gradient systems in view of information geometry abstract dualistic properties of a gradient flow on a manifold m associated with a dualistic structure g is studied from an information geometrical viewpoint some useful applications are also investigated

double adjacency graphs based discriminant neighborhood embedding discriminant neighborhood embedding dne is a typical graph based dimensionality reduction method and has been successfully applied to face recognition by constructing an adjacency graph aiming to keep the local structure for original data in the subspace it is able to find the optimal discriminant direction effectively not for every sample does dne set up a link between it and its heterogeneous samples when constructing the adjacency graph which would result in a small between class scatter motivated by this fact we develop an extension of dne called double adjacency graphs based discriminant neighborhood embedding dag dne by introducing two adjacency graphs or homogeneous and heterogeneous neighbor adjacency graphs in dag dne neighbors belonging to the same class are compact while neighbors belonging to different classes become separable in the subspace thus dag dne could keep the local structure of a given data and find a good projection matrix for them to investigate the performance of dag dne we compare it with the state of the art dimensionality reduction techniques such as dne and mfa on several publicly available datasets experimental results show the feasibility and effectiveness of the proposed method highlightsa supervised dimensionality method dag dne is proposed for classification tasks dag dne constructs two neighbor adjacency graphs to balance links dag dne keeps the local structure of data and finds a good projection for them experimental results show the feasibility and effectiveness of dag dne

enhanced graph based dimensionality reduction with repulsion laplaceans graph based methods for linear dimensionality reduction have recently attracted much attention and research efforts the main goal of these methods is to preserve the properties of a graph representing the affinity between data points in local neighborhoods of the high dimensional space it has been observed that in general supervised graph methods outperform their unsupervised peers in various classification tasks supervised graphs are typically constructed by allowing two nodes to be adjacent only if they are of the same class however such graphs are oblivious to the proximity of data from different classes in this paper we propose a novel methodology which builds on repulsion graphs i e graphs that model undesirable proximity between points the main idea is to repel points from different classes that are close by in the input high dimensional space the proposed methodology is generic and can be applied to any graph based method for linear dimensionality reduction we provide ample experimental evidence in the context of face recognition which shows that the proposed methodology i offers significant performance improvement to various graph based methods and ii outperforms existing solutions relying on repulsion forces

o n3 noniterative heuristic algorithm for linear programs with error free implementation we present an o n 3 noniterative heuristic algorithm and describe an implementation of p adic arithmetic to solve the linear programming problems error free an optimality test as well as numerical examples depict the scope of such an algorithm in the few cases where it did not lead to the optimal solution it did provide a nonnegative solution of the system ax b which could be used as the basic feasible solution for any one of the widely used simplex methods say the revised simplex algorithm without the need to include artificial variables in ax b for consistency check thus the algorithm could be at least a pre processor for any simplex algorithm and avoids the enhancement of dimensions of the constraint matrix a and consequent additional space computational complexities and errors

a derivative free optimization algorithm based on conditional moments in this paper we propose a derivative free optimization algorithm based on conditional moments for finding the maximizer of an objective function the proposed algorithm does not require calculation or approximation of any order derivative of the objective function the step size in iteration is determined adaptively according to the local geometrical feature of the objective function and a pre specified quantity representing the desired precision the theoretical properties including convergence of the method are presented numerical experiments comparing with the newton quasi newton and trust region methods are given to illustrate the effectiveness of the algorithm

conical projection algorithms for linear programming the linear programming problem is manipulated to be stated as a non linear programming problem in which karmarkar s logarithmic potential function is minimized in the positive cone generated by the original feasible set the resulting problem is then solved by a master algorithm that iteratively rescales the problem and calls an internal unconstrained non linear programming algorithm several different procedures for the internal algorithm are proposed giving priority either to the reduction of the potential function or of the actual cost we show that karmarkar s algorithm is equivalent to this method in the special case in which the internal algorithm is reduced to a single steepest descent iteration all variants of the new algorithm have the same complexity as karmarkar s method but the amount of computation is reduced by the fact that only one projection matrix must be calculated for each call of the internal algorithm

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

design and hybrid realization of fir nyquist filters with quantized coefficients and low sensitivity to timing jitter a design of finite impulse response fir nyquist filters with zero intersymbol interference isi and low sensitivity to timing jitter is presented using an affine scaling linear programming algorithm a near optimum quantized coefficient set can be obtained in a feasible computational time by varying a parameter the design provides a tradeoff between the tail energy of the impulse response in the time domain and the stopband of the magnitude response in the frequency domain we also present a pipelined multiplier free fir filter realization with periodically time varying ptv coefficients based on a hybrid form the realizations exploit the coefficient symmetry to reduce the hardware by about one half by placing most of the shifts followed by addition toward the back end of the structure hardware is reduced due to the shorter wordlength of the adders the proposed structure has a provision to increase the speed by adjusting a design parameter but at the expense of more hardware

karmarkar s linear programming algorithm and newton s method this paper describes a full dimensional version of karmarkar s linear programming algorithm theprojective scaling algorithm which is defined for any linear program in n having a bounded full dimensional polytope of feasible solutions if such a linear program hasm inequality constraints then it is equivalent under an injective affine mappingj n m to karmarkar s original algorithm for a linear program in m havingm n equality constraints andm inequality constraints karmarkar s original algorithm minimizes a potential functiong x and the projective scaling algorithm is equivalent to that version of karmarkar s algorithm whose step size minimizes the potential function in the step direction

an ol n 3 primal interior point algorithm for convex quadratic programming we present a primal interior point method for convex quadratic programming which is based upon a logarithmic barrier function approach this approach generates a sequence of problems each of which is approximately solved by taking a single newton step it is shown that the method requires o sqrt n l iterations and o n3 5l arithmetic operations by using modified newton steps the number of arithmetic operations required by the algorithm can be reduced to o n3l

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

minimum seeking properties of analog neural networks with multilinear objective functions in this paper we study the problem of minimizing a multilinear objective function over the discrete set

an interior newton method for quadratic programming quadratic programming represents an extremely important class of optimization problem in this paper we propose a new interior approach for the general quadratic programming problem we establish that our new method is globally and quadratically convergent published alternative interior approaches do not share such strong convergence properties for the nonconvex case we also report on the results of preliminary numerical experiments the results indicate that the proposed method has considerable prcatival potential

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

an extension of karmarkar projective algorithm for convex quadratic programming we present an extension of karmarkar s linear programming algorithm for solving a more general group of optimization problems convex quadratic programs this extension is based on the iterated application of the objective augmentation and the projective transformation followed by optimization over an inscribing ellipsoid centered at the current solution it creates a sequence of interior feasible points that converge to the optimal feasible solution in o ln iterations each iteration can be computed in o ln3 arithmetic operations wheren is the number of variables andl is the number of bits in the input in this paper we emphasize its convergence property practical efficiency and relation to the ellipsoid method

a polynomial time predictor corrector algorithm for a class of linear complementarity problems a polynomial time algorithm for a class of linear complementarity problems with positive semidefinite matrices is presented the method is based on a one step euler s prediction and one step newton s correction procedure to follow the homotopy path defined as the set x y in mathbb r n x i y i mu i 1 cdots n mu 0 y mx q and solves the problem in o n 3 5 l arithmetic operations moreover after one iteration the value x t y decreases with the ratio at least 1 2 5 sqrt n

a quadratically convergent o k 1 n l iteration algorithm for the p k matrix linear complementarity problem an interior point predictor corrector algorithm for thep matrix linear complementarity problem is proposed the algorithm is an extension of mizuno todd ye s predictor corrector algorithm for linear programming problem the extended algorithm is quadratically convergent with iteration complexity o kappa 1 sqrt n l it is the first polynomially and quadratically convergent algorithm for a class of lcps that are not necessarily monotone

a self correcting version of karmarkar s algorithm a relaxed version of karmarkar s algorithm is developed that does not require the direction of movement to be in the null space of the constraint matrix the algorithm is proved to have the same rate of convergence as karmarkar s algorithm conditions are also given guaranteeing that the approach suggested by karmarkar for transforming an inequality form linear program into the form required by his algorithm is well defined

formulating two stage stochastic programs for interior point methods this paper describes an approach for modeling two stage stochastic programs that yields a form suitable for interior point algorithms a staircase constraint structure is created by replacing first stage variables with sparse split variables in conjunction with side constraints dense columns are thereby eliminated the resulting model is larger than traditional stochastic programs but computational savings are substantial over a tenfold improvement for the problems tested a series of experiments with stochastic networks drawn from financial planning demonstrates the attained efficiencies comparisons with minos and the dual block angular stochastic programming model are provided as benchmarks the split variable approach is applicable to general two stage stochastic programs and other dual block angular models

discriminant neighborhood embedding for classification in this paper a novel subspace learning method called discriminant neighborhood embedding dne is proposed for pattern classification we suppose that multi class data points in high dimensional space tend to move due to local intra class attraction or inter class repulsion and the optimal embedding from the point of view of classification is discovered consequently after being embedded into a low dimensional subspace data points in the same class form compact submanifod whereas the gaps between submanifolds corresponding to different classes become wider than before experiments on the umist and mnist databases demonstrate the effectiveness of our method

theoretical convergence of large step primal dual interior point algorithms for linear programming this paper proposes two sets of rules rule g and rule p for controlling step lengths in a generic primal dual interior point method for solving the linear programming problem in standard form and its dual theoretically rule g ensures the global convergence while rule p which is a special case of rule g ensures the o nl iteration polynomial time computational complexity both rules depend only on the lengths of the steps from the current iterates in the primal and dual spaces to the respective boundaries of the primal and dual feasible regions they rely neither on neighborhoods of the central trajectory nor on potential function these rules allow large steps without performing any line search rule g is especially flexible enough for implementation in practically efficient primal dual interior point algorithms

on monotonicity in the scaled potential algorithm for linear programming in this note we show that a simple modification of ye s affinely scaled potential reduction algorithm makes the method monotone in the true objective on primal steps based on computational experience with the standard form projective algorithm the monotonicity modification should substantially improve the performance of the algorithm when it is initialized with a lower bound much less than the optimal objective value imposing monotonicity on primal steps also results in stronger lower bound updates which is not the case with the standard form projective algorithm

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

a continuation algorithm for a class of linear complementarity problems using an extrapolation technique abstract a polynomial time continuation algorithm is presented for a class of linear complementarity problems with positive semidefinite matrices the linear extrapolation technique is combined with the newton iteration in the predictor corrector procedure of the algorithm to numerically follow the solution curve of the homotopy equations arising from the perturbed karush kuhn tucker condition the convergence rate of the method is proved to be 1 4 7 n after each cycle consisting of one extrapolation between two newton steps

using objective values to start multiple objective linear programming algorithms we introduce in this paper a new starting mechanism for multiple objective linear programming molp algorithms this makes it possible to start an algorithm from any solution in objective space the original problem is first augmented in such a way that a given starting solution is feasible the augmentation is explicitly or implicitly controlled by one parameter during the search process which verifies the feasibility efficiency of the final solution this starting mechanism can be applied either to traditional algorithms which search the exterior of the constraint polytope or to algorithms moving through the interior of the constraints we provide recommendations on the suitability of an algorithm for the various locations of a starting point in objective space numerical considerations illustrate these ideas

a relaxed revision of karmarkar s method a relaxed version of karmarkar s method is developed this method is proved to have the same polynomial time complexity as karmarkar s method and its efficient implementation using inexact projections is discussed computational results obtained using a preliminary implementation of the method are presented which indicate that the method is practicable

on the complexity of a class of projective interior point methods we present a generic projective interior point algorithm for linear programming which includes modified versions of karmarkar s original algorithm and most other projective algorithms that have been proposed as special cases we show that this class of algorithms has a worst case iteration bound of o nl and is closely related to the class of potential reduction interior point methods

face recognition using laplacianfaces we propose an appearance based face recognition method called the laplacianface approach by using locality preserving projections lpp the face images are mapped into a face subspace for analysis different from principal component analysis pca and linear discriminant analysis lda which effectively see only the euclidean structure of face space lpp finds an embedding that preserves local information and obtains a face subspace that best detects the essential face manifold structure the laplacianfaces are the optimal linear approximations to the eigenfunctions of the laplace beltrami operator on the face manifold in this way the unwanted variations resulting from changes in lighting facial expression and pose may be eliminated or reduced theoretical analysis shows that pca lda and lpp can be obtained from different graph models we compare the proposed laplacianface approach with eigenface and fisherface methods on three different face data sets experimental results suggest that the proposed laplacianface approach provides a better representation and achieves lower error rates in face recognition

metric learning by discriminant neighborhood embedding in this paper we learn a distance metric in favor of classification task from available labeled samples multi class data points are supposed to be pulled or pushed by discriminant neighbors we define a discriminant adjacent matrix in favor of classification task and learn a map transforming input data into a new space such that intra class neighbors become even more nearby while extra class neighbors become as far away from each other as possible our method is non parametric non iterative and immune to small sample size sss problem target dimensionality of the new space is selected by spectral analysis in the proposed method experiments on real world data sets demonstrate the effectiveness of our method

a convergence proof for an affine scaling algorithm for convex quadratic programming without nondegeneracy assumptions this paper presents a theoretical result on convergence of a primal affine scaling method for convex quadratic programs it is shown that as long as the stepsize is less than a threshold value which depends on the input data only ye and tse s interior ellipsoid algorithm for convex quadratic programming is globally convergent without nondegeneracy assumptions in addition its local convergence rate is at least linear and the dual iterates have an ergodically convergent property

a globally and quadratically convergent affine scaling method for linear e 1 problems recently various interior point algorithms related to the karmarkar algorithm have been developed for linear programming in this paper we first show how this interior point philosophy can be adapted to the linear l1 problem in which there are no feasibility constraints to yield a globally and linearly convergent algorithm we then show that the linear algorithm can be modified to provide aglobally and ultimatelyquadratically convergent algorithm this modified algorithm appears to be significantly more efficient in practise than a more straightforward interior point approach via a linear programming formulation we present numerical results to support this claim

a polynomial method of approximate centers for linear programming we present a path following algorithm for the linear programming problem with a surprisingly simple and elegant proof of its polynomial behaviour this is done both for the problem in standard form and for its dual problem we also discuss some implementation strategies

computation of heterogeneous object co embeddings from relational measurements dimensionality reduction and data embedding methods generate low dimensional representations of a single type of homogeneous data objects in this work we examine the problem of generating co embeddings or pattern representations from two different types of objects within a joint common space of controlled dimensionality where the only available information is assumed to be a set of pairwise relations or similarities between instances of the two groups we propose a new method that models the embedding of each object type symmetrically to the other type subject to flexible scale constraints and weighting parameters the embedding generation relies on an efficient optimization dispatched using matrix decomposition that is also extended to support multidimensional co embeddings we also propose a scheme of heuristically reducing the parameters of the model and a simple way of measuring the conformity between the original object relations and the ones re estimated from the co embeddings in order to achieve model selection by identifying the optimal model parameters with a simple search procedure the capabilities of the proposed method are demonstrated with multiple synthetic and real world datasets from the text mining domain the experimental results and comparative analyses indicate that the proposed algorithm outperforms existing methods for co embedding generation analysis of arbitrary relational information between heterogeneous objects explicit generation of co embeddings directly from relational measurements high efficiency due to simple matrix decomposition and a small set of parameters simple scheme to implement the model identification of the parameters multiple comparisons with synthetic and real world datasets and existing algorithms

multi manifold discriminant isomap for visualization and classification isomap aims to recover the intrinsic geometric structure of manifold by preserving geodesic distances between all pairs of data points however it is an unsupervised dimensionality reduction method usually using class label information can increase the discriminating capability hence a new supervised isomap is proposed in this paper dubbed multi manifold discriminant isomap mmd isomap first data points are partitioned into different manifolds according to their class label information then mmd isomap aims at seeking an optimal nonlinear subspace to preserve the geometrical structure of each manifold according to the isomap criterion meanwhile to enhance the discriminating capability by maximizing the distances between data points of different manifolds finally the corresponding optimization problem is solved by using a majorization algorithm furthermore two new numerical metrics are designed to measure the performance of dimensionality reduction method in both visualization and classification experiments mmd isomap achieves improved performance over many state of the art methods preserve the geometrical structure and enhance the discriminating capability smacof algorithm is introduced to solve the optimization problem two metrics are designed to evaluate dimensionality reduction methods experiments show the effectiveness of the method in visualization and classification

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

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

comparative analysis of affine scaling algorithms based on simplifying assumptions we analyze several affine potential reduction algorithms for linear programming based on simplifying assumptions we show that under a strong probabilistic assumption regarding the distribution of the data in an iteration the decrease in the primal potential function will be omega rho sqrt log n with high probability compared to the guaranteed 1 2n is a parameter in the potential function andn is the number of variables under the same assumption we further show that the objective reduction rate of dikin s affine scaling algorithm is 1 1 sqrt log n with high probability compared to no guaranteed convergence rate

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

a dantzig wolfe like variant of karmarkar s interior point linear programming algorithm we show that a variant of karmarkar s projective algorithm for linear programming can be viewed as following the approach of dantzig wolfe decomposition at each iteration the current primal feasible solution generates prices which are used to form a simple subproblem the solution to the subproblem is then incorporated into the current feasible solution with a suitable choice of stepsize a constant reduction in potential function is achieved at each iteration we also use our analysis to motivate a new primal simplex pivot rule that is closely related to rules used by e klotz and l schrage

large step path following methods for linear programming part ii potential reduction method the algorithm proposed in this paper is the affine polynomial potential reduction method with new procedures for updating the lower bounds for an optimal solution of the linear programming problem a method is developed for updating the lower bounds by large steps with strict control over the duality gaps associated with each iterate two algorithms are obtained by this approach the first one has complexity o nl iterations and a very simple updating procedure the second one updates the lower bounds at points very near the central trajectory and achieves a complexity of o sqrt n l iterations

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

combining phase i and phase ii in a potential reduction algorithm for linear programming this paper describes an affine potential reduction algorithm for linear programming that simultaneously seeks feasibility and optimality the algorithm is closely related to a similar method of anstreicher the new features are that we use a two dimensional programming problem to derive better lower bounds than anstreicher that our direction finding subproblem treats phase i and phase ii more symmetrically and that we do not need an initial lower bound our method also allows for the generation of a feasible solution so that phase i is terminated during the course of the iterations and we describe two ways to encourage this behavior

recovering optimal dual solutions on karmarkar s polynomial algorithm for linear programming this paper presents a standard form variant of karmarkar s algorithm for linear programming the tecniques of using duality and cutting objective are combined in this variant to maintain polynomial time complexity and to bypass the difficulties found in karmarkar s original algorithm the variant works with problems in standard form and simultaneously generates sequences of primal and dual feasible solutions whose objective function values converge to the unknown optimal value some computational results are also reported

locality preserved maximum information projection dimensionality reduction is usually involved in the domains of artificial intelligence and machine learning linear projection of features is of particular interest for dimensionality reduction since it is simple to calculate and analytically analyze in this paper we propose an essentially linear projection technique called locality preserved maximum information projection lpmip to identify the underlying manifold structure of a data set lpmip considers both the within locality and the between locality in the processing of manifold learning equivalently the goal of lpmip is to preserve the local structure while maximize the out of locality global information of the samples simultaneously different from principal component analysis pca that aims to preserve the global information and locality preserving projections lpps that is in favor of preserving the local structure of the data set lpmip seeks a tradeoff between the global and local structures which is adjusted by a parameter alpha so as to find a sub space that detects the intrinsic manifold structure for classification tasks computationally by constructing the adjacency matrix lpmip is formulated as an eigenvalue problem lpmip yields orthogonal basis functions and completely avoids the singularity problem as it exists in lpp further we develop an efficient and stable lpmip qr algorithm for implementing lpmip especially on high dimensional data set theoretical analysis shows that conventional linear projection methods such as weighted pca maximum margin criterion mmc linear discriminant analysis lda and lpp could be derived from the lpmip framework by setting different graph models and constraints extensive experiments on face digit and facial expression recognition show the effectiveness of the proposed lpmip method

exploiting special structure in a primal dual path following algorithm a primal dual path following algorithm that applies directly to a linear program of the form min ctx ax b hx u x 0 x n is presented this algorithm explicitly handles upper bounds generalized upper bounds variable upper bounds and block diagonal structure we also show how the structure of time staged problems and network flow problems can be exploited especially on a parallel computer finally using our algorithm we obtain a complexity bound of o sqrt n ds2 log nk fortransportation problems withs origins d destinations s

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

label propagation based supervised locality projection analysis for plant leaf classification the label propagation has the benefits of nearly linear running time and easy implementation in this paper we make use of the label propagation to propose a new weight measure and present a supervised locality projection analysis slpa method for plant leaf classification firstly we apply warshall algorithm to label propagation and get the label matrix then incorporate it into the weight which has a clear physical meaning secondly multi class data points in high dimensional space are to be pulled or pushed by discriminant neighbors to form an optimum projecting to low dimensionality finally the experimental results on two plant leaf databases show that the proposed method is quite effective and feasible

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