exact penalty results for mathematical programs with vanishing constraints a mathematical program with vanishing constraints mpvc is a constrained optimization problem arising in certain engineering applications the feasible set has a complicated structure so that the most familiar constraint qualifications are usually violated this in turn implies that standard penalty functions are typically non exact for mpvcs we therefore develop a new mpvc tailored penalty function which is shown to be exact under reasonable assumptions this new penalty function can then be used to derive or recover suitable optimality conditions for mpvcs

on the status of stability of interconnected systems in the present paper we report on the status of stability of complex dynamical systems we consider those systems which may be viewed as an interconnection of lower order subsystems we emphasize those types of results which are phrased in terms of the stability properties of the subsystems and in terms of the qualitative properties of the system interconnecting structure we consider both lyapunov stability and input output stability results in our presentation we concentrate on selected sample results and we show how some of these results are related we then use these sample results to identify modifications extensions generalizations and applications in this way we are able to provide a reasonably good picture of the entire subject on hand

strong duality and kkt conditions in nonconvex optimization with a single equality constraint and geometric constraint some topological and geometric characterizations of strong duality for a non convex optimization problem under a single equality and geometric constraints are established in particular a hidden convexity of the conic hull of joint range of the pair of functions associated to the original problem is obtained applications to derive a characterization of the validity of kkt conditions without standard constraints qualification are also discussed it goes beyond the exact penalization technique several examples showing our results provide much more information than those appearing elsewhere are given finally the standard quadratic problem involving a non necessarily polyhedral cone is analyzed in detail

on the natural vibrations of linear structures with constraints the undamped natural vibrations of a constrained linear structure are given by the solutions to a generalized eigenvalue problem derived from the equations of motion for the constrained system involving lagrangian multipliers the eigenvalue problem derived is defined by the mass matrix of the unconstrained structure and a non symmetric and singular stiffness matrix for the constrained system the character of the solution of the eigenvalue problem of the constrained system is stated and proved in a theorem applications of the constrained eigenvalue problem to some simple structures are demonstrated finally a condition for the calculation of the damped natural vibrations for the constrained structure in terms of the undamped mode shapes is formulated

stationary conditions for mathematical programs with vanishing constraints using weak constraint qualifications abstract we consider a class of optimization problems that is called a mathematical program with vanishing constraints mpvc for short this class has some similarities to mathematical programs with equilibrium constraints mpecs for short and typically violates standard constraint qualifications hence the well known karush kuhn tucker conditions do not provide necessary optimality criteria in order to obtain reasonable first order conditions under very weak assumptions we introduce several mpvc tailored constraint qualifications discuss their relation and prove an optimality condition which may be viewed as the counterpart of what is called m stationarity in the mpec field

bounds for singular values based on the eigenvectors abstract for a nondefective matrix a c n n the known inequalities k k k express a bound on the possible ratio between the kth singular value k of a and its kth eigenvalue k based only on the condition number of the associated eigenvector matrix we obtain and analyse several refinements to this bound

a new regularization method for mathematical programs with complementarity constraints with strong convergence properties mathematical programs with equilibrium or complementarity constraints mpecs form a difficult class of optimization problems the feasible set has a very special structure and violates most of the standard constraint qualifications therefore one typically applies specialized algorithms in order to solve mpecs one very prominent class of specialized algorithms are the regularization or relaxation methods the first regularization method for mpecs is due to scholtes siam j optim 11 2001 pp 918 936 but in the meantime there exist a number of different regularization schemes which try to relax the difficult constraints in different ways however almost all regularization methods converge to c stationary points only which is a very weak stationarity concept an exception is a recent method by kadrani dussault and benchakroun siam j optim 20 2009 pp 78 103 whose limit points are shown to be m stationary here we provide a new regularization method which also converges to m sta

block scaling with optimal euclidean condition let m denote the set of all complex n n matrices whose columns span certain given linear subspaces the minimal euclidean condition number of matrices in m is given in terms of the canonical angles between the linear subspaces and optimal matrices in m are described the result is also stated in terms of norms of certain projections

the coc algorithm part ii topological geometrical and generalized shape optimization abstract after outlining analytical methods for layout optimization and illustrating them with examples the coc algorithm is applied to the simultaneous optimization of the topology and geometry of trusses with many thousand potential members the numerical results obtained are shown to be in close agreement up to twelve significant digits with analytical results finally the problem of generalized shape optimization finding the best boundary topology and shape is discussed

computing stable eigendecompositions of matrices abstract if a matrix t is known only to within a tolerance because of measurement or roundoff errors then it may be difficult to compute an eigendecomposition of t since its invariant subspaces are discontinuous functions of its entries in this paper we show how to compute a stable decomposition of an uncertain matrix t which varies continuously and boundedly as t varies in a ball of radius

multiple eigenvalue sensitivity analysis abstract this paper surveys the development of the sensitivity analysis of multiple eigenvalues and the associated invariant subspaces of two kinds of matrix eigenvalue problems in which the matrices are analytically dependent on several parameters

on the selection of poles in the single input pole placement problem abstract it is well known that the single input pole placement problem can be quite sensitive to perturbations in the data recent results by mehrmann and xu show how this sensitivity depends on the location of the poles in many applications it suffices to prescribe a set k in the complex plane that contains the poles mehrmann and xu formulated a minimization problem for allocating the poles in a given set k so that the sensitivity to perturbations is reduced the present paper uses methods of potential theory to derive simple algorithms that yield approximate solutions of this minimization problem

a note on optimal block scaling of matrices after pointing out that two recent results on optimal blockscaling are equivalent a new short and simple proof of both results is given

error bounds for approximate eigenvalues of periodic coefficient linear delay differential equations we describe a new chebyshev spectral collocation method for systems of variable coefficient linear delay differential equations with a single fixed delay computable uniform a posteriori bounds are given for this method when the coefficients are periodic the system has a unique compact nonnormal monodromy operator whose spectrum determines the stability of the system the spectral method approximates this operator by a dense matrix of modest size in cases where the coefficients are smooth we observe spectral convergence of the eigenvalues of that matrix to those of the operator our main result is a computable a posteriori bound on the eigenvalue approximation error in the case that the coefficients are analytic

robust truss topology design via semidefinite programming we present and motivate a new model of the truss topology design problem where the rigidity of the resulting truss with respect both to given loading scenarios and small occasional loads is optimized it is shown that the resulting optimization problem is a semidefinite program we derive and analyze several equivalent reformulations of the problem and present illustrative numerical examples

optimization of structures with uncertain constraints based on convex model and satisfaction degree of interval an optimization method for uncertain structures is suggested based on convex model and a satisfaction degree of interval in the investigated problem the uncertainty only exists in constraints convex model is used to describe the uncertainty in which the intervals of the uncertain parameters are only needed not necessarily to know the precise probability distributions a satisfaction degree of interval which represents the possibility that one interval is smaller than another is employed to deal with the uncertain constraints based on a predetermined satisfaction degree level the uncertain constraints are transformed to deterministic ones and the transformed optimization problem can be solved by traditional optimization methods for complex structural problems that the optimization model cannot be expressed in an explicit form the interval analysis method is adopted to calculate the intervals of the constraints efficiently and whereby eliminate the optimization nesting two numerical examples have been presented to demonstrate the efficiency of the suggested method

scalable preconditioned conjugate gradient inversion of vector finite element mass matrices mass matrices arise in the numerical solution of time dependent partial differential equations by the galerkin method since these systems must be inverted at each time step rapid inversion algorithms for these systems are important when nodal finite elements are used as basis functions it is known that the mass matrices can be consistently approximated by a diagonal matrix or solved by a scalable conjugate gradient method this may not be the case for other basis functions in this paper we show that the preconditioned conjugate gradient method is scalable when used to invert mass matrices that arise from vector finite element basis functions these basis functions are particularly important for solving maxwell s equations on unstructured grids by the galerkin method

robust partial quadratic eigenvalue assignment with time delay using the receptance and the system matrices abstract in this paper we consider the robust partial quadratic eigenvalue assignment problem in vibration by active feedback control based on the receptance measurements and the system matrices we propose an optimization method for the robust and minimum norm partial quadratic eigenvalue assignment problem we provide a new cost function and the closed loop eigenvalue sensitivity and the feedback norms can be minimized simultaneously our method is also extended to the case of time delay between measurements of state and actuation of control numerical tests demonstrate the effectiveness of our method

on a theorem of feingold abstract we discuss an extension of a theorem which is apparently due to feingold both the theorem and its extension follow mutatis mutandis from the corresponding material associated with the bauer fike theorem feingold made no claims for his result and indeed referred to it as trivial although it proves to be rather disappointing an understanding of the source of its weakness is instructive and leads to related techniques which are of greater value

characterizing fj and kkt conditions in nonconvex mathematical programming with applications in this paper we analyze the fritz john and karush kuhn tucker kkt conditions for a g teaux differentiable nonconvex optimization problem with inequality constraints and a geometric constraint set the fritz john condition is characterized in terms of an alternative theorem which covers beyond standard situations while characterizations of kkt conditions without assuming constraints qualifications are related to strong duality of a suitable linear approximation of the given problem and the properties of its associated image mapping such characterizations are suitable for dealing with some problems in structural optimization where most of the known constraint qualifications fail in particular several examples are given showing the usefulness and optimality in a certain sense of our results which provide much more information than those including the mordukhovich normal cone or clarke s appearing elsewhere the case with a single inequality constraint is discussed in detail by establishing

on the time transformation of mixed integer optimal control problems using a consistent fixed integer control function nonlinear control systems with instantly changing dynamical behavior can be modeled by introducing an additional control function that is integer valued in contrast to a control function that is allowed to have continuous values the discretization of a mixed integer optimal control problem miocp leads to a non differentiable optimization problem and the non differentiability is caused by the integer values the paper is about a time transformation method that is used to transform a miocp with integer dependent constraints into an ordinary optimal control problem differentiability is achieved by replacing a variable integer control function with a fixed integer control function and a variable time allows to change the sequence of active integer values in contrast to other contributions so called control consistent fixed integer control functions are taken into account here it is shown that these control consistent fixed integer control functions allow a better accuracy in the resulting trajectories in particular in the computed switching times the method is verified on analytical and numerical examples

least squares residuals and minimal residual methods we study krylov subspace methods for solving unsymmetric linear algebraic systems that minimize the norm of the residual at each step minimal residual mr methods mr methods are often formulated in terms of a sequence of least squares ls problems of increasing dimension we present several basic identities and bounds for the ls residual these results are interesting in the general context of solving ls problems when applied to mr methods they show that the size of the mr residual is strongly related to the conditioning of different bases of the same krylov subspace using different bases is useful in theory because relating convergence to the characteristics of different bases offers new insight into the behavior of mr methods different bases also lead to different implementations which are mathematically equivalent but can differ numerically our theoretical results are used for a finite precision analysis of implementations of the gmres method y saad and m h schultz siam j sci statist comput 7 1986 pp 856 869 we explain that the choice of the basis is fundamental for the numerical stability of the implementation as demonstrated in the case of simpler gmres h f walker and l zhou numer linear algebra appl 1 1994 pp 571 581 the best orthogonalization technique used for computing the basis does not compensate for the loss of accuracy due to an inappropriate choice of the basis in particular we prove that simpler gmres is inherently less numerically stable than the classical gmres implementation due to saad and schultz siam j sci statist comput 7 1986 pp 856 869

extreme structural response analysis of truss structures under material uncertainty via linear mixed 0 1 programming this paper concerns the extreme structural response analysis of truss structures taking the possible uncertainties of young s moduli of the bars into consideration the uncertainty of young s modulus is described by a non probability but bounded interval model with the use of this uncertain model it is shown that the structural responses such as nodal displacement and bar stress will attain their extreme values when young s moduli of the bars take either their upper or lower bound value based on this conclusion and simultaneous analysis and design formulation a linear mixed 0 1 programming problem can be constructed to find the extreme values of the structural response by the well established branch and bound approach with global optimality the proposed approach is applied to several test problems for the demonstration of its validity

stable polefinding and rational least squares fitting via eigenvalues a common way of finding the poles of a meromorphic function f in a domain where an explicit expression of f is unknown but f can be evaluated at any given z is to interpolate f by a rational function pq such that r i f i at prescribed sample points i i 1l and then find the roots of q this is a two step process and the type of the rational interpolant needs to be specified by the user many other algorithms for polefinding and rational interpolation or least squares fitting have been proposed but their numerical stability has remained largely unexplored in this work we describe an algorithm with the following three features 1 it automatically finds an appropriate type for a rational approximant thereby allowing the user to input just the function f 2 it finds the poles via a generalized eigenvalue problem of matrices constructed directly from the sampled values f i in a one step fashion and 3 it computes rational approximants p q in a numerically stable manner in that p p q q f with small p q at the sample points making it the first rational interpolation or approximation algorithm with guaranteed numerical stability our algorithm executes an implicit change of polynomial basis by the qr factorization and allows for oversampling combined with least squares fitting through experiments we illustrate the resulting accuracy and stability which can significantly outperform existing algorithms

mathematical programs with vanishing constraints optimality conditions and constraint qualifications we consider a difficult class of optimization problems that we call a mathematical program with vanishing constraints problems of this kind arise in various applications including optimal topology design problems of mechanical structures we show that some standard constraint qualifications like licq and mfcq usually do not hold at a local minimum of our program whereas the abadie constraint qualification is sometimes satisfied we also introduce a suitable modification of the standard abadie constraint qualification as well as a corresponding optimality condition and show that this modified constraint qualification holds under fairly mild assumptions we also discuss the relation between our class of optimization problems with vanishing constraints and a mathematical program with equilibrium constraints

pencils of complex and real symmetric and skew matrices abstract this expository paper establishes the canonical forms under congruence for pairs of complex or real symmetric or skew matrices the treatment is in the spirit of the well known book of gantmacher on matrix theory and may be regarded as a supplement to gantmacher s chapters on pencils of matrices

some new search directions for primal dual interior point methods in semidefinite programming search directions for primal dual path following methods for semidefinite programming sdp are proposed these directions have the properties that 1 under certain nondegeneracy and strict complementarity assumptions the jacobian matrix of the associated symmetrized newton equation has a bounded condition number along the central path in the limit as the barrier parameter mu tends to zero and 2 the schur complement matrix of the symmetrized newton equation is symmetric and the cost for computing this matrix is 2mn3 0 5m2n2 flops where n and m are the dimension of the matrix and vector variables of the semidefinite program respectively these two properties imply that a path following method using the proposed directions can achieve the high accuracy typically attained by methods employing the direction proposed by alizadeh haeberly and overton currently the best search direction in terms of accuracy but each iteration requires at most half the amount of flops to leading order

stationarity conditions and constraint qualifications for mathematical programs with switching constraints in optimal control switching structures demanding at most one control to be active at any time instance appear frequently discretizing such problems a so called mathematical program with switching constraints is obtained although these problems are related to other types of disjunctive programs like optimization problems with complementarity or vanishing constraints their inherent structure makes a separate consideration necessary since standard constraint qualifications are likely to fail at the feasible points of switching constrained optimization problems stationarity notions which are weaker than the associated karush kuhn tucker conditions need to be investigated in order to find applicable necessary optimality conditions furthermore appropriately tailored constraint qualifications need to be formulated in this paper we introduce suitable notions of weak mordukhovich and strong stationarity for mathematical programs with switching constraints and present some associated constraint qualifications our findings are exploited to state necessary optimality conditions for discretized optimal control problems with switching constraints furthermore we apply our results to optimization problems with either or constraints first a novel reformulation of such problems using switching constraints is presented second the derived surrogate problem is exploited to obtain necessary optimality conditions for the original program

a mixed integer programming for robust truss topology optimization with stress constraints this paper presents a mixed integer programming mip formulation for robust topology optimization of trusses subjected to the stress constraints under the uncertain load a design dependent uncertainty model of the external load is proposed for dealing with the variation of truss topology in the course of optimization for a truss with the discrete member cross sectional areas it is shown that the robust topology optimization problem can be reduced to an mip problem which is solved globally numerical examples illustrate that the robust optimal topology of a truss depends on the magnitude of uncertainty

local stress constrained and slope constrained sand topology optimisation summary we study the alternative simultaneous analysis and design sand formulation of the local stress and slope constrained topology design problem it is demonstrated that a standard trust region lagrange newton sequential quadratic programming sqp type algorithm based in this case on strictly convex and separable approximate subproblems may converge to singular optima of the local stress constrained problem without having to resort to relaxation or perturbation techniques moreover due to the negation of the sensitivity analyses in sand the density and displacement variables are independent and the immense sparsity of the sand problem solutions to large scale problem instances may be obtained in a reasonable amount of computation time this article is protected by copyright all rights reserved

approximation of pseudospectra of block triangular matrices abstract we approximate the pseudospectra of a block triangular matrix with two blocks on the diagonal in terms of the pseudospectra of its diagonal blocks

robust design of structures using optimization methods the robust design of structures with stochastic parameters is studied using optimization techniques the first two statistical moments of the stochastic parameters including design variables are considered in conjunction with the second order perturbation method for the approximation of mean value and variance of the structural response in this framework the sensitivities of the mean values and variances of the structural performance function with respect to the design variables are calculated for use in the optimization task the robust design of structures is formulated as a multi criteria optimization problem in which both the expected value and the standard deviation of the objective function are to be minimized the robustness of the feasibility is also taken into account by involving the variability of the structural response in the constraints the two criteria optimization problem is converted into a scalar one and is then solved by a gradient based optimization algorithm to demonstrate the applicability of the presented method numerical examples are given involving static and dynamic response

sampling algebraic sets in local intrinsic coordinates numerical data structures for positive dimensional solution sets of polynomial systems are sets of generic points cut out by random planes of complementary dimension we may represent the linear spaces defined by those planes either by explicit linear equations or in parametric form these descriptions are respectively called extrinsic and intrinsic representations while intrinsic representations lower the cost of the linear algebra operations we observe worse condition numbers in this paper we describe the local adaptation of intrinsic coordinates to improve the numerical conditioning of sampling algebraic sets local intrinsic coordinates also lead to a better step size control we illustrate our results with maple experiments and computations with phcpack on some benchmark polynomial systems

optimal design and relaxation of variational problems iii abstract this paper has attempted a synthesis of three subjects that have developed separately optimal design homogenization and relaxation in the calculus of variations the underlying theme is that of oscillations they occur with certain definite patterns in minimizing sequences for the variational problem the same patterns recur in nearly optimal designs and in the coefficients of the equation being homogenized each problem is resolved by determining the structure of those oscillations and above all their macroscopic averages it is those averages that solve the relaxed problem they also determine the optimal design and the solution of the homogenized equation we emphasize also the role of optimization the homogenized equations and composite materials of interest here come not from arbitrary mixtures but from those that are extremal since they correspond to minimizing sequences for the variational problem the effective properties of these composites attain the optimal bounds for given materials in fixed proportions

pole placement preconditioning the partial pole placement problem has received considerable attention in control theory where it is applied to stabilize time invariant linear control systems this paper discusses application of the partial pole placement problem to the construction of preconditioners for linear systems of equations numerical examples show that these preconditioners can improve the rate of convergence of the restarted gmres methods significantly

the condition number of equivalence transformations that block diagonalize matrix pencils how ill conditioned must a matrix s be if its columns are constrained to span certain subspaces we answer this question in order to find nearly best conditioned matrices sr and sl that block diagonalize a given matrix pencil t a b i e s l 1 tsr is bloc diagonal we show that the best conditioned sr has a condition number approximately equal to the cosecant of the smallest angle between right subspaces belonging to different diagonal blocks of thus the more nearly the right subspaces overlap the more ill conditioned sr must be the same is true of sl and the left subspaces for the standard eigenproblem t a i sl sr and the cosecant of the angle between subspaces turns out equal to an earlier estimate of the smallest condition number namely the norm of the projection matrix associated with one of the subspaces we apply this result to bound the error in an algorithm to compute analytic functions of matrices for instance exp t

primal dual interior point methods for semidefinite programming in finite precision recently a number of primal dual interior point methods for semidefinite programming have been developed to reduce the number of floating point operations each iteration of these methods typically performs block gaussian elimination with block pivots that are close to singular near the optimal solution as a result these methods often exhibit complex numerical properties in practice we consider numerical issues related to some of these methods our error analysis indicates that these methods could be numerically stable if certain coefficient matrices associated with the iterations are well conditioned but are unstable otherwise with this result we explain why one particular method the one introduced by alizadeh haeberly and overton is in general more stable than others we also explain why the so called least squares variation introduced for some of these methods does not yield more numerical accuracy in general finally we present results from our numerical experiments to support our analysis

minimum norm partial quadratic eigenvalue assignment with time delay in vibrating structures using the receptance and the system matrices abstract the partial quadratic eigenvalue assignment problem pqeap is to compute a pair of feedback matrices such that a small number of unwanted eigenvalues in a structure are reassigned to suitable locations while keeping the remaining large number of eigenvalues and the associated eigenvectors unchanged the problem arises in active vibration control of structures for real life applications it is not enough just to compute the feedback matrices they should be computed in such a way that both closed loop eigenvalue sensitivity and feedback norms are as small as possible also for practical effectiveness the time delay between the measurement of the state and implementation of the feedback controller should be considered while solving the pqeap these problems are usually solved using only system matrices and do not necessarily take advantage of the receptances which are available by measurements in this paper we propose hybrid methods combining the system matrices and measured receptances for solutions of the multi input pqeap and the minimum norm pqeap both for systems with and without time delay these hybrid methods are more efficient than the standard methods which only use the system matrices and not the receptances these hybrid methods also offer several other computational advantages over the standard methods our results generalize the recent work by ram et al partial pole placement with time delay in structures using the receptance and the system matrices linear algebra and its applications 434 2011 1689 1696 the results of numerical experiments demonstrate the effectiveness of the proposed methods

numerical methods in control we study classical control problems like pole assignment stabilization linear quadratic control and h control from a numerical analysis point of view we present several examples that show the difficulties with classical approaches and suggest reformulations of the problems in a more general framework we also discuss some new algorithmic approaches

robust optimization methodology and applications robust optimization ro is a modeling methodology combined with computational tools to process optimization problems in which the data are uncertain and is only known to belong to some uncertainty set the paper surveys the main results of ro as applied to uncertain linear conic quadratic and semidefinite programming for these cases computationally tractable robust counterparts of uncertain problems are explicitly obtained or good approximations of these counterparts are proposed making ro a useful tool for real world applications we discuss some of these applications specifically antenna design truss topology design and stability analysis synthesis in uncertain dynamic systems we also describe a case study of 90 lps from the netlib collection the study reveals that the feasibility properties of the usual solutions of real world lps can be severely affected by small perturbations of the data and that the ro methodology can be successfully used to overcome this phenomenon

the perturbation bounds for eigenspaces of a definite matrix pair leta andb be hermitian matrices the matrix pair a b is called definite pair and the corresponding eigenvalue problem ax bx is definite if c a b equiv mathop min limits left x right 1 x h a ib x 0 the perturbation bounds for eigenspaces of a definite pair on every unitary invariant matrix norm were obtained by imposing additional restrictions on the location of the generalized eigenvalues thus it gives a positive answer for an open question proposed by stewart 7 the famous davis kahan sin theorems and sin 2 theorem 2 can also be deduced from the present results

orthogonal polynomial expansions for the matrix exponential many different algorithms have been suggested for computing the matrix exponential in this paper we put forward the idea of expanding in either chebyshev legendre or laguerre orthogonal polynomials in order for these expansions to converge quickly we cluster the eigenvalues into diagonal blocks and accelerate using shifting and scaling

modal analysis of structures with holonomic constraints a formulation is presented to predict the modal parameters of constrained structural systems the formula tion employs the lagrange multiplier technique to investigate structural vibration before and after holonomic constraints are imposed holonomic constraint can either be of the linear scleronomic type that accounts for structure compatibility or of the rheonomic nonlinear type for structures with deployable or articulated components the vibration and stability of an integrated constrained structural system can be predicted from the modal parameters of the corresponding unconstrained system and the characteristics of holonomic con straints ecent developments in aerospace technology have focused attention on the technical aspects of large scale complex and constrained structural systems aircraft and spacecraft launch vehicles consist of a number of subsystems and components verification of structural design is typically performed using a limited series of tests on subsystem and component prototypes to overcome the increasing size of the structural model and to ease the complexity of structural modification during the development phase a method is needed wherein the vibration and stability of the integrated con strained structural systems can be evaluated large complicated systems are often solved by either de composition into components or subsystems and subsequent analysis or system modeling with kinematic constraints the former technique is similar to the so called component mode synthesis method 5 previous research applied these techniques to mechanism dynamic analysis 2 control system stability anal ysis 3 and modal synthesis 4 in which modal parameters such as natural frequency and mode shape of each subsystem or component are first determined modal analysis of an inte grated constrained structural system can then be conducted based on the predetermined modal parameters of the subsys tems and components a number of subsystem and component coupling procedures have been proposed for solving undamped structural systems fuh and chen5 developed a procedure to construct a transfor mation matrix for formulating structure modal synthesis with constraints kuang and tsuei6 presented a model reduction procedure for substructure mode synthesis although both pror cedures require an orthogonal complement of the constraint jacobiano matrix the matrix construction is neither efficient nor unique moreover most of the structure analyses to date consider only linear scleronomic constraints that account for structure compatibility at the interface of two adjacent com ponents little attempt has been made to include holonomic constraint at the interface or to predict its effect on the struc ture s natural frequencies when constraints are imposed nonlinear and or rheonomic constraints have been applied to the area of mechanism dynamics where numerical integra tion of system response is required 7 sohoni and whitesell8 developed a procedure for calculating the natural frequencies of a machine dynamics system however their formulation demands numerical identification of the state variables before

optimally conditioned block matrices this paper provides a description of block 2 2 hermitian positive definite matrices that are optimally conditioned with respect to block diagonal similarity transformations bibliography 7 titles

lipschitz and holder stability of optimization problems and generalized equations this paper studies stability aspects of solutions of parametric mathematical programs and generalized equations respectively with disjunctive constraints we present sufficient conditions that under some constraint qualifications ensuring metric subregularity of the constraint mapping continuity results of upper lipschitz and upper holder type respectively hold furthermore we apply the above results to parametric mathematical programs with equilibrium constraints and demonstrate how some classical results for the nonlinear programming problem can be recovered and even improved by our theory

modelling topology optimization problems as linear mixed 0 1 programs this paper deals with topology optimization of discretized continuum structures it is shown that a large class of non linear 0 1 topology optimization problems including stress and displacement constrained minimum weight problems can equivalently be modelled as linear mixed 0 1 programs the modelling approach is applied to some test problems which are solved to global optimality