adjoint based predictor corrector sequential convex programming for parametric nonlinear optimization this paper proposes an algorithmic framework for solving parametric optimization problems which we call adjoint based predictor corrector sequential convex programming after presenting the algorit

a modified broyden like quasi newton method for nonlinear equations abstract in this paper we introduce a modified broyden like quasi newton method for solving nonlinear equations which is an acceleration version of the broyden like method presented by li and fukushima 2000 we show that the proposed method has global and superlinear convergence properties we report some numerical results to show its efficiency

a decomposition property of polyhedra a polyhedronp x n d ax e is said to have the real decomposition property rdp if for any positivet and any realx tp there are positive coefficients 1 r and integers1 sr p withx lsl rsr andt l r we give a constructive proof that this property holds for a polyhedronp iffp is integral this construction is used to show that some classes of polyhedra have an integral decomposition property furthemore the rdp provides a generalization of the theorem of birkhoff von neumann so rdp may be used in some scheduling problems on parallel processors with preemptions

a numerical study of the limited memory bfgs method and the truncated newton method for large scale optimization this paper examines the numerical performances of two methods for large scale optimization a limited memory quasi newton method l bfgs and a discrete truncated newton method tn various ways of classifying test problems are discussed in order to better understand the types of problems that each algorithm solves well the l bfgs and tn methods are also compared with the polak ribiere conjugate gradient method

on a class of totally unimodular matrices we examine the class of totally unimodular matrices that contain no odd cycles which we call restricted totally unimodular rtum we show that a matrix is rtum if and only if it can be decomposed in a very simple way into the incidence matrices or their transposes of bipartite graphs or directed graphs and give a linear time algorithm to perform this task based on this decomposition we show that the 0 1 integer programming problem with an rtum matrix of constraints has the same time complexity as the b matching and the max flow problems

on the global convergence of the bfgs method for nonconvex unconstrained optimization problems this paper is concerned with the open problem of whether the bfgs method with inexact line search converges globally when applied to nonconvex unconstrained optimization problems we propose a cautious bfgs update and prove that the method with either a wolfe type or an armijo type line search converges globally if the function to be minimized has lipschitz continuous gradients

arithmetic of normal rees algebras two reasons drive this study of the arithmetical properties of blow up like algebras first one still lacks effective criteria for the rees algebra of an ideal to be normal and second there are several new phenomena related to torsionfree symmetric algebras of modules notably bundlesthat need clarification as yet another related question there is the description of the canonical modules of such algebras a necessary step in determining their gorenstein loci the main thread for these questions is the understanding of the divisor class group essentially defined by a pair made up of a morphism and a free subgroup of finite rank the behaviour of the mapping has puzzling

variable metric methods for unconstrainted optimization and nonlinear least squares variable metric or quasi newton methods are well known and commonly used in connection with unconstrained optimization since they have good theoretical and practical convergence properties although these methods were originally developed for small and moderate size dense problems their modifications based either on sparse partitioned or limited memory updates are very efficient on large scale sparse problems very significant applications of these methods also appear in nonlinear least squares approximation and nonsmooth optimization in this contribution we give an extensive review of variable metric methods and their use in various optimization fields

a tool for the analysis of quasi newton methods with application to unconstrained minimization the bfgs update formula is shown to have an important property that is inde pendent of the algorithmic context of the update and that is relevant to both constrained and unconstrained optimization the bfgs method for unconstrained optimization using a variety of line searches including backtracking is shown to be globally and superlinearly convergent on uniformly convex problems the analysis is particularly simple due to the use of some new tools introduced in this paper

m sequences graph ideals and ladder ideals of linear type in this paper we study monomial ideals and ladder determinantal ideals of linear type and their blow up algebras our main tools are grobner bases and sagbi bases deformations and the notion of m sequence of monomials w x let r s k x be a polynomial ring over a field k equipped with a monomial order t let i be an ideal of r generated by polynomials f f consider the presentation 1 s w x w x w x c r t s r t t a r i s r f t f t 1 s i s

global convergence of a cass of quasi newton methods on convex problems we study the global convergence properties of the restricted broyden class of quasi newton methods when applied to a convex objective function we assume that the line search satisfies a standard sufficient decrease condition and that the initial hessian approximation is any positive definite matrix we show global and superlinear convergence for this class of methods except for dfp this generalizes powell s well known result for the bfgs method the analysis gives us insight into the properties of these algorithms in particular it shows that dfp lacks a very desirable self correcting property possessed by bfgs

sparse sums of squares on finite abelian groups and improved semidefinite lifts let g be a finite abelian group this paper is concerned with nonnegative functions on g that are sparse with respect to the fourier basis we establish combinatorial conditions on subsets mathcal s s and mathcal t t of fourier basis elements under which nonnegative functions with fourier support mathcal s s are sums of squares of functions with fourier support mathcal t t our combinatorial condition involves constructing a chordal cover of a graph related to g and mathcal s s the cayley graph mathrm cay widehat g mathcal s cay g s with maximal cliques related to mathcal t t our result relies on two main ingredients the decomposition of sparse positive semidefinite matrices with a chordal sparsity pattern as well as a simple but key observation exploiting the structure of the fourier basis elements of g the characters of g we apply our general result to two examples first in the case where g mathbb z 2 n g z2n by constructing a particular chordal cover of the half cube graph we prove that any nonnegative quadratic form in n binary variables is a sum of squares of functions of degree at most left lceil n 2 right rceil n 2 establishing a conjecture of laurent second we consider nonnegative functions of degree d on mathbb z n zn when d divides n by constructing a particular chordal cover of the dth power of the n cycle we prove that any such function is a sum of squares of functions with at most 3d log n d 3dlog n d nonzero fourier coefficients dually this shows that a certain cyclic polytope in mathbb r 2d r2d with n vertices can be expressed as a projection of a section of the cone of positive semidefinite matrices of size 3d log n d 3dlog n d putting n d 2 n d2 gives a family of polytopes in mathbb r 2d r2d with linear programming extension complexity varomega d 2 d2 and semidefinite programming extension complexity o d log d o dlog d to the best of our knowledge this is the first explicit family of polytopes p d pd in increasing dimensions where mathrm xc text psd p d o mathrm xc text lp p d xcpsd pd o xclp pd where mathrm xc text psd xcpsd and mathrm xc text lp xclp are respectively the sdp and lp extension complexity

the equivalence of strict convexity and injectivity of the gradient in bounded level sets it is shown that lipschitzian functions are strictly convex if and only if their generalized gradients are disjoint at distinct interior points of a given bounded level set

the convergence of variable metric matrices in unconstrained optimization it is proved that if the dfp or bfgs algorithm with step lengths of one is applied to a functionf x that has a lipschitz continuous second derivative and if the calculated vectors of variables converge to a point at which f is zero and 2f is positive definite then the sequence of variable metric matrices also converges the limit of this sequence is identified in the case whenf x is a strictly convex quadratic function

a quasi newton method for estimating the parameter in a nonlinear hyperbolic system a quasi newton method qnm in infinite dimensional spaces for estimating parameters involved in distributed parameter systems is presented in this paper next the global convergence of a sequence generated by the algorithm qnm is also proved we apply the algorithm qnm to an identification problem of a distributed parameter system governed by a nonlinear hyperbolic partial differential equation

a sampling kaczmarz motzkin algorithm for linear feasibility we combine two iterative algorithms for solving large scale systems of linear inequalities the relaxation method of agmon motzkin et al and the randomized kaczmarz method we obtain a family of algorithms that generalize and extend both projection based techniques we prove several convergence results and our computational experiments show our algorithms often outperform the original methods

proximal level bundle methods for convex nondifferentiable optimization saddle point problems and variational inequalities we study proximal level methods for convex optimization that use projections onto successive approximations of level sets of the objective corresponding to estimates of the optimal value we show that they enjoy almost optimal efficiency estimates we give extensions for solving convex constrained problems convex concave saddle point problems and variational inequalities with monotone operators we present several variants establish their efficiency estimates and discuss possible implementations in particular our methods require bounded storage in contrast to the original level methods of lemarechal nemirovskii and nesterov

successive refinement of large multicell models this paper presents and analyzes a new relaxation method for minimizing a function of many variables the minimized function is associated with a model that is composed of a large number of cells in some applications the terms elements mesh points or facilities are more appropriate such problems often arise in models that are based on finite difference or finite element methods the new method which we call the successive refinement method utilizes the special structure of the model to save on both storage and computations although the s r method has many features in common with the nonlinear s o r method there is a basic difference between the two methods the difference is that the steps of the s r method are not using relaxation parameters instead each of its steps is aimed to reduce the objective function value as much as possible this gives us a robust algorithm that is able to solve problems that the s o r method cannot handle such problems are for example minimization of a

superlinear convergence of broyden s bounded θ class of methods in this paper the so called broyden s bounded class of methods is considered it contains as a subclass broyden s restricted class of methods in which the updating matrices retain symmetry and positive definiteness these iteration methods are used for solving unconstrained minimization problems of the following form f hat x min x in r n f x it is assumed that the step size coefficient k 1 in each iteration and the functionalf rn r1 satisfies the standard assumptions viz f is twice continuously differentiable and the hessian matrix is uniformly positive definite and bounded there exist constantsm m 0 such that m y 2 y m parallel y parallel 2 leqslant left langle y f hat x left y right rangle right leqslant m parallel y parallel 2 for ally rn and satisfies a lipschitz like condition at the optimal point hat x the gradient vanishes at hat x under these assumptions the local convergence of broyden s methods is proved furthermore the q superlinear convergence is shown

on the smoothness of convex envelopes we examine differentiability properties of the convex envelope conv e of a given function e rn cc cc in terms of properties of e it is shown that c1 as well as optimal c1 a regularity results 0 a 1 can be obtained under general conditions

decomposition in conic optimization with partially separable structure decomposition techniques for linear programming are difficult to extend to conic op timization problems with general nonpolyhedral convex cones because the conic inequalities introduce an additional nonlinear coupling between the variables however in many applications the convex cones have a partially separable structure that allows them to be characterized in terms of simpler lower dimensional cones the most important example is sparse semidefinite programming with a chordal sparsity pattern here partial separability derives from the clique decomposition theorems that characterize positive semidefinite and positive semidefinite completable matrices with chordal sparsity patterns the paper describes a decomposition method that exploits partial separability in conic linear optimization the method is based on spingarn s method for equality constrained convex optimization combined with a fast interior point method for evaluating proximal operators

on the flatness of models of certain shimura varieties of pel type consider a pel shimura variety associated to a unitary group that splits over an unramified extension of q p rapoport and zink have defined a model of the shimura variety over the ring of integers of the completion of the reflex field at a place lying over p with parahoric level structures at p we show that this model is flat as conjectured by rapoport and zink and that its special fibre is reduced

the global convergence of partitioned bfgs on problems with convex decompositions and lipschitzian gradients the main purpose of this paper is the extension of powell s 1976 global convergence result to the partitioned bfgs method introduced by griewank and toint 1982 even in the unpartitioned case the original result is strengthened because the search directions need not be computed exactly and the gradient is only required to be lipschitzian rather than differentiable using the functional of byrd and nocedal 1987 a strong form of r superlinear convergence is obtained if the element functions are uniformly convex and their gradients are strictly differentiable at the minimizerx only in order to deal with the possibility of singular functions we utilize a damping of the bfgs update that becomes inactive if the problem turns out to be regular nearx

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

a globally and superlinearly convergent gauss newton based bfgs method for symmetric nonlinear equations in this paper we present a gauss newton based bfgs method for solving symmetric nonlinear equations which contain as a special case an unconstrained optimization problem a saddle point problem and an equality constrained optimization problem a suitable line search is proposed with which the presented bfgs method exhibits an approximate norm descent property under appropriate conditions global convergence and superlinear convergence of the method are established the numerical results show that the proposed method is successful

iterative solution of integral equations by a quasi newton method abstract when broyden s type methods are applied to the solution of integral equations they can be specialized in such a way to generate in the course of the iteration low rank degenerate kernel equations we describe the features of this approach

sequential quadratic programming sqp for optimal control in direct numerical simulation of turbulent flow the current work focuses on the development and application of an efficient algorithm for optimization of three dimensional turbulent flows simulated using direct numerical simulation dns or large eddy simulations and further characterized by large dimensional optimization parameter spaces the optimization algorithm is based on sequential quadratic programming sqp in combination with a damped formulation of the limited memory bfgs method the latter is suitable for solving large scale constrained optimization problems whose hessian matrices cannot be computed and stored at a reasonable cost we combine the algorithm with a line search merit function based on an l 1 norm to enforce the convergence from any remote point it is first shown that the proposed form of the damped l bfgs algorithm is suitable for solving equality constrained rosenbrock type functions then we apply the algorithm to an optimal control test problem that consists of finding the optimal initial perturbations to a turbulent temporal mixing layer such that mixing is improved at the end of a simulation time horizon t the controls are further subject to a non linear equality constraint on the total control energy dnss are used to resolve all turbulent scales of motion and a continuous adjoint formulation is employed to calculate the gradient of the cost functionals we compare the convergence speed of the sqp l bfgs algorithm to a conventional non linear conjugate gradient method i e the current standard in dns based optimal control and find that the sqp algorithm is more than an order of magnitude faster than the conjugate gradient method

second order methods for optimal control of time dependent fluid flow second order methods for open loop optimal control problems governed by the two dimensional instationary navier stokes equations are investigated optimality systems based on a lagrangian formulation and adjoint equations are derived the newton and quasi newton methods as well as various variants of sqp methods are developed for applications to optimal flow control and their complexity in terms of system solves is discussed local convergence and rate of convergence are proved a numerical example illustrates the feasibility of solving optimal control problems for two dimensional instationary navier stokes equations by second order numerical methods in a standard workstation environment

on the relation between two local convergence theories of least change secant update methods in this paper we show that the main results of the local convergence theory for least change secant update methods of dennis and walker siam j numer anal 18 1981 949 987 can be proved using the theory introduced recently by martinez math comp 55 1990 143 167 in addition we exhibit two generalizations of well known methods whose local convergence can be easily proved using martinez s theory

algebras defined by powers of determinantal ideals let x be an m x n matrix of indeterminates over a field k a k x and i z x the ideal generated by the t minors of x the main objects of this article are the rees algebra 2 y these characteristics will be called non exceptional we show how this result generalizes to an arbitrary integral domain of coefficients the intersection of primary ideals which gives i in non exceptional characteristics always is the integral closure of i it follows immediately from the primary decomposition that the powers i are integrally closed in non exceptional characteristics therefore is a normal domain and the primary decomposition of i turns out easy giving some insight into the structure of y an interesting observation the primary decomposition of the ideals i can be computed very quickly if one knows in advance that all these ideals are integrally closed the best results are obtained in characteristic 0 since one has a multiplicity free action of the linearly reductive group gl m k x gl n k on k x under which z is stable cf dep or bv sect 111 applying the theory of u invariants kraft kr one shows that g has rational singularities in particular and hence 9 are cohen macaulay rings we have no doubt that and 3 are cohen macaulay in arbitrary non exceptional characteristic it seems however that in exceptional characteristic they are as far as possible from this property the case t 2

a partitioned psb method for partially separable unconstrained optimization problems in this paper we propose a partitioned psb method for solving partially separable unconstrained optimization problems by using a projection technique we construct a sufficient descent direction under appropriate conditions we show that the partitioned psb method with projected direction is globally and superlinearly convergent for uniformly convex problems in particular the unit step length is accepted after finitely many iterations finally some numerical results are presented which show that the partitioned psb method is effective and competitive

on support varieties for lie superalgebras and finite supergroup schemes abstract we study the spectrum of the cohomology rings of cocommutative hopf superalgebras restricted and non restricted lie superalgebras and finite supergroup schemes we also investigate support varieties in these settings and demonstrate that they have the desirable properties of such a theory we completely characterize support varieties for finite supergroup schemes over algebraically closed fields of characteristic zero while for non restricted lie superalgebras in positive characteristic we obtain descriptions that are suggestive of varieties previously defined by duflo and serganova in characteristic zero but which have no known cohomological interpretation our computations for restricted lie superalgebras and infinitesimal supergroup schemes provide natural generalizations of foundational results of friedlander and parshall and of bendel friedlander and suslin in the classical setting

bfgs with update skipping and varying memory we give conditions under which limited memory quasi newton methods with exact line searches will terminate in n steps when minimizing n dimensional quadratic functions we show that although all broyden family methods terminate in n steps in their full memory versions only bfgs does so with limited memory additionally we show that full memory broyden family methods with exact line searches terminate in at most n p steps when p matrix updates are skipped we introduce new limited memory bfgs variants and test them on nonquadratic minimization problems

superlinear convergence of affine scaling interior point newton methods for infinite dimensional nonlinear problems with pointwise bounds we develop and analyze a superlinearly convergent affine scaling interior point newton method for infinite dimensional problems with pointwise bounds in lp space the problem formulation is motivated by optimal control problems with lp controls and pointwise control constraints the finite dimensional convergence theory by coleman and li siam j optim 6 1996 pp 418 445 makes essential use of the equivalence of norms and the exact identifiability of the active constraints close to an optimizer with strict complementarity since these features are not available in our infinite dimensional framework algorithmic changes are necessary to ensure fast local convergence the main building block is a newton like iteration for an affine scaling formulation of the kkt condition we demonstrate in an example that a stepsize rule to obtain an interior iterate may require very small stepsizes even arbitrarily close to a nondegenerate solution using a pointwise projection instead we prove superlinear convergence under a weak strict complementarity condition and convergence with mbox q rate 1 under a slightly stronger condition if a smoothing step is available we discuss how the algorithm can be embedded in the class of globally convergent trust region interior point methods recently developed by m heinkenschloss and the authors numerical results for the control of a heating process confirm our theoretical findings

a pde sensitivity equation method for optimal aerodynamic design the use of gradient based optimization algorithms in inverse design is well established as a practical approach to aerodynamic design a typical procedure uses a simulation scheme to evaluate the objective function from the approximate states and its gradient then passes this information to an optimization algorithm once the simulation scheme cfd flow solver has been selected and used to provide approximate function evaluations there are several possible approaches to the problem of computing gradients one popular method is to differentiate the simulation scheme and compute design sensitivities that are then used to obtain gradients although this black box approach has many advantages in shape optimization problems one must compute mesh sensitivities in order to compute the design sensitivity in this paper we present an alternative approach using the pde sensitivity equation to develop algorithms for computing gradients this approach has the advantage that mesh sensitivities need not be computed moreover when it is possible to use the cfd scheme for both the forward problem and the sensitivity equation then there are computational advantages an apparent disadvantage of this approach is that it does not always produce consistent derivatives however for a proper combination of discretization schemes one can showasymptotic consistencyunder mesh refinement which is often sufficient to guarantee convergence of the optimal design algorithm in particular we show that when asymptotically consistent schemes are combined with a trust region optimization algorithm the resulting optimal design method converges we denote this approach as thesensitivity equation method the sensitivity equation method is presented convergence results are given and the approach is illustrated on two optimal design problems involving shocks

a double parameter scaled bfgs method for unconstrained optimization abstract a double parameter scaled bfgs method for unconstrained optimization is presented in this method the first two terms of the known bfgs update formula are scaled with a positive parameter while the third one is scaled with another positive parameter these parameters are selected in such a way as to improve the eigenvalues structure of the bfgs update the parameter scaling the first two terms of the bfgs update is determined by clustering the eigenvalues of the scaled bfgs matrix on the other hand the parameter scaling the third term is determined as a preconditioner to the hessian of the minimizing function combined with the minimization of the conjugacy condition from conjugate gradient methods under the inexact wolfe line search the global convergence of the double parameter scaled bfgs method is proved in very general conditions without assuming the convexity of the minimizing function using 80 unconstrained optimization test functions with a medium number of variables the preliminary numerical experiments show that this double parameter scaled bfgs method is more efficient than the standard bfgs update or than some other scaled bfgs methods

sequentially decomposed programming model based decomposition is a powerful tool for breaking design problems into smaller subproblems establishing hierarchical structure and analyzing the interrelations in engineering design problems however the theoretical foundation for solving decomposed nonlinear optimization problems requires further work we show that theformulation of the coordination problem is critical in quickly identifying thecorrectactiveconstraints and that solving subproblems independently may hinder the local convergence of algorithms tailored to hierarchical coordination yet hierarchical decomposition algorithms can have excellent global convergence properties and can be expected to exhibit superior improvement in the rst few iterations when compared to the undecomposed case basedontheseinsights agenericsequentially decomposedprogramming sdp algorithmisoutlined sdphastwo phases far from the solution rst phase decomposition is used close to the solution second phase subproblems are not solved separately the generic sdp is applied to sequential quadratic programming sqp to de ne an sdp sqp implementation a global convergence proof and a simple example are given

a quasi newton method for solving fixed point problems in hilbert spaces we study the convergence properties of a projective variant of newton s method for the approximation of fixed points of completely continuous operators in hilbert spaces we then discuss applications to nonlinear integral equations and we produce some numerical examples

on the variety of complexes publisher summary this chapter focuses on the characteristic free study of the buchsbaum eisenbud varieties of complexes in detail with the use of young diagrams and young tableaux to give an explicit basis for buchsbaum and eisenbud have studied the varieties of complexes from the point of view of resolutions and have proven the results for the variety w k 1 a young tableau is a filling of the boxes of a young diagram with integers out of 1 2 and is called the shape of the young tableau the chapter presents a proposition that states that ach element in can be written as a linear combination of standard multitableaux

additively decomposed quasiconvex functions letf be a real valued function defined on the product ofm finite dimensional open convex setsx1 xm assume thatf is quasiconvex and is the sum of nonconstant functionsf1 fm defined on the respective factor sets then everyfi is continuous with at most one exception every functionfi is convex if the exception arises all the other functions have a strict convexity property and the nonconvex function has several of the differentiability properties of a convex function we define the convexity index of a functionfi appearing as a term in an additive decomposition of a quasiconvex function and we study the properties of that index in particular in the case of two one dimensional factor sets we characterize the quasiconvexity of an additively decomposed functionf either in terms of the nonnegativity of the sum of the convexity indices off1 andf2 or equivalently in terms of the separation of the graphs off1 andf2 by means of a logarithmic function we investigate the extension of these results to the case ofm factor sets of arbitrary finite dimensions the introduction discusses applications to economic theory

properties and numerical performance of quasi newton methods with modified quasi newton equations quasi newton qn equation plays a core role in contemporary nonlinear optimization the usual qn equation employs only the gradients but ignores the available function value information in this paper we derive a class of modified qn equations with a vector parameter which use both available gradient and function value information the modified quasi newton methods maintain most properties of the usual quasi newton methods meanwhile they achieve a higher order accuracy in approximating the second order curvature of the problem functions than the usual ones do numerical experiments are reported which support the theoretical analyses and show the advantages of the modified qn methods over the usual ones

on the limited memory bfgs method for large scale optimization we study the numerical performance of a limited memory quasi newton method for large scale optimization which we call the l bfgs method we compare its performance with that of the method developed by buckley and lenir 1985 which combines cycles of bfgs steps and conjugate direction steps our numerical tests indicate that the l bfgs method is faster than the method of buckley and lenir and is better able to use additional storage to accelerate convergence we show that the l bfgs method can be greatly accelerated by means of a simple scaling we then compare the l bfgs method with the partitioned quasi newton method of griewank and toint 1982a the results show that for some problems the partitioned quasi newton method is clearly superior to the l bfgs method however we find that for other problems the l bfgs method is very competitive due to its low iteration cost we also study the convergence properties of the l bfgs method and prove global convergence on uniformly convex problems

the convergence of matrices generated by rank 2 methods from the restricted β class of broyden it is shown that the matricesb k generated by any method from the restricted class of broyden converge if the method is applied to the unconstrained minimization of a functionf c 2 r n with lipschitz continuous 2 f x and if the method is such that it generates vectorsx k converging sufficiently fast sum limits k x k x infty to a local minimumx off with positive definite 2 f x this result not only holds for constant step sizes k 1 in each iterationx k x k 1 x k k b k 1 f x k of these methods but also for step sizes determined by asymptotically exact line searches the paper generalizes corresponding results of ge ren pu and powell 6 for the dfp and bfgs methods used in conjunction with step sizes k 1

on categories of equivariant d modules abstract let x be a variety with an action by an algebraic group g in this paper we discuss various properties of g equivariant d modules on x such as the decompositions of their global sections as representations of g when g is reductive and descriptions of the categories that they form when g acts on x with finitely many orbits the category of equivariant d modules is equivalent to the category of finite dimensional representations of a finite quiver with relations we describe explicitly these categories for irreducible g modules x that are spherical varieties and show that in such cases the quivers are almost always representation finite i e with finitely many indecomposable representations

polynomial algorithms for a class of linear programs this paper describes several algorithms for solution of linear programs the algorithms are polynomial when the problem data satisfy certain conditions

intersection theory of linear embeddings we study intersection theoretic properties of subschemes defined by ideal sheaves of linear type in particular their behavior with respect to blowups and segre classes

max flow min cut matroids polynomial testing and polynomial algorithms for maximum flow and shortest routes a polynomial test for deciding whether or not a given binary matroid has the max flow min cut property is described in addition polynomial algorithms for finding shortest routes and extreme integral max flows in max flow min cut matroids are developed the algorithms make use of a decomposition result for the max flow min cut matroids that was previously established in joint work with f t tseng the max flow algorithm also implies a strengthened version of p d seymour s original characterization of the max flow min cut matroids

the global convergence of a modified bfgs method for nonconvex functions abstract the standard bfgs method plays an important role among the quasi newton algorithms for constrained un constrained optimization problems however dai 2003 constructed a counterexample to demonstrate that this method may fail for non convex functions with inexact wolfe line searches and mascarenhas 2004 showed the nonconvergence of this method and other methods in the broyden family even with exact line search techniques these works motivate us to try to find another way to obtain another quasi newton method with better convergence based on the standard bfgs formula in this paper four approaches are used in the designed algorithm 1 a modified weak wolfe powell line search technique is introduced 2 if one defined condition is satisfied then the search direction and its associated step length are accepted and the next point is designed 3 otherwise a parabolic will be presented and it is regarded as the projection surface to avoid using the failed direction and the next point x k 1 is generated by a projection technique and 4 to easily obtain the global convergence of the proposed algorithm the projection point is not used in the current bfgs update but rather for all the next iterations the new algorithm possesses global convergence for general functions under the inexact modified weak wolfe powell line search technique and it is shown that other methods in the broyden class also have this property numerical results are reported to show the performance of the given algorithm and other similar methods

sparse quasi newton updates with positive definite matrix completion quasi newton methods are powerful techniques for solving unconstrained minimization problems variable metric methods which include the bfgs and dfp methods generate dense positive definite approximations and therefore are not applicable to large scale problems to overcome this difficulty a sparse quasi newton update with positive definite matrix completion that exploits the sparsity pattern e i j nabla 2 f x ij neq 0 rm for some x in r n of the hessian is proposed the proposed method first calculates a partial approximate hessian h qn ij i j in f where f supseteq e using an existing quasi newton update formula such as the bfgs or dfp methods next a full matrix h k 1 which is a maximum determinant positive definite matrix completion of h qn ij i j in f is obtained if the sparsity pattern e or its extension f has a property related to a chordal graph then the matrix h k 1 can be expressed as products of some sparse matrices the time and space requirements of the proposed method are lower than those of the bfgs or the dfp methods in particular when the hessian matrix is tridiagonal the complexities become o n the proposed method is shown to have superlinear convergence under the usual assumptions

a pointwise quasi newton method for integral equations for fully nonlinear integral equations conventional quasi newton methods such as broyden s method may not possess good convergence properties in this paper we present a new method that gives superlinear convergence for such integral equations and provides for savings over newton s method in the cost of solution of the linear equation for the quasi newton step

convergence of quasi newton matrices generated by the symmetric rank one update quasi newton algorithms for unconstrained nonlinear minimization generate a sequence of matrices that can be considered as approximations of the objective function second derivatives this paper gives conditions under which these approximations can be proved to converge globally to the true hessian matrix in the case where the symmetric rank one update formula is used the rate of convergence is also examined and proven to be improving with the rate of convergence of the underlying iterates the theory is confirmed by some numerical experiments that also show the convergence of the hessian approximations to be substantially slower for other known quasi newton formulae

numerical solution of large sets of algebraic nonlinear equations this paper describes the application of the partitioned updating quasi newton methods for the solution of high dimensional systems of algebraic nonlinear equations this concept was introduced and successfully tested in nonlinear optimization of partially separable functions see 6 here its application to the case of nonlinear equations is explored nonlinear systems of this nature arise in many large scale applications including finite elements and econometry it is shown that the method presents some advantages in efficiency over competing algorithms and that use of the partially separable structure of the system can lead to significant improvements also in the more classical discrete newton method

preconditioned conjugate and secant newton methods for non linear problems the preconditioned conjugate gradient cg method is becoming accepted as a powerful tool for solving the linear systems of equations resulting from the application of the finite element method applications of the non linear algorithm are mainly confined to the diagonally scaled cg in this study the coupling of preconditioning techniques with non linear versions of the conjugate gradient and quasi newton methods creates a set of conjugate and secant newton methods for the solution of non linear problems the preconditioning matrices used to improve the ellipticity of the problem and to reduce the computer storage requirements are obtained by the application of the partial preconditioning and the partial elimination techniques both techniques use a drop off parameter to control the computer storage demands of the method making it more versatile for any computer hardware environment consideration is given to the development of a highly effective stability test for the line search minimization routine which computes accurate values without much effort this results in a beneficiary effect not only on the convergence properties of the methods but on their efficiency as well

on representing the positive semidefinite cone using the second order cone the second order cone plays an important role in convex optimization and has strong expressive abilities despite its apparent simplicity second order cone formulations can also be solved more efficiently than semidefinite programming problems in general we consider the following question posed by lewis and glineur parrilo saunderson is it possible to express the general positive semidefinite cone using second order cones we provide a negative answer to this question and show that the 3 times 3 positive semidefinite cone does not admit any second order cone representation in fact we show that the slice consisting of 3 times 3 positive semidefinite hankel matrices does not admit a second order cone representation our proof relies on exhibiting a sequence of submatrices of the slack matrix of the 3 times 3 positive semidefinite cone whose second order cone rank grows to infinity

a note on the superlinear convergence of gmres in the paper it is shown how the rate of convergence of the gmres method for solving a linear operator equation lambda i k u f in a hilbert space is related to the degree of compactness of k measured by the products of its singular values

the local convergence of broyden like methods on lipschitzian problems in hilbert spaces the classical local convergence analysis for broyden s method requires a finite number of real variables holder continuity of the jacobian and a rather accurate initial approximation b 0 here these three conditions are relaxed to allow for lipschitz continuous hilbert space operators that can be reasonably well approximated by a linear operator b that is close to b 0 up to an arbitrary compact perturbation to achieve local and r linear convergence broyden s method is modified by a very mild line search condition in case of a continuously differentiable system in euclidean space we obtain r superlinear convergence without restricting b 0 or requiring the jacobian to be holder continuous

unconstrained duals to partially separable constrained programs in this note partially separable convex programs are dualized in such a way that under certain assumptions unconstrained concave duals arise a return formula is given by which the solution of the primal is directly computed if a solution of the dual is known further the solvability of both the primal and the dual is shown to depend essentially on the behaviour of the lower dimensional programs for determining the fenchel conjugates

convergence properties of minimization algorithms for convex constraints using a structured trust region in this paper we present a class of trust region algorithms for minimization problems within convex feasible regions in which the structure of the problem is explicitly used in the definition of the trust region this development is intended to reflect the possibility that some parts of the problem may be more accurately modelled than others a common occurrence in large scale nonlinear applications after describing the structured trust region mechanism we prove global convergence for all algorithms in our class

on the conormal bundle of the determinantal variety call w such variety if we choose bases b v a a b u v i b for each module we can identify w with the variety of pairs of matrices y x with entries in r x being an m x n matrix y an ii x m matrix such that x y omx y x onxm now let a be the coordinate ring of the affine space a of dimension 2 m x n i e the polynomial ring r x yji i l m j i n and

images of connected sets by semicontinuous multifunctions abstract the image of a connected set by an upper semicontinuous or a lower semicontinuous multifunction whose values are nonempty and connected is connected we prove this theorem in its most general setting and show its usefulness in various examples from optimization and nonlinear analysis

a globally convergent bfgs method for nonlinear monotone equations without any merit functions r since 1965 there has been significant progress in the theoretical study on quasi newton methods for solving nonlinear equations especially in the local convergence analysis however the study on global convergence of quasi newton methods is relatively fewer especially for the bfgs method to ensure global convergence some merit function such as the squared norm merit function is typically used in this paper we propose an algorithm for solving nonlinear monotone equations which combines the bfgs method and the hyperplane projection method we also prove that the proposed bfgs method converges globally if the equation is monotone and lipschitz continuous without differentiability requirement on the equation which makes it possible to solve some nonsmooth equations an attractive property of the proposed method is that its global convergence is independent of any merit function we also report some numerical results to show efficiency of the proposed method

the approximation of generalized turning points newline by krylov subspace methods certain types of singular solutions of nonlinear parameter dependent operator equations were characterized by griewank and reddien 5 6 as regular solutions of suitable augmented systems for their numerical approximation an approach based on the use of krylov subspaces is here presented the application to boundary value problems is illustrated by numerical examples

projection algorithms for linear programming abstract based on the nearest point projection of geometric convexity we give a general projection approach for solving the feasibility problem of linear programming application of shor s method of space dilation gives rise to a family of polynomial time ellipsoidal algorithms with improved termination criteria in case of infeasibility moreover the approach renders possible application of various techniques from nonlinear programming in particular using a variable metric algorithm with exact line search we obtain a fast and practically well behaving algorithm for linear programming

on the existence of convex decompositions of partially separable functions the concept of a partially separable functionf developed in 4 is generalized to include all functionsf that can be expressed as a finite sum of element functionsf i whose hessians have nontrivial nullspacesn i such functions can be efficiently minimized by the partitioned variable metric methods described in 5 provided that each element functionf i is convex if this condition is not satisfied we attempt toconvexify the given decomposition by shifting quadratic terms among the originalf i such that the resulting modified element functions are at least locally convex to avoid tests on the numerical value of the hessian we study the totally convex case where all locally convexf with the separability structuren i 1 have a convex decomposition it is shown that total convexity only depends on the associated linear conditions on the hessian matrix in the sparse case when eachn i is spanned by cartesian basis vectors it is shown that a sparsity pattern corresponds to a totally convex structure if and only if it allows a permuted ldlt factorization without fill in

chordal decomposition in operator splitting methods for sparse semidefinite programs we employ chordal decomposition to reformulate a large and sparse semidefinite program sdp either in primal or dual standard form into an equivalent sdp with smaller positive semidefinite psd constraints in contrast to previous approaches the decomposed sdp is suitable for the application of first order operator splitting methods enabling the development of efficient and scalable algorithms in particular we apply the alternating direction method of multipliers admm to solve decomposed primal and dual standard form sdps each iteration of such admm algorithms requires a projection onto an affine subspace and a set of projections onto small psd cones that can be computed in parallel we also formulate the homogeneous self dual embedding hsde of a primal dual pair of decomposed sdps and extend a recent admm based algorithm to exploit the structure of our hsde the resulting hsde algorithm has the same leading order computational cost as those for the primal or dual problems only with the advantage of being able to identify infeasible problems and produce an infeasibility certificate all algorithms are implemented in the open source matlab solver cdcs numerical experiments on a range of large scale sdps demonstrate the computational advantages of the proposed methods compared to common state of the art solvers

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

broyden s method in hilbert space broyden s method is formulated for the solution of nonlinear operator equations in hilbert spaces the algorithm is proven to be well defined and a linear rate of convergence is shown under an additional assumption on the initial approximation for the derivative we prove the superlinear rate of convergence

a modified bfgs method and its global convergence in nonconvex minimization in this paper we propose a modification of the bfgs method for unconstrained optimization a remarkable feature of the proposed method is that it possesses a global convergence property even without convexity assumption on the objective function under certain conditions we also establish superlinear convergence of the method

sums of squares and semidefinite program relaxations for polynomial optimization problems with structured sparsity unconstrained and inequality constrained sparse polynomial optimization problems pops are considered a correlative sparsity pattern graph is defined to find a certain sparse structure in the objective and constraint polynomials of a pop based on this graph sets of the supports for sums of squares sos polynomials that lead to efficient sos and semidefinite program sdp relaxations are obtained numerical results from various test problems are included to show the improved performance of the sos and sdp relaxations

quasi newton methods for solving underdetermined nonlinear simultaneous equations abstract we analyze iterative processes of type xk 1 xk xk ek f xk for solving f x 0 f r n r m m n parameters ek are updated at each iteraction using least change secant update procedures we prove local linear and superlinear convergence results we introduce two new superlinearly convergent methods of this type and one linearly convergent quasi newton generalization of cimmino s parallel algorithm for solving linear systems some numerical experiments are presented

element by element preconditioners for large partially separable optimization problems we study the solution of large scale nonlinear optimization problems by methods which aim to exploit their inherent structure in particular we consider the property of partial separability first studied by griewank and toint nonlinear optimization 1981 pp 301 312 a typical minimization method for nonlinear optimization problems approximately solves a sequence of simplified linearized subproblems in this paper we explore how partial separability may be exploited by iterative methods for solving these subproblems we particularly address the issue of computing effective preconditioners for such iterative methods we concentrate on element by element preconditioners which reflect the structure of the problem we find that the performance of these methods can be considerably improved by amalgamating elements before applying the preconditioners we report the results of numerical experiments which demonstrate the effectiveness of this approach

a note about sparsity exploiting quasi newton updates this paper is concerned with two questions relating to quasi newton updates for unconstrained optimization that exploit any sparsity present in the second derivative matrix of the objective function first a family of such updates is derived that reduces to any a priori known dense update formula when no sparsity is imposed this family uses the frobenius projection of the desired update on the subspace of matrices that satisfy all the needed conditions in the second part we prove that under mild assumptions a positive definite sparse quasi newton update always exists the proof of this result includes the explicit determination of such an update

recognizing underlying sparsity in optimization exploiting sparsity is essential to improve the efficiency of solving large optimization problems we present a method for recognizing the underlying sparsity structure of a nonlinear partially separable problem and show how the sparsity of the hessian matrices of the problem s functions can be improved by performing a nonsingular linear transformation in the space corresponding to the vector of variables a combinatorial optimization problem is then formulated to increase the number of zeros of the hessian matrices in the resulting transformed space and a heuristic greedy algorithm is applied to this formulation the resulting method can thus be viewed as a preprocessor for converting a problem with hidden sparsity into one in which sparsity is explicit when it is combined with the sparse semidefinite programming relaxation by waki et al for polynomial optimization problems the proposed method is shown to extend the performance and applicability of this relaxation technique preliminary numerical results are presented to illustrate this claim

a quasi newton method for elliptic boundary value problems for some boundary value problems for equations of the form nabla 2 u f x u nabla u 0 with linear boundary conditions we give a convergence analysis for a quasi newton method that converges superlinearly in the c 1 norm this scheme for infinite dimensional problems becomes a method for the finite dimensional problems that are produced by discretizing the differential equation this method is a generalization of a method proposed by hart and soul for discretizations of two point boundary value problems