exterior point algorithms for nearest points and convex quadratic programs we consider the problem of finding the nearest point by euclidean distance in a simplicial cone to a given point and develop an exterior penalty algorithm for it each iteration in the algorithm consists of a single newton step following a reduction in the value of the penalty parameter proofs of convergence of the algorithm are given various other versions of exterior penalty algorithms for nearest point problems in nonsimplicial polyhedral cones and for convex quadratic programs all based on a single descent step following a reduction in the value of the penalty parameter per iteration are discussed the performance of these algorithms in large scale computational experiments is very encouraging it shows that the number of iterations grows very slowly if at all with the dimension of the problem

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

an affine scaling interior trust region method for lc1 minimization subject to bounds on variables in this literature we extend the classical affine scaling interior trust region algorithm for smooth bounded constrained nonlinear programming to the nonsmooth case where the objective function is only locally lipschitzian we propose and analyze a new affine scaling trust region method in association with nonmonotonic interior backtracking line search technique for solving the linear equality constrained lc 1 optimization subject to bounds on variables at each iteration the trust region subproblem is defined by minimizing a quadratic function which is required both first and second order information of the objective function and a quadratic affine scaling matrix model subject only to an ellipsoidal constraint in a null space of the affine scaling linear equality constraints the second order derivative of the objective function is explicitly required to be lipschitzian each iterate switches to a very general setting of computing trial backtracking steps generated by the general trust region subproblem and then the line search technique projects the trial backtracking steps onto the strictly feasible set of the bounded constraints on variables the global convergence and fast local convergence rate of the proposed algorithm are established under some reasonable conditions where twice smoothness of the objective function is not required a nonmonotonic criterion should bring about speeding up the convergence progress in some ill conditioned cases applications of the algorithm to various nonsmooth optimization problems are discussed

a class of linear complementarity problems solvable in polynomial time we describe a condition number for the linear complementarity problem lcp which characterizes the degree of difficulty for its solution when a potential reduction algorithm is used consequently we develop a class of lcps solvable in polynomial time the result suggests that the convexity or positive semidefiniteness of the lcp may not be the basic issue that separates lcps solvable and not solvable in polynomial time

improved complexity using higher order correctors for primal dual dikin affine scaling in this paper we show that the primal dual dikin affine scaling algorithm for linear programming of jansen roos and terlaky enhances an asymptotical o sqrt n l complexity by using corrector steps we also show that the result remains valid when the method is applied to positive semi definite linear complementarity problems

nonmonotonic back tracking trust region interior point algorithm for linear constrained optimization in this paper we modify the trust region interior point algorithm proposed by bonnans and pola in siam j optim 7 3 1997 717 for linear constrained optimization a mixed strategy using both trust region and line search techniques is adopted which switches to back tracking steps when a trial step produced by the trust region subproblem may be unacceptable the global convergence and local convergence rate of the improved algorithm are established under some reasonable conditions a nonmonotonic criterion is used to speed up the convergence progress in some ill conditioned cases the results of numerical experiments are reported to show the effectiveness of the proposed algorithm

the three dimensional ising model revisited numerically several basic universal amplitude ratios are studied afresh for three dimensional nearest neighbor ising models in revising earlier work modern estimates of the critical temperature and exponents are used in conjunction with biased inhomogeneous differential approximants to extrapolate the longest available series expansions to find the critical amplitudes c for the susceptibility 1 for the correlation length 1 a for the specific heat c t and b for the spontaneous magnetization m0 we find c c 4 95 0 15 1 1 1 960 0 01 a a 0 523 0 009 a c b 0 0581 0 0010 while a 1 3 verifies hyperuniversality to within 0 8 a method for calculating amplitude ratios which allows for corrections to scaling yields estimates for c c and 1 1 in excellent agreement with those derived from the individual amplitudes finally explicit formulae are given for the numerical evaluation of t 1 t c t and m0 t over the full temperature range from criticality to t 0 and corresponding plots and convenient near critical representations are also presented

chaotic behavior of the affine scaling algorithm for linear programming the affine scaling algorithm for linear programming involves a step size parameter t that must be chosen in the interval 0 1 it is known that the algorithm converges to an optimal solution for values of t le 2 3 in this paper we examine the behavior of the algorithm for values of t 2 3 we show that for certain values of t in this range the algorithm can exhibit chaotic behavior

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

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

projective transformations for interior point algorithms and a superlinearly convergent algorithm for the w center problem the purpose of this study is to broaden the scope of projective transformation methods in mathematical programming both in terms of theory and algorithms we start by generalizing the concept of the analytic center of a polyhedral system of constraints to the w center of a polyhedral system which stands for weighted center where there is a positive weight on the logarithmic barrier term for each inequality constraint defining the polyhedronx we prove basic results regarding contained and containing ellipsoids centered at the w center of the systemx we next shift attention to projective transformations and we exhibit an elementary projective transformation that transforms the polyhedronx to another polyhedronz and that transforms the current interior point to the w center of the transformed polyhedronz we work throughout with a polyhedral system of the most general form namely both inequality and equality costraints

on affine scaling algorithms for nonconvex quadratic programming we investigate the use of interior algorithms especially the affine scaling algorithm to solve nonconvex indefinite or negative definite quadratic programming qp problems although the nonconvex qp with a polytope constraint is a hard problem we show that the problem with an ellipsoidal constraint is easy when the hard qp is solved by successively solving the easy qp the sequence of points monotonically converge to a feasible point satisfying both the first and the second order optimality conditions

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

a logarithmic barrier approach for linear programming this paper presents a logarithmic barrier method for solving a linear programming problem we are interested in computation of the direction by the newtons method and in computation of the displacement step using majorant functions instead line search methods in order to reduce the computation cost this purpose is confirmed by numerical experiments showing the efficiency of our approach which are presented in the last section of this paper

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

approximation algorithms for indefinite quadratic programming we consider approximation schemes for indefinite quadratic programming we argue that such an approximation can be found in polynomial time for fixed andt wheret denotes the number of negative eigenvalues of the quadratic term our algorithm is polynomial in 1 for fixedt and exponential int for fixed we next look at the special case of knapsack problems showing that a more efficient polynomial int approximation algorithm exists

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

series analysis of the 3 state potts model in 2 1 dimensions both high temperature and low temperature series are used to locate and characterize the first order transition in the 3 state potts model in 2 1 dimensions on both square and triangular lattices estimates are presented for the vacuum energy latent heat magnetization susceptibility and mass gap at the transition the spontaneous magnetization and latent heat appear to display an approximate universality at this weak first order transition

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

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

primal dual affine scaling algorithms fail for semidefinite programming in this paper we give an example of a semidefinite programming problem in which primal dual affine scaling algorithms using the hrvw ksh m mt and aho directions fail we prove that each of these algorithms can generate a sequence converging to a non optimal solution and that for the aho direction even its associated continuous trajectory can converge to a non optimal point in contrast with these directions we show that the primal dual affine scaling algorithm using the nt direction for the same semidefinite programming problem always generates a sequence converging to the optimal solution both primal and dual problems have interior feasible solutions and unique optimal solutions which satisfy strict complementarity and are nondegenerate everywhere

extension of a projective interior point method for linearly constrained convex programming in this paper we propose an interior point method of type projective to minimize a nonlinear function under linear constraints we combine the approach of linearization with ingredients brought by karmarkar the theoretical results deduct of those established later

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

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

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

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

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

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

affine scaling for linear programs with free variables the affine scaling modification of karmarkar s algorithm is extended to solve problems with free variables this extended primal algorithm is used to prove two important results first the geometrically elegant feasibility algorithm proposed by chandru and kochar is the same algorithm as the one obtained by appending a single column of residuals to the constraint matrix second the dual algorithm as first described by adler et al is the same as the extended primal algorithm applied to the dual

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

computation of the collapse state in limit analysis using the lp primal affine scaling algorithm abstract the object of this work is twofold the first goal is to demonstrate that for the duality problem of limit analysis with linearized yield condition the lp primal affine scaling algorithm shows properties which are significantly different from those of the simplex method and that as a consequence of this better results can be obtained the second goal is to compute the collapse state for a classical and hard problem in plane strain tension of a rectangular bar with symmetric thin cuts both the collapse multiplier and collapse fields are computed to a better accuracy and detail than previously

a polynomial arc search interior point algorithm for convex quadratic programming abstract arc search is developed for linear programming in 24 25 the algorithms search for optimizers along an ellipse that is an approximation of the central path in this paper the arc search method is applied to primal dual path following interior point method for convex quadratic programming a simple algorithm with iteration complexity o n log 1 is devised several improvements on the simple algorithm which improve computational efficiency increase step length and further reduce duality gap in every iteration are then proposed and implemented it is intuitively clear that the iteration with these improvements will reduce the duality gap more than the iteration of the simple algorithm without the improvements though it is hard to show how much these improvements reduce the complexity bound the proposed algorithm is implemented in matlab and tested on quadratic programming problems originating from 13 the result is compared to the one obtained by loqo in 22 the proposed algorithm uses fewer iterations in all these problems and the number of total iterations is 27 fewer than the one obtained by loqo this preliminary result shows that the proposed algorithm is promising

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

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 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

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

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

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

duality and sensitivity in nonconvex quadratic optimization over an ellipsoid abstract in this paper a duality framework is discussed for the problem of optimizing a nonconvex quadratic function over an ellipsoid additional insight is obtained from the observation that this nonconvex problem is in a sense equivalent to a convex problem of the same type from which known necessary and sufficient conditions for optimality readily follow based on the duality results some existing solution procedures are interpreted as in fact solving the dual the duality relations are also shown to provide a natural framework for sensitivity analysis

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

on some properties of quadratic programs with a convex quadratic constraint in this paper we consider the problem of minimizing a possibly nonconvex quadratic function with a quadratic constraint we point out some new properties of the problem in particular in the first part of the paper we show that i given a kkt point that is not a global minimizer it is easy to find a better feasible point ii strict complementarity holds at the local nonglobal minimizer in the second part of this paper we show that the original constrained problem is equivalent to the unconstrained minimization of a piecewise quartic merit function using the unconstrained formulation we give in the nonconvex case a new second order necessary condition for global minimizers in the third part of this paper algorithmic applications of the preceding results are briefly outlined and some preliminary numerical experiments are reported

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

an interior point algorithm of o m ln e iterations for c 1 convex programming we present a theoretical result on a path following algorithm for convex programs the algorithm employs a nonsmooth newton subroutine it starts from a near center of a restricted constraint set performs a partial nonsmooth newton step in each iteration and converges to a point whose cost is within accuracy of the optimal cost in o sqrt m left ln varepsilon right iterations wherem is the number of constraints in the problem unlike other interior point methods the analyzed algorithm only requires a first order lipschitzian condition and a generalized hessian similarity condition on the objective and constraint functions therefore our result indicates the theoretical feasibility of applying interior point methods to certainc1 optimization problems instead ofc2 problems since the complexity bound is unchanged compared with similar algorithms forc2 convex programming the result shows that the smoothness of functions may not be a factor affecting the complexity of interior point methods

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

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

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

an affine scaling projective reduced hessian algorithm for minimum optimization with nonlinear equality and linear inequality constraints abstract in this paper we propose a nonmonotonic interior point backtracking strategy to modify the reduced projective affine scaling trust region algorithm for solving minimum optimization subject to both nonlinear equality and linear inequality constraints the general full trust region subproblem for solving the minimum optimization is decomposed to a pair of trust region subproblems in horizontal and vertical subspaces of linearize equality constraints and extended affine scaling equality constraints by qr decomposition of an affine scaling matrix and an orthonormal basis on the null subspace the horizontal subproblem in the proposed algorithm is defined by minimizing a quadratic projective reduced hessian function subject only to an ellipsoidal trust region constraint while the vertical subproblem is also defined by the least squares subproblem subject only to an ellipsoidal trust region constraint combining trust region strategy with line search technique will switch to strictly feasible interior point step generated by a component direction of the two trust region subproblems by adopting the l1 penalty function as the merit function the global convergence and fast local convergence rate of the proposed algorithm are established under some reasonable conditions the second order correction step and a nonmonotonic criterion are used to overcome maratos effect and speed up the convergence progress in some ill conditioned cases respectively

partial affine scaling for linearly constrained minimization we propose an interior point method for finding a stationary point of a nonlinear program with linear equality and nonnegativity constraints this method maintains a basis at each iteration and updates the iterate by taking an affine scaling step in the space of nonbasic variables in the general case we propose to choose the basis according to a rule that maximizes the basic components of the iterate in the case where the feasible region is the product of simplices we propose an alternative rule that minimizes the basic components of the cost gradient we analyze the convergence of the method under each rule a key feature of this method is that it can be implemented much like the simplex method

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

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

approximate universality at a first order transition the three state potts model in 3 1 dimensions both high and low temperature perturbation series are used to locate and characterize the first order transition in the three state potts model in 3 1 dimensions on the simple cubic body centred cubic and face centred cubic lattices estimates are presented for the vacuum energy latent heat magnetization susceptibility and mass gap at the transition the results show a remarkable degree of universality between the different lattices and a law of corresponding states is followed very closely

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

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

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

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

an o n 3 l primal dual potential reduction algorithm for solving convex quadratic programs in this paper we combine partial updating and an adaptation of anstreicher s safeguarded linesearch of the primal dual potential function with kojima mizuno and yoshise s potential reduction algorithm for the linear complementarity problem to obtain an o n3l algorithm for convex quadratic programming our modified algorithm is a long step method that requires at most o sqrt n l steps

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

best k digit rational approximation of irrational numbers pre computer versus computer era abstract the best k digit rational approximation for a given irrational number where the numerator has k digits is presented such an approximation allows better error free computation in the real world scenario than that permitted by the k digit decimal approximation of the irrational number it is pointed out that such a best k digit rational approximation though exponential has been made possible due to current high speed computation performed by a digital computer to appreciate the importance of such an approximation we have also focused on the rational approximation not always the best one of some of the famous irrational numbers such as the and the exponential function e by the ingenuity of super mathematicians in several parts of the world during the pre computer era spanning over centuries

a strategy of global convergence for the affine scaling algorithm for convex semidefinite programming the affine scaling algorithm is one of the earliest interior point methods developed for linear programming this algorithm is simple and elegant in terms of its geometric interpretation but it is notoriously difficult to prove its convergence it often requires additional restrictive conditions such as nondegeneracy specific initial solutions and or small step length to guarantee its global convergence this situation is made worse when it comes to applying the affine scaling idea to the solution of semidefinite optimization problems or more general convex optimization problems in math program 83 1 3 393 406 1998 muramatsu presented an example of linear semidefinite programming for which the affine scaling algorithm with either short or long step converges to a non optimal point this paper aims at developing a strategy that guarantees the global convergence for the affine scaling algorithm in the context of linearly constrained convex semidefinite optimization in a least restrictive manner we propose a new rule of step size which is similar to the armijo rule and prove that such an affine scaling algorithm is globally convergent in the sense that each accumulation point of the sequence generated by the algorithm is an optimal solution as long as the optimal solution set is nonempty and bounded the algorithm is least restrictive in the sense that it allows the problem to be degenerate and it may start from any interior feasible point

a new penalty function algorithm for convex quadratic programming in this paper we develop an exterior point algorithm for convex quadratic programming using a penalty function approach each iteration in the algorithm consists of a single newton step followed by a reduction in the value of the penalty parameter the points generated by the algorithm follow an exterior path that we define convergence of the algorithm is established the proposed algorithm was motivated by the work of al sultan and murty on nearest point problems a special quadratic program a preliminary implementation of the algorithm produced encouraging results in particular the algorithm requires a small and almost constant number of iterations to solve the small to medium size problems tested

latent heats from finite size scaling a method of obtaining the latent heat at a first order transition from finite lattice matrix elements is discussed and demonstrated for the case of the 1 1 d potts model the method is shown to provide smoothly convergent finite lattice sequences the structure of the finite lattice hamiltonian eigenvalues is exhibited and a characteristic signal of the first order transition is proposed the finite size scaling behaviour at the transition is also discussed

convergence analysis of the projective scaling algorithm based on a long step homogeneous affine scaling algorithm in this paper we give a new convergence analysis of a projective scaling algorithm we consider a long step affine scaling algorithm applied to a homogeneous linear programming problem obtained from the original linear programming problem this algorithm takes a fixed fraction 2 3 of the way towards the boundary of the nonnegative orthant at each iteration the iteration sequence for the original problem is obtained by pulling back the homogeneous iterates onto the original feasible region with a conical projection which generates the same search direction as the original projective scaling algorithm at each iterate the recent convergence results for the long step affine scaling algorithm by the authors are applied to this algorithm to obtain some convergence results on the projective scaling algorithm specifically we will show i polynomiality of the algorithm with complexities of o nl and o n2l iterations for 2 3 and 2 3 respectively ii global covnergence of the algorithm when the optimal face is unbounded iii convergence of the primal iterates to a relative interior point of the optimal face iv convergence of the dual estimates to the analytic center of the dual optimal face and v convergence of the reduction rate of the objective function value to 1

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

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

algorithms for the solution of quadratic knapsack problems we consider the problem of minimizing a quadratic function with a knapsack constraint quadratic knapsack problems have numerous applications including the least distance problem quadratic programming defined on the convex hull of a set of points and the maximum clique problem we propose and analyze three algorithms for solving quadratic knapsack problems two algorithms are based on recently developed interior point methods the first solves convex quadratic problems and the second computes a stationary point for the indefinite case for both computational results on a variety of test problems are presented the third algorithm based on simplicial partitioning and convex underestimating functions can be used to compute the global optimum of indefinite quadratic knapsack problems

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

a hybrid method for the nonlinear least squares problem with simple bounds abstract this paper presents a new method with trust region technique for solving the nonlinear least squares problem with lower and upper bounds on the variables the proposed method constructs trust region constraints that are ellipses centered at the iterative points in such a way that they lie in the interior of the feasible region thus the method belongs to the class of interior point methods and hence we may expect that the generated sequence approaches a solution smoothly without the combinatorial complications inherent to traditional active set methods we establish a convergence theorem for the proposed method and show its practical efficiency by numerical experiments