density evolution for deterministic generalized product codes on the binary erasure channel at high rates generalized product codes gpcs are extensions of product codes pcs where code symbols are protected by two component codes but not necessarily arranged in a rectangular array we consider a deterministic construction of gpcs as opposed to randomized code ensembles and analyze the asymptotic performance over the binary erasure channel under iterative decoding our code construction encompasses several classes of gpcs previously proposed in the literature such as irregular pcs blockwise braided codes and staircase codes it is assumed that the component codes can correct a fixed number of erasures and that the length of each component code tends to infinity we show that this setup is equivalent to studying the behavior of a peeling algorithm applied to a sparse inhomogeneous random graph using a convergence result for these graphs we derive the density evolution equations that characterize the asymptotic decoding performance as an application we discuss the design of irregular gpcs employing a mixture of component codes with different erasure correcting capabilities

on the global convergence of trust region algorithms for unconstrained minimization many trust region algorithms for unconstrained minimization have excellent global convergence properties if their second derivative approximations are not too large 2 we consider how large these approximations have to be if they prevent convergence when the objective function is bounded below and continuously differentiable thus we obtain a useful convergence result in the case when there is a bound on the second derivative approximations that depends linearly on the iteration number

second order design sensitivity analysis of structural systems esign sensitivity analysis methods for calculation of first derivatives of structural performance with respect to design parameters have become quite well developed and applied 1 2 it is the purpose of this note to extend the adjoint method of design sensitivity analysis to calculation of second derivatives of performance with respect to design parameters the value of second deriva tive information in design optimization is well known using only first order gradient information optimization algorithms must resort to onedimensional search or heuristic step size selection methods if second derivative information is available at a reasonable computing cost much more powerful iterative optimization algorithms are possible 3 finite dimensional matrix structural models are considered here even though the methods presented can be extended to distributed parameter boundary value problem structural models structural design is presumed to be described by a vector of design parameters b b1 b2 bk t structural response is described by a vector of nodal displacement coordinates z zi z2 zn 7

extensions and traces of functions of bounded variation on metric spaces abstract in the setting of a metric space equipped with a doubling measure and supporting a poincare inequality and based on results by bjorn and shanmugalingam 2007 7 we show that functions of bounded variation can be extended from any bounded uniform domain to the whole space closely related to extensions is the concept of boundary traces which have previously been studied by hakkarainen et al 2014 17 on spaces that satisfy a suitable locality condition for sets of finite perimeter we establish some basic results for the traces of functions of bounded variation our analysis of traces also produces novel pointwise results on the behavior of functions of bounded variation in their jump sets

neumann problem for p laplace equation in metric spaces using a variational approach existence boundedness and boundary regularity we employ a variational approach to study the neumann boundary value problem for the p laplacian on bounded smooth enough domains in the metric setting and show that solutions exist and are bounded the boundary data considered are borel measurable bounded functions we also study boundary continuity properties of the solutions one of the key tools utilized is the trace theorem for newton sobolev functions and another is an analog of the de giorgi inequality adapted to the neumann problem

a new first order interior feasible direction method for structural optimization a new interior feasible direction method for optimization problems with linear objective functions but non linear constraints is proposed the method is particularly suitable for weight minimization problems in structural design where the design variables are the cross sectional areas of members and arbitrary stress and displacement constraints are applied four truss structures are analysed and in each case the minimum weight consistent with or lower than that currently available in the literature is achieved although still expensive in its current version the method is comparable in cost and robustness with other methods such as the trust region multiplier method apart from one tolerance value a reconnaissance parameter needs to be specified the fact that the same value of the parameter was used for all the examples and that the method is relatively insensitive to the choice thereof confirms the robustness of the method

trust region model management in multidisciplinary design optimization a common engineering practice is the use of approximation models in place of expensive computer simulations to drive a multidisciplinary design process based on nonlinear programming techniques the use of approximation strategies is designed to reduce the number of detailed costly computer simulations required during optimization while maintaining the pertinent features of the design problem this paper overviews the current state of the art in model management strategies for approximate optimization model management strategies coordinate the interaction between the optimization and the fidelity of the approximation models so as to ensure that the process converges to a solution of the original design problem approximations play an important role in multidisciplinary design optimization mdo by offering system behavior information at a relatively low cost most approximate mdo strategies are sequential in which an optimization of an approximate problem subject to design variable move limits is iteratively repeated until convergence the move limits or trust region are imposed to restrict the optimization to regions of the design space in which the approximations provide meaningful information in order to insure convergence of the sequence of approximate optimizations to a karush kuhn tucker solution a trust region model management or move limit strategy is required in this paper recent developments in approximate mdo strategies and issues of trust region model management in mdo are reviewed

multiplier methods for engineering optimization multiplier methods used to solve the constrained engineering optimization problem are described these methods solve the problem by minimizing a sequence of unconstrained problems defined using the cost and constraint functions the methods proposed in 1969 have been determined to be quite robust although not as efficient as other algorithms they can be more effective for some engineering applications such as optimum design and control oflarge scale dynamic systems since 1969 several modifications and extensions of the methods have been developed therefore it is important to review the theory and computational procedures of these methods so that more efficient and effective ones can be developed for engineering applications recent methods that are similar to the multiplier methods are also discussed these are continuous multiplier update exact penalty and exponential penalty methods

efficient erasure correcting codes we introduce a simple erasure recovery algorithm for codes derived from cascades of sparse bipartite graphs and analyze the algorithm by analyzing a corresponding discrete time random process as a result we obtain a simple criterion involving the fractions of nodes of different degrees on both sides of the graph which is necessary and sufficient for the decoding process to finish successfully with high probability by carefully designing these graphs we can construct for any given rate r and any given real number spl epsiv a family of linear codes of rate r which can be encoded in time proportional to ln 1 spl epsiv times their block length n furthermore a codeword can be recovered with high probability from a portion of its entries of length 1 spl epsiv rn or more the recovery algorithm also runs in time proportional to n ln 1 spl epsiv our algorithms have been implemented and work well in practice various implementation issues are discussed

non monotone trust region algorithms for nonlinear optimization subject to convex constraints this paper presents two new trust region methods for solving nonlinear optimization problems over convex feasible domains these methods are distinguished by the fact that they do not enforce strict monotonicity of the objective function values at successive iterates the algorithms are proved to be convergent to critical points of the problem from any starting point extensive numerical experiments show that this approach is competitive with the lancelot package

optimal structural design considering flexibility a topology optimization method based on the homogenization method the homogenization design method has been successfully applied to a variety of optimization problems such as the stiffness maximization problem and the eigen frequency maximization problem in this study a methodology to obtain the optimal structure design considering flexibility is developed as a new extension of the homogenization design method first flexibility is formulated using the mutual energy concept second a new multi objective function is proposed to obtain optimal solutions incorporating flexibility and stiffness next the topology optimization procedure is constructed using the homogenization method and sequential linear programming slp finally some examples are presented to confirm that the methodology presented here can provide flexible structures satisfying the problem specifications

generalized ldpc codes and generalized stopping sets a generalized low density parity check code gldpc is a low density parity check code in which the constraint nodes of the code graph are block codes rather than single parity checks in this paper we study gldpc codes which have bch or reed solomon codes as subcodes under bounded distance decoding bdd the performance of the proposed scheme is investigated in the limit case of an infinite length cycle free code used over a binary erasure channel bec and the corresponding thresholds for iterative decoding are derived the performance of the proposed scheme for finite code lengths over a bec is investigated as well structures responsible for decoding failures are defined and a theoretical analysis over the ensemble of gldpc codes which yields exact bit and block error rates of the ensemble average is derived unfortunately this study shows that gldpc codes do not compare favorably with their ldpc counterpart over the bec fortunately it is also shown that under certain conditions objects identified in the analysis of gldpc codes over a bec and the corresponding theoretical results remain useful to derive tight lower bounds on the performance of gldpc codes over a binary symmetric channel bsc simulation results show that the proposed method yields competitive performance with a good decoding complexity trade off for the bsc

optimal design of frame structures using multivariate spline approximation the objective of this work is to demonstrate the efficiency and applicability of the optimization algorithm using a multivariate spline approximation in the present algorithm based on the function values and first order derivatives of the constraints available at the intermediate points of optimization an explicit approximation of constraint functions is created by using the least squares spline algorithm the nonlinearity of the function is adaptively updated using feedback information from the previous two iterations for finding the order of spline approximation in addition constraint deletion and design variable linking concepts are employed in solving the approximate problem by using the quasianalytical sequential quadratic programming and dual methods the behavior constraints include stresses displacements and local buckling in the optimum design of frame structures to demonstrate the broad applicability of the optimization algorithm the cross sectional dimensions are directly selected as the design variables for frame problems having thin walled rectangular and tube cross sectional members and the cross sectional areas are selected as design variables for an section problem

on the robustness and efficiency of the sam algorithm for structural optimization a parameter study involving a feasibility factor adaptive tolerance refinement and active set and move limits is conducted on the interior feasible direction method with successive approximations sam the study involves the sizing design of five structural problems of which one is a large power transmission tower to be designed in adherence with a standard design code it is concluded that a move limit is dispensable for this application while adaptive tolerance refinement makes the algorithm more efficient optimal ranges for the feasibility factor and active set limits are suggested this study reaffirms and quantifies the exceptional stability and robustness of the sam algorithm

higher order failure probability calculation using nonlinear approximations use of approximations for calculating the structural failure probability under uncertainties is a necessary aspect but efficient use of suitable approximations at different stages of reliability prediction and design makes this valuable tool practical for many large scale structures the objective of this paper is to address the modern approximation concepts and their utility in structural reliability estimation emphasis is placed on building the higher order approximations using intervening variables to closely represent the nonlinear limit state functions the most probable failure point mpp search can be carried either in x space or u space for a wide variety of random variable distributions several test problems with highly nonlinear performance functions are used to demonstrate the accuracy of the proposed method

braided block codes a new class of binary iteratively decodable codes with good decoding performance is presented these codes called braided block codes bbcs operate on continuous data streams and are constructed by interconnection of two component block codes bbcs can be considered as convolutional or sliding version of either elias product codes or expander codes in this paper we define bbcs describe methods of their construction analyze code properties and study asymptotic iterative decoding performance

trace theorems for functions of bounded variation in metric spaces abstract in this paper we show existence of traces of functions of bounded variation on the boundary of a certain class of domains in metric measure spaces equipped with a doubling measure supporting a 1 poincare inequality and obtain l 1 estimates of the trace functions in contrast with the treatment of traces given in other papers on this subject the traces we consider do not require knowledge of the function in the exterior of the domain we also establish a maz ya type inequality for functions of bounded variation that vanish on a set of positive capacity

a recursive approach to low complexity codes a method is described for constructing long error correcting codes from one or more shorter error correcting codes referred to as subcodes and a bipartite graph a graph is shown which specifies carefully chosen subsets of the digits of the new codes that must be codewords in one of the shorter subcodes lower bounds to the rate and the minimum distance of the new code are derived in terms of the parameters of the graph and the subeodes both the encoders and decoders proposed are shown to take advantage of the code s explicit decomposition into subcodes to decompose and simplify the associated computational processes bounds on the performance of two specific decoding algorithms are established and the asymptotic growth of the complexity of decoding for two types of codes and decoders is analyzed the proposed decoders are able to make effective use of probabilistic information supplied by the channel receiver e g reliability information without greatly increasing the number of computations required it is shown that choosing a transmission order for the digits that is appropriate for the graph and the subcodes can give the code excellent burst error correction abilities the construction principles

general formalism for inhomogeneous random graphs we present and investigate an extension of the classical random graph to a general class of inhomogeneous random graph models where vertices come in different types and the probability of realizing an edge depends on the types of its terminal vertices this approach provides a general framework for the analysis of a large class of models the generic phase structure is derived using generating function techniques and relations to other classes of models are pointed out

first and second order sensitivity analysis of linear and nonlinear structures this paper employs the principle of virtual work to derive sensitivity derivatives of structural response with respect to stiffness parameters using both direct and adjoint approaches the computations required are based on additional load conditions characterized by imposed initial strains body forces or surface tractions as such they are equally applicable to numerical or analytical solution techniques the relative efficiency of various approaches for calculating first and second derivatives is assessed it is shown that for the evaluation of second derivatives the most efficient approach is one that makes use of both the first order sensitivities and adjoint vectors two example problems are used for demonstrating the various approaches

a trust region strategy for minimization on arbitrary domains we present a trust region method for minimizing a general differentiable function restricted to an arbitrary closed set we prove a global convergence theorem the trust region method defines difficult subproblems that are solvable in some particular cases we analyze in detail the case where the domain is a euclidean ball for this case we present numerical experiments where we consider different hessian approximations

design of irregular ldpc codes with optimized performance complexity tradeoff the optimal performance complexity tradeoff for error correcting codes at rates strictly below the shannon limit is a central question in coding theory this paper proposes a numerical approach for the minimization of decoding complexity for long block length irregular low density parity check ldpc codes the proposed design methodology is applicable to any binary input memoryless symmetric channel and any iterative message passing decoding algorithm with a parallel update schedule a key feature of the proposed optimization method is a new complexity measure that incorporates both the number of operations required to carry out a single decoding iteration and the number of iterations required for convergence this paper shows that the proposed complexity measure can be accurately estimated from a density evolution and extrinsic information transfer chart analysis of the code a sufficient condition is presented for convexity of the complexity measure in the variable edge degree distribution when it is not satisfied numerical experiments nevertheless suggest that the local minimum is unique the results presented herein show that when the decoding complexity is constrained the complexity optimized codes significantly outperform threshold optimized codes at long block lengths within the ensemble of irregular codes

global convergence of a class of trust region algorithms for optimization with simple bounds this paper extends the known excellent global convergence properties of trust region algorithms for unconstrained optimization to the case where bounds on the variables are present weak conditions on the accuracy of the hessian approximations are considered it is also shown that when the strict complementarily condition holds the proposed algorithms reduce to an unconstrained calculation after finitely many iterations allowing a fast asymptotic rate of convergence

explicit constraint approximation forms in structural optimization part 1 analyses and projections abstract an analytical study of several alternative forms of constraint approximation relevant to problems in structural design is presented limitations of the existing forms of the constraint approximations are stated and methods are outlined to improve them a new generalized form of constraint approximation is proposed for optimal design problems with and without design variable linking a power form of constraint approximation is derived for bending types of finite elements having nonlinear size stiffness relationships a detailed analysis to determine the relative effectiveness of the various forms is included a rating guideline is proposed to quantify the characteristics of chosen approximations or family of approximations on grounds of accuracy and of conservatism of remaining feasible

sequential subspace robustness assessment and sensitivity analysis to avoid an overconservative design and to ensure desired performance in an optimal way the product quality and robustness are considered in terms of the product performance mean and variance in this paper to facilitate robust design exploration under uncertainty a new sequential subspace robustness assessment method is presented to assess not only the mean and variance of performance but also their sensitivities with respect to design parameters the proposed method is based on the computational framework that integrates the univariate revolving integration and surrogate modeling of univariate integral functions the proposed framework enables consideration of bivariate interaction effects approximately by the aggregation of multiple revolving integration terms in a partial set of bivariate subspaces it is found that the proposed method provides better accuracy with comparable computational cost in assessing the statistical moments and sensitivities of product performance than existing methods suc

improved sequential linear programming formulation for structural weight minimization abstract this paper presents an optimization algorithm for weight minimization of structures the algorithm denoted as lstrlp line search trust region linear programming combines sequential linear programming slp and trust region methods trm lstrlp solves a linearized sub problem in each design cycle and accepts or rejects intermediate designs based on a line search strategy which detects if the eventual improvement in cost is actually the largest possible it is to be noticed that the present work is the closure to several studies carried out by the present authors in order to improve the overall efficiency and robustness of the sequential linear programming method the lstrlp algorithm is implemented by an optimization code written in fortran 90 the optimization code is tested in eight cases of weight minimization of bar truss and frame structures the test cases include examples of large scale and configuration optimization the results obtained here are compared to those presented in literature the optimizations are run also with sequential quadratic programming sqp routines implemented in commercial software the results indicate that lstrlp is competitive with recently published algorithms and commercial software

on the convergence of a new trust region algorithm in this paper we present a new trust region algorithm for general nonlinear constrained optimization problems the algorithm is based on the l infty exact penalty function under very mild conditions global convergence results for the algorithm are given local convergence properties are also studied it is shown that the penalty parameter generated by the algorithm will be eventually not less than the l 1 norm of the lagrange multipliers at the accumulation point it is proved that the method is equivalent to the sequential quadratic programming method for all large k hence superlinearly convergent results of the sqp method can be applied numerical results are also reported

high quality approximation of eigenvalues in structural optimization a new function for approximating natural frequency constraints during structural optimization is presented the rayleigh quotient approximation rqa presented here icreases the quality of the approximate frequency constraint by approximating the modal strain and kinetic energies instead of the frequency eigenvalue itself

geometry of interpolation sets in derivative free optimization we consider derivative free methods based on sampling approaches for nonlinear optimization problems where derivatives of the objective function are not available and cannot be directly approximated we show how the bounds on the error between an interpolating polynomial and the true function can be used in the convergence theory of derivative free sampling methods these bounds involve a constant that reflects the quality of the interpolation set the main task of such a derivative free algorithm is to maintain an interpolation sampling set so that this constant remains small and at least uniformly bounded this constant is often described through the basis of lagrange polynomials associated with the interpolation set we provide an alternative more intuitive definition for this concept and show how this constant is related to the condition number of a certain matrix this relation enables us to provide a range of algorithms whilst maintaining the interpolation set so that this condition number or the geometry constant remain uniformly bounded we also derive bounds on the error between the model and the function and between their derivatives directly in terms of this condition number and of this geometry constant

a sequential quadratic programming based algorithm for optimization of gas networks abstract british gas uses a complex heavily looped network of pipes and controllable units compressors and regulators to transmit gas from coastal supply terminals to regional demand points computer algorithms are required for efficient management of the system this paper describes an algorithm for optimal control over periods of up to a day the problem is large scale and highly nonlinear in both objective function and constraints the method is based on sequential quadratic programming and takes account of the structure of the pipeflow equations by means of a reduced gradient technique which eliminates most of the variables from the quadratic subproblems the latter involve only simple bound constraints which are handled efficiently by a conjugate gradient active set algorithm trust region techniques permit use of the exact hessian preserving sparsity more general constraints are handled at an outer level by a truncated augmented lagrangian method results are included for some realistic problems the algorithm is generally applicable to problems with a control structure

a computational study of transformation methods for optimal design in this paper computational aspects of transformatio n methods are studied transformation methods which include sequential unconstrained minimization techniques sumts and multiplier methods are based on solving a sequence of unconstrained minimization problems an efficient technique is given to compute gradients of the transformation function an operations count is given to demonstrate savings of the suggested technique over two other techniques used in the literature computer programs implementing the use of this technique in sumts and multiplier algorithms are developed applications of these programs on a set of structural design problems are given multiplier methods are found to be very stable and reliable even on some relatively difficult problems and they perform better than sumts some ways of improving the efficiency of transformation methods are given together with possible extensions of using the suggested approach to compute gradients of implicit functions

non convex dual forms based on exponential intervening variables with application to weight minimization we study the weight minimization problem in a dual setting we propose new dual formulations for non linear multipoint approximations with diagonal approximate hessian matrices which derive from separable series expansions in terms of exponential intervening variables these generally nonconvex approximations are formulated in terms of intervening variables with negative exponents and are therefore applicable to the solution of the weight minimization problem in a sequential approximate optimization sao framework problems in structural optimization are traditionally solved using sao algorithms like the method of moving asymptotes which require the approximate subproblems to be strictly convex hence during solution the nonconvex problems are approximated using convex functions and this process may in general be inefficient we argue based on falk s definition of the dual that it is possible to base the dual formulation on nonconvex approximations to this end we reintroduce a nonconvex approach to the weight minimization problem originally due to fleury and we explore certain convex and nonconvex forms for subproblems derived from the exponential approximations by the application of various methods of mixed variables we show in each case that the dual is well defined for the form concerned which may consequently be of use to the future code developers

approaching capacity at high rates with iterative hard decision decoding a variety of low density parity check ldpc ensembles have now been observed to approach capacity with message passing decoding however all of them use soft i e non binary messages and a posteriori probability decoding of their component codes in this paper we show that one can approach capacity at high rates using iterative hard decision decoding hdd of generalized product codes specifically a class of spatially coupled generalized ldpc codes with bose chaudhuri hocquengham component codes is considered and it is observed that in the high rate regime they can approach capacity under the proposed iterative hdd these codes can be seen as generalized product codes and are closely related to braided block codes an iterative hdd algorithm is proposed that enables one to analyze the performance of these codes via density evolution

enhanced two point diagonal quadratic approximation methods for design optimization abstract based on two point diagonal quadratic approximation referred to as tdqa developed by kim et al min soo kim jong rip kim jae young jeon dong hoon choi efficient mechanical system optimization using two point diagonal approximation in the nonlinear intervening variable space j ksme 15 2001 1257 126 enhanced two point approximation methods are proposed in this paper the suggested methods reinforce tdqa with new quadratic correction terms using the concept of tana 3 these methods overcome the disadvantage of tdqa when the derivatives at two design points have the same sign in addition the values and derivatives of the proposed approximation functions are completely equal to those of an original function at the two design points whether the derivatives at those points have the same sign or not several examples show the numerical performance and accuracy of the proposed methods compared to previous work and optimization examples show that these methods can successfully reach the optimum via sequential approximation optimization

a note on the extension of bv functions in metric measure spaces in this note a classical extension result for bv functions due to yu d burago and v g maz ja yu d burago v g maz ja potential theory and function theory for irregular regions seminars in math vol 3 v a steklov math inst leningrad 1969 translated from russian is generalized to the abstract setting of doubling metric measure spaces endowed with a differentiation structure called d structure

on the resolution of the generalized nonlinear complementarity problem minimization of a differentiable function subject to box constraints is proposed as a strategy to solve the generalized nonlinear complementarity problem gncp defined on a polyhedral cone it is not necessary to calculate projections that complicate and sometimes even disable the implementation of algorithms for solving these kinds of problems theoretical results that relate stationary points of the function that is minimized to the solutions of the gncp are presented perturbations of the gncp are also considered and results are obtained related to the resolution of gncps with very general assumptions on the data these theoretical results show that local methods for box constrained optimization applied to the associated problem are efficient tools for solving the gncp numerical experiments are presented that encourage the use of this approach

structural reliability optimization using an efficient safety index calculation procedure the objective of this paper is to conduct reliability based structural optimization in a multidisciplinary environment an efficient reliability analysis is developed by expanding the limit functions in terms of intermediate design variables the design constraints are approximated using multivariate splines in searching for the optimum the reduction in computational cost realized in safety index calculation and optimization are demonstrated through several structural problems this paper presents safety index computation analytical sensitivity analysis of reliability constraints and optimization using truss frame and plate examples

newton s method with a model trust region modification a modified newton method for unconstrained minimization is presented and analyzed the modification is based upon the model trust region approach this report contains a thorough analysis of the locally constrained quadratic minimizations that arise as subproblems in the modified newton iteration several promising alternatives are presented for solving these subproblems in ways that overcome certain theoretical difficulties exposed by this analysis very strong convergence results are presented concerning the minimization algorithm in particular the explicit use of second order information is justified by demonstrating that the iterates converge to a point which satisfies the second order necessary conditions for minimization with the exception of very pathological cases this occurs whenever the algorithm is applied to problems with continuous second partial derivatives

new adaptive move limit management strategy for approximate optimization part 2 approximations play an important role in multidisciplinary design optimization by offering system behavior information at a relatively low cost most approximate optimization strategies are sequential in which an optimization of an approximate problem subject to design variable move limits is iteratively repeated until convergence the move limits are imposed to restrict the optimization to regions of the design space in which the approximations provide meaningful information to ensure convergence of the sequence of approximate optimizations to a karush kuhn tucker solution a move limit management strategy is required in a companion paper issues of move limit management are reviewed and a new adaptive strategy for move limit management is developed wujek b a and renaud j e new adaptive move limit management strategy for approximate optimization part 1 aiaa journal vol 36 no 10 1998 pp 1911 1921 with its basis in the provably convergent trust region methodology the trust region ratio approximation method tram strategy utilizes available gradient information and employs a backtracking process using various two point approximation techniques to provide a flexible move limit adjustment factor the tram strategy is successfully implemented in application to several multidisciplinary design optimization test problems in addition a comprehensive study comparing the performance of the tram strategy to existing move limit strategies is conducted results indicate that application of the tram strategy results in increased efficiency for approximate optimization processes these implementation studies highlight the ability of the tram strategy to control the amount of approximation error and efficiently manage the convergence to a karush kuhn tucker solution

weight minimisation of displacement constrained truss structures using a strain energy criterion this paper discusses a new method for the solution of the general truss weight minimisation problem under simultaneous stress and displacement constraints the method introduces a novel strain energy density criterion that refers to the ratio of the virtual strain energy per unit volume in each structural member to its average value on the whole structure the virtual strain energy comes from the unit load theorem and it is proportional to the product of the axial member forces due to both the actual loads and a virtual unit load that is applied at the node with the maximum displacement a simple recursive formula for updating the cross sectional areas based on displacement constraints is presented a general subsequent algorithm applicable to both single and multiple load cases follows this formula the results are encouraging since in all test cases the method was found to be robust and generally led to the same weight level as the literature in both small and large structures

trust region methods for structural optimization using exact second order sensitivity the performance of multiplier algorithms for structural optimization has been significantly improved by using trust regions the trust regions are constructed using analytical second order sensitivity and within this region the augmented lagrangian is minimized subject to bounds evaluation of first and second derivatives of by the adjoint method does not require derivations of individual implicit constraint functions which makes the method economical eight test problems are considered and a vast improvement over previously used multiplier algorithms has been noted also the algorithm is robust with respect to scaling input parameters and starting designs

a new three point approximation approach for design optimization problems in this paper a newly developed three point approximation scheme is proposed the expression of this scheme consists of a linear combination of the direct and reciprocal linear taylor expansions as well as of the lumped diagonal terms of the second order direct and inverse terms the unknown parameters of the expression are computed on the basis of the function and gradient values at three points in the design space based on this approach the accuracy of the existing constraint approximation methods can be improved the e ectiveness of the proposed approach is demonstrated on a number of numerical examples the numerical results are also compared with those of the previously published work copyright 2001 john wiley sons ltd

a penalty free type nonmonotone trust region method for nonlinear constrained optimization in this paper we propose a penalty free type nonmonotone trust region method for solving general nonlinear programming problem the algorithmic framework yields global convergence without using any penalty function we analyze the global convergence of the main algorithm for the degenerate problems and also give the globally convergent results under the linear independence constraint qualification the preliminary numerical tests are reported

second order nonsmooth optimization for h synthesis the standard way to compute h feedback controllers uses algebraic riccati equations and is therefore of limited applicability here we present a new approach to the h output feedback control design problem which is based on nonlinear and nonsmooth mathematical programming techniques our approach avoids the use of lyapunov variables and is therefore flexible in many practical situations

iterative decoding threshold analysis for ldpc convolutional codes an iterative decoding threshold analysis for terminated regular ldpc convolutional ldpcc codes is presented using density evolution techniques the convergence behavior of an iterative belief propagation decoder is analyzed for the binary erasure channel and the awgn channel with binary inputs it is shown that for a terminated ldpcc code ensemble the thresholds are better than for corresponding regular and irregular ldpc block codes

a recursive quadratic programming method with active set strategy for optimal design recursive quadratic programming methods have become popular in the field of mathematical programming owing to their excellent convergence characteristics there are two recursive quadratic programming methods that have been published in the literature one is by han and the other is by pshenichny published in 1977 and 1970 respectively the algorithm of pshenichny had been undiscovered until now and is examined here for the first time it is found that the proof of global convergence by han requires computing sensitivity coefficients derivatives of all constraint functions of the problem at every iteration this is prohibitively expensive for large scale applications in optimal design in contrast pshenichny has proved global convergence of his algorithm using only an active set strategy this is clearly preferable for large scale applications the method of pshenichny has been coded into a fortran program applications of this method to four example problems are presented the method is found to be very reliable however the method is found to be very sensitive to local minima i e it converges to a local minimum nearest to the starting design thus for optimal design problems which usually possess multiple local minima it is suggested that pshenichny s method be used as part of a hybrid method

newton s method for large bound constrained optimization problems we analyze a trust region version of newton s method for bound constrained problems our approach relies on the geometry of the feasible set not on the particular representation in terms of constraints the convergence theory holds for linearly constrained problems and yields global and superlinear convergence without assuming either strict complementarity or linear independence of the active constraints we also show that the convergence theory leads to an efficient implementation for large bound constrained problems

an active set quasi newton method with projected search for bound constrained minimization we analyze an active set quasi newton method for large scale bound constrained problems our approach combines the accurate active set identification function and the projected search both of these strategies permit fast change in the working set the limited memory method is employed to update the inactive variables while the active variables are updated by simple rules a further division of the active set enables the global convergence of the new algorithm numerical tests demonstrate the efficiency and performance of the present strategy and its comparison with some existing active set strategies

finite length scaling for iteratively decoded ldpc ensembles we investigate the behavior of iteratively decoded low density parity check ldpc codes over the binary erasure channel in the so called ldquowaterfall region rdquo we show that the performance curves in this region follow a simple scaling law we conjecture that essentially the same scaling behavior applies in a much more general setting and we provide some empirical evidence to support this conjecture the scaling law together with the error floor expressions developed previously can be used for a fast finite length optimization

finite element model updating for structural applications abstract a novel method for performing model updating on finite element models is presented the approach is particularly tailored to modal analyses of buildings by which the lowest frequencies obtained by using sensors and system identification approaches need to be matched to the numerical ones predicted by the model this is done by optimizing some unknown material parameters such as mass density and young s modulus of the materials and or the boundary conditions which are often known only approximately in particular this is the case when considering historical buildings the straightforward application of a general purpose optimizer can be impractical given the large size of the model involved in the paper we show that by slightly modifying the projection scheme used to compute the eigenvalues at the lowest end of the spectrum one can obtain local parametric reduced order models that embedded in a trust region scheme form the basis for a reliable and efficient specialized algorithm we describe an optimization strategy based on this approach and we provide numerical experiments that confirm its effectiveness and accuracy

globally convergent optimization algorithm using conservative convex separable diagonal quadratic approximations we implement and test a globally convergent sequential approximate optimization algorithm based on convexified diagonal quadratic approximations the algorithm resides in the class of globally convergent optimization methods based on conservative convex separable approximations developed by svanberg at the start of each outer iteration the initial curvatures of the diagonal quadratic approximations are estimated using historic objective and or constraint function value information or by building the diagonal quadratic approximation to the reciprocal approximation at the current iterate during inner iterations these curvatures are increased if no feasible descent step can be made although this conditional enforcement of conservatism on the subproblems is a relaxation of the strict conservatism enforced by svanberg global convergence is still inherited from the conservative convex separable approximations framework developed by svanberg a numerical comparison with the globally convergent version of the method of moving asymptotes and the nonconservative variants of both our algorithm and method of moving asymptotes is made

new successive approximation method for optimum structural design a new procedure using only first order explicit information and consisting of the successive solution of approximate subproblems is proposed the method differs from established sequential quadratic programming methods by not utilizing the lagrangian form and not attempting to store full and explicit hessian information instead each constraint is individually approximated by a quadratic function involving only one coefficient in the quadratic term this coefficient is determined from a two point collocation a novel feature of the method is that each quadratic subproblem is solved by a recently proposed interior feasible direction method an auxiliary problem is formulated to generate a feasible starting point for each subproblem

successive matrix inversion method for reanalysis of engineering structural systems over the past several decades numerous structural analysis techniques have been developed to represent physical systems behavior more realistically and the structural models therefore have become larger and more complex even though modern computer power has increased significantly cost of computational analysis has been a major restrictive factor in a structural system design involving multiple disciplines and repetitive simulations a new reanalysis technique is developed the successive matrix inversion smi method is most suitable for reanalysis of structures the smi method reproduces exact solutions for any localized modification of the initial system several numerical examples are given to demonstrate the efficiency of this method

conditions for convergence of trust region algorithms for nonsmooth optimization this paper discusses some properties of trust region algorithms for nonsmooth optimization the problem is expressed as the minimization of a functionh f x whereh is convex andf is a continuously differentiable mapping from to bounds for the second order derivative approximation matrices are discussed it is shown that powel s 7 8 results hold for nonsmooth optimization

modified product codes by modifying product codes a new coding scheme and its decoding method are proposed compared to a product code a the first stage code a 1 of the new code a m is constructed in the same way as that of the code a except that it has at least one subcode while the second stage codes a 2 j of the code a m are a set of codes with the same length and different rates the new coding scheme has a smaller upper bound on the probability of decoding error than the original product coding scheme for any given nonzero rate less than the capacity of a binary symmetric channel an example is given for which the rate is increased compared with the original product code at a fixed probability of decoding error for a relatively short code length

calculation of the move limits for the sequential linear programming method the sequential linear programming slp method is one of the easiest optimization techniques used to treat non linear optimization problems the successful application of the method is based on the proper choice of the move limits which are unfortunately uncertain in the slp algorithm to enhance the efficiency and reliability of the slp algorithm six methods are proposed to calculate the move limits the first three methods use the constraint equations to compute the minimum variations needed from the current design point to reach the constraint boundaries the fourth method incorporates the cost function information as well as the constraints to figure out the move limits the last two methods form a quadratic programming qp subproblem and solve for the change of the quadratic cost function the move limits used in slp are then computed from the estimated change of the linear cost function from the qp solution four numerical examples are employed to demonstrate the solution process by the methods proposed the advantages and disadvantages of these methods are discussed

effective two point function approximation for design optimization anew two pointapproximationofafunctionthatisverystraightforwardtobuildandthatprovidesaccurateand stable approximation is presented earlier developments of the two point approximations had either incomplete matches at two data points or needed the solution of additional equations to get all of the parameters the present two point approximation is an incomplete second order taylor series expansion in terms of intervening variables thehessianmatrixhasonlydiagonalelements anditdependsondesignvariables theexponentofeachintervening design variable and the unknown constant of the second order terms are evaluated by matching the derivatives and the value of the approximation with the previous data point gradients and the value of the original function respectively all of theunknownsareidentie ed in aclosedform both thefunctionandthegradientof thetwo point approximation are equal to those of the original function at two data points several examples are given to show the accuracy and efe ciency of this method

convergence to second order stationary points in inequality constrained optimization we propose a new algorithm for the nonlinear inequality constrained minimization problem and prove that it generates a sequence converging to points satisfying the kkt second order necessary conditions for optimality the algorithm is a line search algorithm using directions of negative curvature and it can be viewed as a nontrivial extension of corresponding known techniques from unconstrained to constrained problems the main tools employed in the definition and in the analysis of the algorithm are a differentiable exact penalty function and results from the theory of lc1 functions

threshold saturation via spatial coupling why convolutional ldpc ensembles perform so well over the bec convolutional low density parity check ldpc ensembles introduced by felstrom and zigangirov have excellent thresholds and these thresholds are rapidly increasing functions of the average degree several variations on the basic theme have been proposed to date all of which share the good performance characteristics of convolutional ldpc ensembles we describe the fundamental mechanism that explains why convolutional like or spatially coupled codes perform so well in essence the spatial coupling of individual codes increases the belief propagation bp threshold of the new ensemble to its maximum possible value namely the maximum a posteriori map threshold of the underlying ensemble for this reason we call this phenomenon threshold saturation this gives an entirely new way of approaching capacity one significant advantage of this construction is that one can create capacity approaching ensembles with an error correcting radius that is increasing in the blocklength although we prove the threshold saturation only for a specific ensemble and for the binary erasure channel bec empirically the phenomenon occurs for a wide class of ensembles and channels more generally we conjecture that for a large range of graphical systems a similar saturation of the dynamical threshold occurs once individual components are coupled sufficiently strongly this might give rise to improved algorithms and new techniques for analysis

on the accurate identification of active constraints we consider nonlinear programs with inequality constraints and we focus on the problem of identifying those constraints which will be active at an isolated local solution the correct identification of active constraints is important from both a theoretical and a practical point of view such an identification removes the combinatorial aspect of the problem and locally reduces the inequality constrained minimization problem to an equality constrained problem which can be more easily dealt with we present a new technique which identifies active constraints in a neighborhood of a solution and which requires neither complementary slackness nor uniqueness of the multipliers we also present extensions to variational inequalities and numerical examples illustrating the identification technique

poincare inequalities uniform domains and extension properties for newton sobolev functions in metric spaces in the setting of metric measure spaces equipped with a doubling measure supporting a weak p poincare inequality with 1 p 1 we obtain a near characterization of n1 p extension domains with loc

improved two point function approximations for design optimization two point approximations are developed by utilizing both the function and gradient information of two data points the objective of this work is to build a high quality approximation to realize computational savings in solving complex optimization and reliability analysis problems two developments are proposed in the new twopoint approximations 1 calculation of a correction term by matching with the previous known function value and supplementing it to the first order approximation for including the effects of higher order terms and 2 development of a second order approximation without the actual calculation of second order derivatives by using an approximate hessian matrix several highly nonlinear functions and structural examples are used for demonstrating the new two point approximations that improved the accuracy of existing first order methods evelopmentof new and improved constraint approximations has been an active area of research in the mathematical optimization community over the last two decades 1 3 more accurate function approximations reduce the finite element analysis cost and increase the search domain in each iteration of an optimization run recent developments in constraint approximations addressed methods for eigenvalues dynamic response static forces structural reliability etc some of the typical methods are summarized in the following kirsch4 developed a scaling factor for a stiffness matrix and approximated the displacements stresses and forces with respect to sizing and topology variables vanderplaats and salajegheh5 used two levels of approximations for discrete sizing and shape design of structures first for solving the continuous problem and next the discrete one using a dual theory canfield6 developed an approach for eigenvalues by approximating the modal strain and kinetic energies independently thomas et al 7 presented an approximation for the frequency response of damped structures by using the concept of ref 6 a multivariate spline approximation was developed by wang and grandhi8 making use of the previously generated exact finite element analyses similarly a hermite approximation for n dimensional problems used several data points and had the property of reproducing the function and gradient values at the known data points 9 canfield10 used multipoint data in building an approximate hessian matrix and constructed a second order approximation for improving the accuracy toropov et al 11 approximated functions as a linear combination or products of variables and the coefficients of the polynomial were computed by applying a least squared approach on multiple data points fadel et al 12 considered intermediate variables in terms of exponentials and they were computed by matching the approximate function gradients with the previous point s exact values this approach did not compare the function values of the previous point wang and grandhi13 used a single exponential for all of the variables and calculated it by matching the approximate function with the previous point exact value and the gradient information of the previous point was not utilized the proposed paper develops an improved two point approximation utilizing both function and gradient information of two data points in the new two point approximation two developments are proposed 1 calculation of a correction term based on two function values and

adaptive experimental design for construction of response surface approximations sequential approximate optimization sao is a class of methods available for the multidisciplinary design optimization mdo of complex systems that are composed of several disciplines coupled together one of the approaches used for sao is based on a quadratic response surface approximation where zero and first order information are required in these methods designers must generate and query a database of order o n2 in order to compute the second order terms of the quadratic response surface approximation as the number of design variables grows the computational cost of generating the required database becomes a concern in this paper we present an new approach in which we require just o ri parameters for constructing a second order approximation this is accomplished by transforming the matrix of second order terms into the canonical form the method periodically requires an order o n2 update of the second order approximation to maintain accuracy results show that the proposed approach is practical and convenient for engineering design problems by dramatically reducing the total number of calls to the simulation tools

quasiconformal maps in metric spaces with controlled geometry this paper develops the foundations of the theory of quasiconformal maps in metric spaces that satisfy certain bounds on their mass and geometry the principal message is that such a theory is both relevant and viable the first main issue is the problem of definition which we next describe quasiconformal maps are commonly understood as homeomorphisms that distort the shape of infinitesimal balls by a uniformly bounded amount this requirement makes sense in every metric space given a homeomorphism f from a metric space x to a metric space y then for x x and r 0 set

accelerated engineering design optimization using successive matrix inversion method in a design optimization procedure many repeated system analyses are conducted to evaluate certain performances of successively modified structural designs to alleviate the high computational cost of structural reanalysis an augmented preconditioner is introduced by using the successive matrix inversion smi method the smi method as an exact matrix solver is successively applied to an iterative procedure to accelerate the solution convergence by improving the numerical condition of the reanalysis system in this work a new iterative technique the binomial series iterative bsi method is developed from the binomial series expansion by using the same concept as smi as a combined iterative ci method smi is incorporated in the procedure of bsi the ci method combined with smi shows sufficient efficiency and robustness through a stable iterative behaviour due to simple and straightforward computations in the iterative procedure published in 2005 by john wiley sons ltd

partitioned variable metric updates for large structured optimization problems this paper presents a minimization method based on the idea of partitioned updating of the hessian matrix in the case where the objective function can be decomposed in a sum of convex element functions this situation occurs in a large class of practical problems including nonlinear finite elements calculations some theoretical and algorithmic properties of the update are discussed and encouraging numerical results are presented

multifidelity quasi newton method for design optimization multifidelity approaches are frequently used in design when high fidelity models are too expensive to use directly and lower fidelity models of reasonable accuracy exist in optimization corrected

global convergence of trust region interior point algorithms for infinite dimensional nonconvex minimization subject to pointwise bounds a class of interior point trust region algorithms for infinite dimensional nonlinear optimization subject to pointwise bounds in l p banach spaces 2 le p le infty is formulated and analyzed the problem formulation is motivated by optimal control problems with l p controls and pointwise control constraints the interior point trust region algorithms are generalizations of those recently introduced by coleman and li siam j optim 6 1996 pp 418 445 for finite dimensional problems many of the generalizations derived in this paper are also important in the finite dimensional context all first and second order global convergence results known for trust region methods in the finite dimensional setting are extended to the infinite dimensional framework of this paper

a novel multi objective optimization method based on an approximation model management technique in this paper a novel multi objective optimization method is suggested based on an approximation model management technique it is a sequential approximation method in which a multi objective optimization with approximation models subject to design variable move limits is iterated until convergence in each iteration step the approximation models are constructed by the response surface approximations with the samples which are obtained from the design of experiments and a pareto optimal set predicted by the approximations is identified through a multi objective genetic algorithm according to the prediction of the approximation models a move limits updating strategy is employed to determine the design variable move limits for the next iteration at the end of each iteration step some uniform distributed points chosen from the predictive pareto optimal frontier are verified by the high fidelity models and the obtained actual pareto optimal set is stored in an external archive the high efficiency of the present method is demonstrated by four different test functions and two engineering applications

a limited memory quasi newton trust region method for box constrained optimization by means of wolfe conditions strategy we propose a quasi newton trust region method to solve box constrained optimization problems this method is an adequate combination of the compact limited memory bfgs and the trust region direction while the generated point satisfies the wolfe conditions and therefore maintains a positive definite approximation to the hessian of the objective function the global convergence and the quadratic convergence of this method are established under suitable conditions finally we compare our algorithms iwtral and ibwtral with an active set trust region algorithm astral xu and burke 2007 on the cutest box constrained test problems presented by gould et al 2015 numerical results show that the presented method is competitive and totally interesting for solving box constrained optimization

performance of product codes and related structures with iterated decoding several modifications of product codes have been suggested as standards for optical networks we show that the performance exhibits a threshold that can be estimated from a result about random graphs for moderate input bit error probabilities the output error rates for codes of finite length can be found by easy simulations the analysis indicates that the performance curve can be extrapolated until the error floor is reached the analysis allows us to calculate the error floors and avoid time consuming simulations