems and compares favorably with the simplex codeminos 4 0

the affine scaling direction for linear programming is a limit of projective scaling directions abstract there are two ways to convert a standard form linear programming problem to a form suitable for application of a projective scaling interior point algorithm based on karmarkar s method one of these adds a dummy variable which is identically one we show that as the number of dummy variables all identically one added tends to infinity the resulting direction in all the original variables tends to the direction chosen by dikin s affine scaling algorithm

optimizing over three dimensional subspaces in an interior point method for linear programming abstract interior point algorithms for solving linear programming problems are considered a three dimensional method is developed that at each iteration solves a subproblem based on minimizing the cost function on low dimensional cross sections of the feasible region the generators for the three dimensional subproblem include the dual affine search direction and two higher order search directions one of the higher order directions is a third order correction to the newton recentering direction and the other is a correction to the dual affine direction that is motivated by the use of rank one updates of the second derivative information numerical results are presented for this method that indicate a nearly 20 reduction in cpu time compared to our best dual affine implementation

image coding using wavelet transform a scheme for image compression that takes into account psychovisual features both in the space and frequency domains is proposed this method involves two steps first a wavelet transform used in order to obtain a set of biorthogonal subclasses of images the original image is decomposed at different scales using a pyramidal algorithm architecture the decomposition is along the vertical and horizontal directions and maintains constant the number of pixels required to describe the image second according to shannon s rate distortion theory the wavelet coefficients are vector quantized using a multiresolution codebook to encode the wavelet coefficients a noise shaping bit allocation procedure which assumes that details at high resolution are less visible to the human eye is proposed in order to allow the receiver to recognize a picture as quickly as possible at minimum cost a progressive transmission scheme is presented it is shown that the wavelet transform is particularly well adapted to progressive transmission

wavelet based scalable l infinity oriented compression among the different classes of coding techniques proposed in literature predictive schemes have proven their outstanding performance in near lossless compression however these schemes are incapable of providing embedded linfin oriented compression or at most provide a very limited number of potential linfin bit stream truncation points we propose a new multidimensional wavelet based linfin constrained scalable coding framework that generates a fully embedded linfin oriented bit stream and that retains the coding performance and all the scalability options of state of the art l2 oriented wavelet codecs moreover our codec instantiation of the proposed framework clearly outperforms jpeg2000 in linfin coding sense

global convergence property of the affine scaling methods for primal degenerate linear programming problems in this paper we investigate the global convergence property of the affine scaling method under the assumption of dual nondegeneracy the behavior of the method near degenerate vertices is analyzed in detail on the basis of the equivalence between the affine scaling methods for homogeneous lp problems and karmarkar s method it is shown that the step size 1 8 where the displacement vector is normalized with respect to the distance in the scaled space is sufficient to guarantee the global convergence for dual nondegenerate lp problems the result can be regarded as a counterpart to dikin s global convergence result on the affine scaling method assuming primal nondegeneracy

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

convergence and boundary behavior of the projective scaling trajectories for linear programming we analyze the convergence and boundary behavior of the continuous trajectories of the vector field induced by the projective scaling algorithm as applied to possibly degenerate linear programming problems in karmarkar s standard form we show that a projective scaling trajectory tends to an optimal solution which in general depends on the starting point when the optimal solution is unique we show that all projective scaling trajectories approach the optimal solution through the same asymptotic direction our analysis is based on the affine scaling trajectories for the homogeneous standard form linear program that arises from karmarkar s standard form linear program by removing the unique nonhomogeneous constraint

on the continuous trajectories for a potential reduction algorithm for linear programming for a possibly degenerate linear program minx ctx ax b x 0 a is an m n real matrix b rm and c rn whose optimal value is 0 we study the limiting behavior of the trajectories of the family of vector fields i qx i xpax xc q i1ctx1 for all values of q n where x is the diagonal matrix associated with x and pax is the projection operator onto the null space of ax a polynomial algorithm based on the directions i qx has been presented by gonzaga 6 when q n or q n n we show that all trajectories of i qx converge to the unique center of the optimal face of the given linear program when this face consists of a unique vertex it is shown that any trajectory of i qx approaches this vertex along the same direction when the optimal face consists of more than one point we show that there is a threshold value i 0 such that for q i most of the trajectories of i qx converge to the center tangentially to the optimal face and that the direction of approach of a trajectory of i qx depends on the initial condition for q i q i the trajectories of i qx converge to the center along a unique direction along several directions which depend on the initial condition forming a positive angle with the optimal face

image coding with an l sup spl infin norm and confidence interval criteria a new image coding technique based on an l sup spl infin norm criterion and exploiting statistical properties of the reconstruction error is investigated the original image is preprocessed quantized encoded and reconstructed within a given confidence interval two important classes of preprocessing namely linear prediction and iterated filterbanks are used the approach is also shown to be compatible with previous techniques the approach allows a great flexibility in that it can perform lossless coding as well as a controlled lossy one specifications are typically that only p the reconstructed pixels are different from the original ones

the nonlinear geometry of linear programming i affine and projective scaling trajectories this series of papers studies a geometric structure underlying karmarkar s projective scaling algorithm for solving linear programming problems a basic feature of the projective scaling algorithm is a vector field depending on the objective function which is defined on the interior of the polytope of feasible solutions of the linear program the geometric structure studied is the set of trajectories obtained by integrating this vector field which we call ptrajectories we also study a related vector field the affine scaling vector field and its associated trajectories called a trajectories the affine scaling vector field is associated to another linear programming algorithm the affine scaling algorithm affine and projective scaling vector fields are each defined for linear programs of a special form called strict standard form and canonical form respectively this paper derives basic properties of p trajectories and a trajectones it reviews the projective and affine scaling algorithms defines the projective and affine scaling vector fields and gives differential equations for p trajectories and a trajectories it shows that projective transformations map p trajectories into p trajectories it presents karmarkar s interpretation of a trajectories as steepest descent paths of the objective function c x with respect to the riemannian geometry ds2 z dx dx x2 restricted to the relative interior of the polytope of feasible solutions p trajectories of a canonical form linear program are radial projections of a trajectories of an associated standard form linear program as a consequence there is a polynomial time linear programming algorithm using the affine scaling vector field of this associated linear program this algorithm is essentially karmarkar s algorithm these trajectories are studied in subsequent papers by two nonlinear changes of variables called legendre transform coordinates and projective legendre transform coordinates respectively it will be shown that p trajectories have an algebraic and a geometric interpretation they are algebraic curves and they are geodesics actually distinguished chords of a geometry isometric to a hilbert geometry on a polytope combinatorially dual to the polytope of feasible solutions the a trajectories of strict standard form linear programs have similar interpretations they are algebraic curves and are geodesics of a geometry isometric to euclidean geometry received by the editors july 28 1986 and in revised form september 28 1987 and march 21 1988 1980 mathematics subject classification 1985 revision primary 90c05 secondary 52a40 34a34 research of the first author partially supported by onr contract n00014 87 k0214 d 1989 american mathematical society 0002 9947 89 1 00 25 per page

limiting behavior of the affine scaling continuous trajectories for linear programming problems we consider the continuous trajectories of the vector field induced by the primal affine scaling algorithm as applied to linear programming problems in standard form by characterizing these trajectories as solutions of certain parametrized logarithmic barrier families of problems we show that these trajectories tend to an optimal solution which in general depends on the starting point by considering the trajectories that arise from the lagrangian multipliers of the above mentioned logarithmic barrier families of problems we show that the trajectories of the dual estimates associated with the affine scaling trajectories converge to the so called centered optimal solution of the dual problem we also present results related to asymptotic direction of the affine scaling trajectories we briefly discuss how to apply our results to linear programs formulated in formats different from the standard form finally we extend the results to the primal dual amne scaling algorithm

the nonlinear geometry of linear programming ii legendre transform coordinates and central trajectories karmarkar s projective scaling algorithm for solving linear programming problems associates to each objective function a vector field defined in the interior of the polytope of feasible solutions of the problem this paper studies the set of trajectories obtained by integrating this vector field called p trajectories as well as a related set of trajectories called a trajectories the a trajectories arise from another linear programming algorithm the affine scaling algorithm the affine and projective scaling vector fields are each defined for linear programs of a special form called standard form and canonical form respectively these trajectories are studied using a nonlinear change of variables called legendre transform coordinates which is a projection of the gradient of a logarithmic barrier function the legendre transform coordinate mapping is given by rational functions and its inverse mapping is algebraic it depends only on the constraints of the linear program and is a one to one mapping for canonical form linear programs when the polytope of feasible solutions is bounded there is a unique point mapping to zero called the center the a trajectories of standard form linear programs are linearized by the legendre transform coordinate mapping when the polytope of feasible solutions is bounded they are the complete set of geodesics of a riemannian geometry isometric to euclidean geometry each a trajectory is part of a real algebraic curve each p trajectory for a canonical form linear program lies in a plane in legendre transform coordinates the p trajectory through 0 in legendre transform coordinates called the central p trajectory is part of a straight line and is contained in the a trajectory through 0 called the central a trajectory each p trajectory is part of a real algebraic curve the central a trajectory is the locus of centers of a family of linear programs obtained by adding an extra equality constraint of the form c x u it is also the set of minima of a parametrized family of logarithmic barrier functions power series expansions are derived for the central a trajectory which is also the central p trajectory these power series have a simple recursive form and are useful in developing higher order analogues of karmarkar s algorithm a trajectories are defined for a general linear program using this definition it is shown that the limit point x 0 of a central a trajectory on the boundary of the feasible solution polytope p is the center of the unique face of p containing x 0 in its relative interior received by the editors october 8 1986 and in revised form june 9 1987 and march 25 1988 1980 mathematics subject classification 1985 revision primary 90c05 secondary 52a40 34a34 the first author was partially supported by onr contract n00014 87 k0214 1989 american mathematical society 0002 9947 89 1 00 25 per page

global convergence of the affine scaling algorithm for convex quadratic programming in this paper we give a global convergence proof of the second order affine scaling algorithm for convex quadratic programming problems where the new iterate is the point which minimizes the objective function over the intersection of the feasible region with the ellipsoid centered at the current point and whose radius is a fixed fraction beta in 0 1 of the radius of the largest scaled ellipsoid inscribed in the nonnegative orthant the analysis is based on the local karmarkar potential function introduced by tsuchiya for any beta in 0 1 and without assuming any nondegeneracy assumption on the problem it is shown that the sequences of primal iterates and dual estimates converge to optimal solutions of the quadratic program and its dual respectively

long steps in an o n 3 l algorithm for linear programming we consider partial updating in ye s affine potential reduction algorithm for linear programming we show that using a goldstein armijo rule to safeguard a linesearch of the potential function during primal steps is sufficient to control the number of updates we also generalize the dual step construction to apply with partial updating the result is the first o n3l algorithm for linear programming whose steps are not constrained by the need to remain approximately centered the fact that the algorithm has a rigorous primal only initialization actually reduces the complexity to less than o m1 5n1 5l

on the convergence of the affine scaling algorithm the affine scaling algorithm first proposed by dikin is presently enjoying great popularity as a potentially effective means of solving linear programs an outstanding question about this algorithm concerns its convergence in the presence of degeneracy in this paper we give new convergence results for this algorithm that do not require any non degeneracy assumption on the problem in particular we show that if the stepsize choice of either dikin or barnes or vanderbei et al is used then the algorithm generates iterates that converge at least linearly with a convergence ratio of 1 beta sqrt n wheren is the number of variables and 0 1 is a certain stepsize ratio for one particular stepsize choice which is an extension of that of barnes we show that the cost of the limit point is within o 1 of the optimal cost and for sufficiently small roughly proportional to how close the cost of the nonoptimal vertices are to the optimal cost is exactly optimal we prove the latter result by using an unusual proof technique that of analyzing the ergodic convergence of the corresponding dual vectors for the special case of network flow problems with integer data we show that it suffices to take 1 6mc wherem is the number of constraints andc is the sum of the cost coefficients to attain exact optimality

the nonlinear geometry of linear programming iii projective legendre transform coordinates and hilbert geometry this paper studies projective scaling trajectories which are the tra jectories obtained by following the infinitesimal version of karmarkar s linear programming algorithm a nonlinear change of variables projective legendre transform coordinates is introduced to study these trajectories the projective legendre transform mapping has a coordinate free geometric interpretation in terms of the notion of centering by a projective transformation let h be a set of linear programming constraints aj x bj 1 s j s m on rn such that its polytope of feasible solutions ph is bounded and contains 0 in its interior the projective legendre transform mapping i h is given by m a where ph x l b j 1 aj x i x ph x h m ph x x here ph x is the legendre transform coordinate mapping introduced in part ii l h x is a one to one and onto mapping of the interior of the feasible solu tion polytope int ph to the interior of its polar polytope int p the set of projective scaling trajectories with objective function c x co are mappcd under l h to the set of straight line segments in int p passing through the boundary point clco of p as a consequence the projective scaling tra jectories for all objective functions can be interpreted as the complete set of geodesics actually distinguished chords of a projectively invariant metric ge ometry on int ph which is isometric to hilbert geometry on the interior of the polar polytope p

feasible region contraction interior point algorithm abstract in this paper we propose a new interior point algorithm for linear programming by combining the affine scaling algorithm and the primal path following algorithm we define a series of subproblems whose feasible region contracts toward the set of optimal solutions in each iteration we take a step in the steepest descent direction in a transformed space to reduce the objective value our preliminary computational results are very favorable

central paths generalized proximal point methods and cauchy trajectories in riemannian manifolds we study the relationships between three concepts which arise in connection with variational inequality problems central paths defined by arbitrary barriers generalized proximal point methods where a bregman distance substitutes for the euclidean one and cauchy trajectory in riemannian manifolds first we prove that under rather general hypotheses the central path defined by a general barrier for a monotone variational inequality problem is well defined bounded and continuous and converges to the analytic center of the solution set with respect to the given barrier thus generalizing results which deal only with complementarity problems and with the logarithmic barrier next we prove that a sequence generated by the proximal point method with the bregman distance naturally induced by the barrier function converges precisely to the same point furthermore for a certain class of problems including linear programming such a sequence is contained in the central path making the concepts of central path and generalized proximal point sequence virtually equivalent finally we prove that for this class of problems the central path also coincides with the cauchy trajectory in the riemannian manifold defined on the positive orthant by a metric given by the hessian of the barrier i e a curve whose direction at each point is the negative gradient of the objective function at that point in the riemannian metric

best wavelet packet bases in a rate distortion sense a fast rate distortion r d optimal scheme for coding adaptive trees whose individual nodes spawn descendents forming a disjoint and complete basis cover for the space spanned by their parent nodes is presented the scheme guarantees operation on the convex hull of the operational r d curve and uses a fast dynamic programing pruning algorithm to markedly reduce computational complexity applications for this coding technique include r coefman et al s yale univ 1990 generalized multiresolution wavelet packet decomposition iterative subband coders and quadtree structures applications to image processing involving wavelet packets as well as discrete cosine transform dct quadtrees are presented

a polynomial time primal dual affine scaling algorithm for linear and convex quadratic programming and its power series extension we describe an algorithm for linear and convex quadratic programming problems that uses power series approximation of the weighted barrier path that passes through the current iterate in order to find the next iterate if r 1 is the order of approximation used we show that our algorithm has time complexity o n 1 1 r l 1 1 r iterations and o n3 n2r arithmetic operations per iteration where n is the dimension of the problem and l is the size of the input data when r 1 we show that the algorithm can be interpreted as an affine scaling algorithm in the primal dual setup

global convergence of the affine scaling methods for degenerate linear programming problems in this paper we show the global convergence of the affine scaling methods without assuming any condition on degeneracy the behavior of the method near degenerate faces is analyzed in detail on the basis of the equivalence between the affine scaling methods for homogeneous lp problems and karmarkar s method it is shown that the step size 1 8 where the displacement vector is normalized with respect to the distance in the scaled space is sufficient to guarantee the global convergence of the affine scaling methods

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

a wavelet based two stage near lossless coder in this paper we present a two stage near lossless compression scheme it belongs to the class of lossy plus residual coding and consists of a wavelet based lossy layer followed by arithmetic coding of the quantized residual to guarantee a given linfin error bound in the pixel domain we focus on the selection of the optimum bit rate for the lossy layer to achieve the minimum total bit rate unlike other similar lossy plus lossless approaches using a wavelet based lossy layer the proposed method does not require iteration of decoding and inverse discrete wavelet transform in succession to locate the optimum bit rate we propose a simple method to estimate the optimal bit rate with a theoretical justification based on the critical rate argument from the rate distortion theory and the independence of the residual error

an ol n 3 potential reduction algorithm for linear programming we describe a primal dual potential function for linear programming phi x s rho ln x t s sum limits j 1 n ln x j s j where n x is the primal variable ands is the dual slack variable as a result we develop an interior point algorithm seeking reductions in the potential function with rho n sqrt n neither tracing the central path nor using the projective transformation the algorithm converges to the optimal solution set in o sqrt n l iterations and uses o n3l total arithmetic operations we also suggest a practical approach to implementing the algorithm

a variation on karmarkar s algorithm for solving linear programming problems the algorithm described here is a variation on karmarkar s algorithm for linear programming it has several advantages over karmarkar s original algorithm in the first place it applies to the standard form of a linear programming problem and produces a monotone decreasing sequence of values of the objective function the minimum value of the objective function does not have to be known in advance secondly in the absence of degeneracy the algorithm converges to an optimal basic feasible solution with the nonbasic variables converging monotonically to zero this makes it possible to identify an optimal basis before the algorithm converges

on some properties and an application of the logarithmic barrier method we analyze the logarithmic barrier method for nonsmooth convex optimization in the setting of point to set theory this general framework allows us to both extend and include classical results we also propose an application for finding efficient points of nonsmooth constrained convex vector valued problems

hamiltonian structure of dynamical systems which solve linear programming problems abstract a new class of completely integrable hamiltonian systems which solve linear programming problems is introduced the relations with reduced toda flows double bracket equations interior point methods are established an interpretation in terms of total actions on kahler manifolds is given a complete description of the phase portrait including the explicit description of action angle variables is provided

integrability of vector and multivector fields associated with interior point methods for linear programming in the feasible region of a linear programming problem a number of desirably good directions have been defined in connexion with various interior point methods each of them determines a contravariant vector field in the region whose only stable critical point is the optimum point some interior point methods incorporate a two or higher dimensional search which naturally leads us to the introduction of the corresponding contravariant multivector field we investigate the integrability of those multivector fields i e whether a contravariantp vector field isxp forming is enveloped by a family ofxq s q p or envelops a family ofxq s q p in j a schouten s terminology wherexq is aq dimensional manifold immediate consequences of known facts are 1 the directions hitherto proposed arex1 forming with the optimum point of the linear programming problem as the stable accumulation point and 2 there is anx2 forming contravariant bivector field for which the center path is the critical submanifold most of the meaningfulp vector fields withp 3 are notxp forming in general though they envelop that bivector field this observation will add another circumstantial evidence that the bivector field has a kind of invariant significance in the geometry of interior point methods for linear programming for a kind of appendix it is noted that if we have several objectives i e in the case of multiobjective linear programs extension to higher dimensions is easily obtained

improving the rate of convergence of interior point methods for linear programming this paper proposes a procedure for improving the rate of convergence of interior point methods for linear programming if x k is the sequence generated by an interior point method the procedure derives an auxiliary sequence bar x k under the suitable assumptions it is shown that the sequence bar x k converges superlinearly faster to the solution than x k application of the procedure to the projective and afflne scaling algorithms is discussed and some computational illustration is provided

limiting behavior of weighted central paths in linear programming we study the limiting behavior of the weighted central paths x s 0 in linear programming at both 0 and we establish the existence of a partition b n of the index set 1 n such thatx i ands j as fori b andj n andx n s b converge to weighted analytic centers of certain polytopes for allk 1 we show that thekth order derivativesx k ands k converge when 0 and consequently the derivatives of each order are bounded in the interval 0 we calculate the limiting derivatives explicitly and establish the surprising result that all higher order derivatives k 2 converge to zero when

solving convex programs by means of ordinary differential equations within a hilbert space we consider nonsmooth convex programs with sharp constraints examples include all problem instances that are bounded and strictly feasible to solve such programs we pursue an absolutely continuous trajectory generated by a differential inclusion of subgradient type whenever this inclusion offers some freedom of choice we select a steepest descent direction it is shown that the proposed algorithm converges to an optimal solution in finite time

linear programming and the newton barrier flow in this note we report a simple characteristic of linear programming central trajectories which has a surprising consequence specifically we show that given a bounded polyhedral setp with nonempty interior the logarithmic barrier function with no objective component induces a vector field of negative newton directions which flows from the center ofp along central trajectories to solutions of every possible linear program onp

polynomial affine algorithms for linear programming the method of steepest descent with scaling affine scaling applied to the potential functionq logc x i 1n logxi solves the linear programming problem in polynomial time forq n ifq n then the algorithm terminates in no more than o n2l iterations if q n sqrt n withq o n then it takes no more than o nl iterations a modified algorithm using rank 1 updates for matrix inversions achieves respectively o n4l and o n3 5l arithmetic computions

joint moments of the total discounted gains and losses in the renewal risk model with two sided jumps abstract this paper considers a renewal insurance risk model with two sided jumps e g labbe et al 2011 where downward and upward jumps typically represent claim amounts and random gains respectively a generalization of the gerber shiu expected discounted penalty function gerber and shiu 1998 is proposed and analyzed for sample paths leading to ruin in particular we shall incorporate the joint moments of the total discounted costs associated with claims and gains until ruin into the gerber shiu function because ruin may not occur the joint moments of the total discounted claim costs and gain costs are also studied upon ultimate survival of the process general recursive integral equations satisfied by these functions are derived and our analysis relies on the concept of moment based discounted densities introduced by cheung 2013 some explicit solutions are obtained in two examples under different cost functions when the distribution of each claim is exponential or a combination of exponentials while keeping the distributions of the gains and the inter arrival times between successive jumps arbitrary the first example looks at the joint moments of the total discounted amounts of claims and gains whereas the second focuses on the joint moments of the numbers of downward and upward jumps until ruin numerical examples including the calculations of covariances between the afore mentioned quantities are given at the end along with some interpretations