solving problems with semidefinite and related constraints using interior point methods for nonlinear programming in this paper we describe how to reformulate a problem that has second order cone and or semidefiniteness constraints in order to solve it using a general purpose interior point algorithm for nonlinear programming the resulting problems are smooth and convex and numerical results from the dimacs implementation challenge problems and sdplib are provided

fuzzy goal programming complementary slackness conditions and computational schemes abstract goal programming gp is an important category in linear programming and multiobjective versions which minimize the deviation among value of each objective function and its goal with satisfying other constraints of problem in this paper we deal with fuzzy goal programming fgp which is a generalized problem in vague nature where parameters relations and goals are given as fuzzy quantities with respect to imprecise data recently inuiguchi et al m inuiguchi j ramik t tanio m vlach satisficing solutions and duality in interval and fuzzy linear programming fuzzy sets and systems 135 2003 151 177 have proved some important dual theorems on fgp which are very close to the same crisp versions in continuation to their works based on fuzzy extensions of relations we get complementary slackness conditions for fgp which have an essential role in many optimization techniques moreover we obtain the most optimistic and pessimistic satisficing solutions employing these conditions and present a convex combination of them which covers the satisficing solutions for a fixed level of certainty our practical schemes implement an efficient interior point algorithm as sub procedure so they are polynomial time algorithms

complete elastic characterization of viscoelastic materials by dynamic measurements of the complex bulk and young s moduli as a function of temperature and hydrostatic pressure abstract two independent systems to measure the dynamic complex young s and bulk moduli of viscoelastic materials as a function of temperature and hydrostatic pressure are described in the young s modulus system a bar shaped sample is adhered to a piezoelectric shaker and mounted vertically inside an air filled pressure vessel data are obtained using both the traditional resonant approach and a wave speed technique in the bulk modulus system the compressibility of a sample of arbitrary shape immersed in castor oil and placed inside a pressure chamber is measured data can be obtained at frequencies typically ranging from 50 hz to 5 khz at temperatures comprised between 2 and 50 c and under hydrostatic pressures ranging from 0 to 2 mpa young s or 6 5 mpa bulk typical data obtained with both systems are presented and it is shown how these data can be combined to completely characterize the elasticity of the material under investigation in particular they can be used to obtain experimental values of the complex poisson s ratio whose accurate measurement is otherwise quite challenging to perform directly as an example the magnitude and loss tangent of poisson s ratio are presented for a nearly incompressible rubber

simplex and interior point specialized algorithms for solving nonoriented multicommodity flow problems multicommodity network flow models arise in a wide variety of contexts typical among which is the dimensioning of telecommunication networks in this paper we present various approaches based on specialization of the simplex algorithm and interior point methods to solve nonoriented multicommodity flowproblems algorithms are tested with data from the france telecom paris district transmission network first we focus on a specialization for the node arc formulation of the problem a primal simplex and dual affine scaling algorithms exploiting the particular structure of the constraint matrix are presented and compared numerical results are provided for problems up to about 800 000 constraints and 6 000 000 variables however much more powerful approaches based on specialized decomposition methods can be implemented for solving the problem a dantzig wolfe decomposition method is designed and compared with a specialized implementation of the analytic center cutting plane method accpm partitioning techniques are used to exploit the structure of the master programs involved in those methods

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

the poisson s loss factor of solid viscoelastic materials the complex poisson s ratio plays an important role in characterizing the linear dynamic behaviour of solid materials and occurs in a number of equations used for acoustical and vibration calculus the ratio of the imaginary part to the real part of complex poisson s ratio is referred to as poisson s loss factor the magnitude of the poisson s loss factor is investigated in this paper for homogeneous isotropic linear solid viscoelastic materials with positive poisson s ratio the relation of the poisson s loss factor to the material damping is determined it is shown that the magnitude of the poisson s loss factor is approximately proportional to the difference between the shear and bulk loss factors and is a rational fractional function of the dynamic poisson s ratio in addition relationships are developed which enable one to determine the approximate magnitude of the poisson s loss factor from knowledge only of the shear loss factor and the dynamic poisson s ratio it is shown that the poisson s loss factor is smaller than the shear loss factor usually by one order of magnitude at least moreover it is pointed out that the poisson s loss factor of a high loss and a low loss material may be about the same experimental data on two rubbers and a hard plastic are presented to verify the theoretical conclusions

determining the elastic constants of isotropic materials by modal vibration testing of rectangular thin plates abstract the paper presents a procedure whereby poisson s ratio and the dynamic young s modulus of isotropic and homogeneous materials are determined using two of the first four frequencies of natural vibration in thin rectangular plates the procedure is based on suitable approximate relationships relating the resonance frequencies to the elastic constants of the material these relations were derived from those of warburton by taking into account a correction factor obtained by an extensive series of numerical analyses carried out by a finite element code in order to verify the procedure a comparison with reference solutions has been made

an experimental technique for complete dynamic characterization of a viscoelastic material abstract an experimental technique for completely characterizing a viscoelastic material by determining the poisson ratio and the complex dynamic young s modulus of a small beam like specimen subject to seismic excitation is presented in this paper together with the theoretical background the same experimental device is used basically for both kinds of tests the specimen is instrumented placed into a temperature controlled chamber and excited by means of an electrodynamic shaker the longitudinal and the transversal deformations are measured by strain gauges to get the poisson ratio whereas the vertical displacement of the specimen and the acceleration of the support are measured to get young s modulus of the tested material the experimental curves of the poisson ratio and of young s modulus obtained at different temperatures are then gathered into a unique master curve by using the reduced variables method the two master curves respectively represent the poisson ratio and young s modulus for the tested material in a very broad frequency range

splitting dense columns in sparse linear systems abstract we consider systems of equations of the form aa t x b where a is a sparse matrix having a small number of columns which are much denser than the other columns these dense columns in a cause aat to be very or even completely dense which greatly limits the effectiveness of sparse matrix techniques for directly solving the above system of equations in the literature on interior point methods for linear programming the usual technique for dealing with this problem is to split a into a sparse part s and a dense part d a s d and to solve systems involving aat in terms of the solution of systems involving sst using either conjugate gradient techniques or the sherman morrison woodbury formula this approach has the difficulty that sst is often rank deficient even when aat has full rank in this paper we propose an alternative method which avoids the rank deficiency problem and at the same time allows for the effective use of sparse matrix techniques the resulting algorithm is both efficient and robust

generalized fuzzy integrals part i fundamental concepts abstract in this paper we introduce a definition of fuzzy integral which is a generalization of both s fuzzy integral defined by m sugeno and n fuzzy integral due to zhao ruhuaiand discuss the fundamen tal properties of this kind of fuzzy integral

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

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

a note on the space of fuzzy measurable functions for a monotone measure abstract in this paper we give a necessary and sufficient condition to guarantee that the space of all finite measurable functions for a monotonic measure is a topological vector space with a countable local base and in this space convergence with respect to this topology is equivalent to the convergence in measure

interior point algorithms for linear programming with inequality constraints interior methods for linear programming were designed mainly for problems formulated with equality constraints and non negative variables the formulation with inequality constraints has shown to be very convenient for practical implementations and the translation of methods designed for one formulation into the other is not trivial this paper relates the geometric features of both representations shows how to transport data and procedures between them and shows how cones and conical projections can be associated with inequality constraints

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

combining a hybrid preconditioner and a optimal adjustment algorithm to accelerate the convergence of interior point methods abstract in this work the optimal adjustment algorithm for p coordinates which arose from a generalization of the optimal pair adjustment algorithm is used to accelerate the convergence of interior point methods using a hybrid iterative approach for solving the linear systems of the interior point method its main advantages are simplicity and fast initial convergence at each interior point iteration the preconditioned conjugate gradient method is used in order to solve the normal equation system the controlled cholesky factorization is adopted as the preconditioner in the first outer iterations and the splitting preconditioner is adopted in the final outer iterations the optimal adjustment algorithm is applied in the preconditioner transition in order to improve both speed and robustness numerical experiments on a set of linear programming problems showed that this approach reduces the total number of interior point iterations and running time for some classes of problems furthermore some problems were solved only when the optimal adjustment algorithm for p coordinates was used in the change of preconditioners

limit analysis with the dual affine scaling algorithm the collapse state of a rigid plastic material with the linearized mises yield condition is computed we use an infeasible point variant of the dual affine scaling algorithm for linear programming which is extremely efficient for this large sparse and ill conditioned problem for a classical test problem we obtain better results than previously both for the limit load and for the collapse fields thus setting a new standard for computations in limit analysis

toward a general theory of fuzzy variables abstract a general framework for a theory is presented that encompasses both statistical uncertainty which falls within the province of probability theory and nonstatistical uncertainty which relates to the concept of a fuzzy set and possibility theory l a zadeh j fuzzy sets 1 1978 3 28 the concept of a fuzzy integral is used to define the expected value of a random variable properties of the fuzzy expectation are stated and a mean value theorem for the fuzzy integral is proved comparisons between the fuzzy and the lebesgue integral are presented after a new concept of dependence is formulated various convergence concepts are defined and their relationships are studied by using a chebyshev like inequality for the fuzzy integral the possibility of using this theory in bayesian estimation with fuzzy prior information is explored

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

decomposed block cholesky factorization in the karmarkar algorithm solving a class of super large lp problems abstract with the introduction of the karmarkar algorithm hope for optimally solving super large lp problems often encountered in the communication industry had risen high however a general code of the original karmarkar algorithm takes a long computation time to solve such problems employing proper factorization schemes substantially reduces the computation time of the karmarkar algorithm for solving super large linear programming lp problems in this paper a modified version of the karmarkar algorithm is described and the complexities involved in applying the algorithm to a class of super large lp problems are identified a factorization scheme decomposed block cholesky factorization is proposed in order to reduce the complexities of a matrix inversion operation that occurs within every iteration of the karmarkar algorithm the factorization scheme capitalizes on the generally well behaved block structure of the constraint matrix of the super large lp problems the computation time required for solving a class of super large lp problems using this scheme is estimated to be reduced by a factor of two orders of magnitude the proposed factorization scheme is even more promising in solving the super large lp problems on parallel processors using the karmarkar algorithm

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

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

interior point methods for nonconvex nonlinear programming orderings and higher order methods the paper extends prior work by the authors on loqo an interior point algorithm for nonconvex nonlinear programming the specific topics covered include primal versus dual orderings and higher order methods which attempt to use each factorization of the hessian matrix more than once to improve computational efficiency results show that unlike linear and convex quadratic programming higher order corrections to the central trajectory are not useful for nonconvex nonlinear programming but that a variant of mehrotra s predictor corrector algorithm can definitely improve performance

an extension of sugeno integral abstract in this paper we extend the concept of sugeno fuzzy integral from nonnegative fuzzy measurable functions to extended real valued fuzzy measurable functions and discuss the lost genuine properties for this extension several necessary and sufficient conditions of absolute s integrability for extended real valued fuzzy measurable functions are given moreover the space s of all fuzzy measurable functions will be proved to be a pseudo metric space under a necessary and sufficient condition finally as an application of this extension the pettis integral will be established for this kind of fuzzy integral

perfect reconstruction filter banks having linear phase fir filters with equiripple response the design of equiripple linear phase analysis and synthesis fir filters of two channel perfect reconstruction pr filter banks is formulated as the minimization of a weighted peak error under both linear inequality arising from the desired responses of the analysis filters and nonlinear equality pr constraints the effectiveness of a proposed method to solve the design problem a modified dual affine scaling variant of karmarkar s 1989 algorithm and an approximation scheme is illustrated through several design examples

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

a short cut potential reduction algorithm for linear programming as most interior point algorithms iterate they repeatedly perform costly matrix operations such as projections on the entire constraint matrix for large scale linear programming problems such operations consume the great majority of the computation time required however for problems where the number of variables far exceeds the number of constraints operations over the entire constraint matrix are unnecessary we will examine and extend decomposition techniques which greatly reduce the amount of work required by such interior point methods as the dual affine scaling and the dual potential reduction algorithms in an effort to judge the practical viability of the decompositioning we compare the performance of the dual potential reduction algorithm with and without decompositioning over a set of randomly generated transportation problems accompanying a theoretical justification of these techniques we focus on the implementation details and computational results of one such technique

on finding a vertex solution using interior point methods abstract an approach is proposed to generate a vertex solution while using a primal dual interior point method to solve linear programs a controlled random perturbation is made to the cost vector a method to identify the active constraints at the vertex to which the solutions are converging is given this basic method is further refined to save computational effort the proposed approach is tested by using a variation of the primal dual interior point method our method is developed by taking a predictor corrector approach in practice this method takes considerably fewer iterations to solve linear programs than methods described by choi monma and shanno lustig marsten and shanno and domich boggs donaldson and witzgall computational results on problems from the netlib test set are reported to test our approach for finding vertex solutions these results show that one perturbation is enough to force the solutions to converge to a vertex the results indicate that the proposed approach is insensitive to the number of degenerate variables the results also indicate that the effort required to generate a vertex solution is comparable to that required to solve the problem using an interior point method

an extended variant of karmarkar s interior point algorithm abstract in this paper extended algorithm of karmarkar s interior point algorithm is considered theoretical aspects of the new algorithm are discussed we show that the new iterative algorithm converges faster than former karmarkar s algorithms successful convergence for problems of different sizes is obtained numerical results show that when we use a new parameter in the classical karmarkar s algorithm number of iterations is less than the number of iterations for two mentioned methods

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

tatonnement procedures for linearly constrained convex optimization the emphasis in this article is to exploit the fact that precision requirements for solutions of most economic models in practice are moderate only a simple approach is introduced for solving linearly constrained partial equilibrium models based on an iterative scheme similar to the simplex method it allows large scale models to be solved within a practical tolerance efficiently even in a micro computer environment extensions to linearly constrained convex optimization problems are presented finally a set of computational tests on 68 linear programs from the netlib library is reported comparison of our approach with the simplex method using minos 5 1 and with karmarkar s algorithm is reported for moderate precision requirements these preliminary results are highly encouraging

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 primal dual potential reduction method for problems involving matrix inequalities we describe a potential reduction method for convex optimization problems involving matrix inequalities the method is based on the theory developed by nesterov and nemirovsky and generalizes gonzaga and todd s method for linear programming a worst case analysis shows that the number of iterations grows as the square root of the problem size but in practice it appears to grow more slowly as in other interior point methods the overall computational effort is therefore dominated by the least squares system that taust be solved in each iteration a type of conjugate gradient algorithm can be used for this purpose which results in important savings for two reasons first it allows us to take advantage of the special structure the problems often have e g lyapunov or algebraic riccati inequalities second we show that the polynomial bound on the number ofiterations remains valid even if the conjugate gradient algorithm is not run until completion which in practice can greatly reduce the computational effort per iteration we describe in detail how the algorithm works for optimization problems with l lyapunov inequaliues each of size m we prove an overall worst case operation count of o mssll5 the average case complexity appears to be closer to o m4ll5 this estimate is justified by extensive numerical experimentation and is consistent with other researchers experience with the practical performance of interior point algorithms for linear programming this result means that the computational cost of extending current control theory based on the solution of lyapunov or riccati equations to a theory that is based on the solution of multiple coupled lyapunov or riccati inequalities is modest

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

some properties of fuzzy integrable function space l 1 μ abstract we introduce definitions of fuzzy orlicz class l and fuzzy mean convergence we show that fuzzy mean convergence is equivalent to convergence in fuzzy measure and l l1 for any orlicz function this conclusion refines a result of d ralescu toward a general theory of fuzzy variable j math anal appl 86 1982 176 193 moreover a necessary and sufficient condition for l1 s is given it also improves a result of d ralescu and g adams the fuzzy integral j math anal appl 75 1980 562 570

higher order predictor corrector interior point methods with application to quadratic objectives in this paper the authors explore the full utility of mehrotra s predictor corrector method in the context of linear and convex quadratic programs they describe a procedure for doing multiple corrections at each iteration and implement it within the framework of ob1 computational results are provided for the multiple correcting procedure using several strategies for determining the number of corrections in a given iteration the results indicate that iteration counts can be significantly reduced by allowing higher order corrections but at the the cost of extra work per iteration the procedure is shown to be a level m composite newton interior point method where m is the number of corrections performed in an iteration

an efficient method for nonlinearly constrained networks many nonlinear network flow problems in addition to the balance constraints in the nodes and capacity constraints on the arc flows have nonlinear side constraints which specify a flow relationship between several of the arcs in the network flow model the short term hydrothermal coordination of electric power generation is an example of this type in this work we solve this kind of problem using an approach in which the efficiency of the well known techniques for network flow can be preserved it lies in relaxing the side constraints in an augmented lagrangian function and minimizing a sequence of these functions subject only to the network constraints for different estimates of the lagrange multipliers of the side constraints this method gives rise to an algorithm which combines first and superlinear order multiplier methods to estimate these multipliers when the number of free variables is very high we can obtain a superlinear order estimate by means of the limited memory bfgs method fitted to our problem an extensive computational comparison with other methods has been performed the numerical results reported indicate that the algorithm described may be employed advantageously to solve large scale network flow problems with nonlinear side constraints

efficient formulations for pricing under attraction demand models we propose a modeling and optimization framework to cast a broad range of fundamental multi product pricing problems as tractable convex optimization problems we consider a retailer offering an assortment of differentiated substitutable products to a population of customers that are price sensitive the retailer selects prices to maximize profits subject to constraints on sales arising from inventory and capacity availability market share goals bounds on allowable prices and other considerations consumers response to price changes is represented by attraction demand models which subsume the well known multinomial logit mnl and multiplicative competitive interaction demand models our approach transforms seemingly non convex pricing problems both in the objective function and constraints into convex optimization problems that can be solved efficiently with commercial software we establish a condition which ensures that the resulting problem is convex prove that it can be solved in polynomial time under mnl demand and show computationally that our new formulations reduce the solution time from days to seconds we also propose an approximation of demand models with multiple overlapping customer segments and show that it falls within the class of demand models we are able to solve such mixed demand models are highly desirable in practice but yield a pricing problem which appears computationally challenging to solve exactly

minimax design of two channel nonuniform division filterbanks using iir allpass filters the design of two channel linear phase nonuniform division filter ndf banks constructed by infinite impulse response iir digital allpass filters dafs in the sense of l sub spl infin error criteria is considered first the theory of two channel ndf bank structures using two iir dafs is developed then the design problem is appropriately formulated to result in a simple optimization problem utilizing a variant of karmarkar s algorithm we can efficiently solve the optimization problem through a frequency sampling and iterative approximation method to find the coefficients for the iir dafs the resulting two channel ndf banks can possess approximately linear phase response without magnitude distortion the effectiveness of the proposed technique is achieved by forming an appropriate chebyshev approximation of a desired phase response and then to find its solution from a linear subspace in a few iterations several simulation examples are presented for illustration and comparison

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

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

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

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

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

dual versus primal dual interior point methods for linear and conic programming we observe a curious property of dual versus primal dual path following interior point methods when applied to unbounded linear or conic programming problems in dual form while primal dual methods can be viewed as implicitly following a central path to detect primal infeasibility and dual unboundedness dual methods can sometimes implicitly move away from the analytic center of the set of infeasibility unboundedness detectors

generalized fuzzy integrals part 3 convergent theorems abstract in this part we continue the work of preceding papers to consider all kinds of convergent theorems of g fuzzy integrals

a primal dual interior point method for robust optimal control of linear discrete time systems this paper describes how to efficiently solve a robust optimal control problem using recently developed primal dual interior point methods one potential application is model predictive control the optimization problem considered consists of a worst case quadratic performance criterion over a finite set of linear discrete time models subject to inequality constraints on the states and control signals the scheme has been prototyped in matlab to give a rough idea of the efficiencies obtained it is possible to solve problems with more than 10 000 primal variables and 40 000 constraints on a workstation the key to the efficient implementation is an iterative solver in conjunction with a riccati recursion invertible pre conditioner for computing the search directions

fixing variables and generating classical cutting planes when using an interior point branch and cut method to solve integer programming problems abstract branch and cut methods for integer programming problems solve a sequence of linear programming problems traditionally these linear programming relaxations have been solved using the simplex method the reduced costs available at the optimal solution to a relaxation may make it possible to fix variables at zero or one if the solution to a relaxation is fractional additional constraints can be generated which cut off the solution to the relaxation but donot cut off any feasible integer points gomory cutting planes and other classes of cutting planes are generated from the final tableau in this paper we consider using an interior point method to solve the linear programming relaxations we show that it is still possible to generate gomory cuts and other cuts without having to recreate a tableau and we also show how variables can be fixed without using the optimal reduced costs the procedures we develop do not require that the current relaxation be solved to optimality this is useful for an interior point method because early termination of the current relaxation results in an improved starting point for the next relaxation

computational experience with a primal dual interior point method for linear programming abstract a new comprehensive implementation of a primal dual algorithm for linear programming is described it allows for easy handling of simple bounds on the primal variables and incorporates free variables which have not previously been included in a primal dual implementation we discuss in detail a variety of computational issues concerning the primal dual implementation and barrier methods for linear programming in general we show that in a certain way lustig s method for obtaining feasibility is equivalent to newton s method this demonstrates that the method is in some sense the natural way to reduce infeasibility the role of the barrier parameter in computational practice is studied in detail numerical results are given for the entire expanded netlib test set for the basic algorithm and its variants as well as version 5 3 of minos

hopfield neural networks in large scale linear optimization problems abstract hopfield neural networks and affine scaling interior point methods are combined in a hybrid approach for solving linear optimization problems the hopfield networks perform the early stages of the optimization procedures providing enhanced feasible starting points for both primal and dual affine scaling interior point methods thus facilitating the steps towards optimality the hybrid approach is applied to a set of real world linear programming problems the results show the potential of the integrated approach indicating that the combination of neural networks and affine scaling interior point methods can be a good alternative to obtain solutions for large scale optimization problems

on dual convergence of the generalized proximal point method with bregman distances the use of generalized distances e g bregman distances instead of the euclidean one in the proximal point method for convex optimization allows for elimination of the inequality constraints from the subproblems in this paper we consider the proximal point method with bregman distances applied to linearly constrained convex optimization problems and study the behavior of the dual sequence obtained from the optimal multipliers of the linear constraints of each subproblem under rather general assumptions which cover most bregman distances of interest we obtain an ergodic convergence result namely that a sequence of weighted averages of the dual sequence converges to a specific point of the dual optimal set as an intermediate result we prove under the same assumptions that the dual central path generated by a large class of barriers including the generalized bregman distances converges to the same point

frequency dependences of complex moduli and complex poisson s ratio of real solid materials abstract the concept of a complex modulus of elasticity is a powerful and widely used tool for characterizing the linear dynamic elastic and damping properties of solid materials in the frequency domain it is shown in this paper that typical characters of frequency dependences of all complex moduli shear young s etc and complex poisson s ratios of real solid materials can be determined by transforming the causal and real relaxation and creep responses respectively from the time domain into the frequency domain even without having to specify the processes of relaxation and creep it is proved that all dynamic moduli monotonically increase and the dynamic poisson s ratio monotonically decreases with increasing frequency and all respective loss factors pass through at least one maximum these frequency dependences are generally valid for any real solid material regardless of the actual damping mechanisms some experimental results are presented and interpreted in the light of the theory the usefulness of theoretical predictions in materials engineering measurements of dynamic properties and in modelling dynamic behaviour is discussed

computational results of an interior point algorithm for large scale linear programming this paper gives computational results for an efficient implementation of a variant of dual projective algorithm for linear programming the implementation uses the preconditioned conjugate gradient method for computing projections our computational experience reported in this paper indicates that this algorithm has potential as an alternative for solving very large lps in which the direct methods fail due to memory and cpu time requirements the conjugate gradient algorithm was able to find very accurate directions even when the system was ill conditioned the paper also discusses a new mathematical technique called the reciprocal estimates for estimating the primal variables we have conducted extensive computational experiments on problems representative of large classes of applications of current interest we have also chosen instances of the problems of future potential interest which could not be solved in the past due to the weakness of the prior solution methods but which represent a large class of new applications the hypergraph model is such an example comparison of our implementation with minos 5 1 shows that our implementation is orders of magnitude faster than minos 5 1 for these problems