hidden conic quadratic representation of some nonconvex quadratic optimization problems the problem of minimizing a quadratic objective function subject to one or two quadratic constraints is known to have a hidden convexity property even when the quadratic forms are indefinite the equivalent convex problem is a semidefinite one and the equivalence is based on the celebrated s lemma in this paper we show that when the quadratic forms are simultaneously diagonalizable sd it is possible to derive an equivalent convex problem which is a conic quadratic cq one and as such is significantly more tractable than a semidefinite problem the sd condition holds for free for many problems arising in applications in particular when deriving robust counterparts of quadratic or conic quadratic constraints affected by implementation error the proof of the hidden cq property is constructive and does not rely on the s lemma this fact may be significant in discovering hidden convexity in some nonquadratic problems

convergence of nonconforming multigrid methods without full elliptic regularity we consider nonconforming multigrid methods for symmetric positive definite second and fourth order elliptic boundary value problems which do not have full elliptic regularity we prove that there is a bound 1 for the contraction number of the w cycle algorithm which is independent of mesh level provided that the number of smoothing steps is sufficiently large we also show that the symmetric variable v cycle algorithm is an optimal preconditioner

quadratically constrained least squares and quadratic problems we consider the following problem compute a vectorx such that ax b 2 min subject to the constraint x 2 a new approach to this problem based on gauss quadrature is given the method is especially well suited when the dimensions ofa are large and the matrix is sparse it is also possible to extend this technique to a constrained quadratic form for a symmetric matrixa we consider the minimization ofx t a x 2b t x subject to the constraint x 2 some numerical examples are given

multigrid methods for two dimensional maxwell s equations on graded meshes in this paper we investigate the numerical solution for two dimensional maxwell s equations on graded meshes the approach is based on the hodge decomposition the solution u of maxwell s equations is approximated by solving standard second order elliptic problems quasi optimal error estimates for both u and i u in the l 2 norm are obtained on graded meshes we prove the uniform convergence of the w cycle and full multigrid algorithms for the resulting discrete problem the performance of these methods is illustrated by numerical results

on the implementation of mixed methods as nonconforming methods for second order elliptic problems in this paper we show that mixed finite element methods for a fairly general second order elliptic problem with variable coefficients can be given a nonmixed formulation lower order terms are treated so our results apply also to parabolic equations we define an approximation method by incorporating some projection operators within a standard galerkin method which we call a projection finite element method it is shown that for a given mixed method if the projection method s finite element space m h satisfies three conditions then the two approximation methods are equivalent these three conditions can be simplified for a single element in the case of mixed spaces possessing the usual vector projection operator we then construct appropriate nonconforming spaces m h for the known triangular and rectangular elements the lowest order raviart thomas mixed solution on rectangular finite elements in r 2 and r 3 on simplices or on prisms is then implemented as a nonconforming method modified in a simple and computationally trivial manner this new nonconforming solution is actually equivalent to a postprocessed version of the mixed solution a rearrangement of the computation of the mixed method solution through this equivalence allows us to design simple and optimal order multigrid methods for the solution of the linear system

multigrid methods for saddle point problems stokes and lame systems we develop new multigrid methods for a class of saddle point problems that include the stokes system in fluid flow and the lame system in linear elasticity as special cases the new smoothers in the multigrid methods involve optimal preconditioners for the discrete laplace operator we prove uniform convergence of the w w cycle algorithm in the energy norm and present numerical results for w w cycle and v v cycle algorithms

a v cycle multigrid method for trunc plate element in this paper an optimal v cycle multigrid method is presented for the trunc plate element the uniform convergence rate independent of mesh sizes and levels is established the method is extended to bergan s energy orthogonal plate element

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

a multigrid algorithm for the lowest order raviart thomas mixed triangular finite element method an optimal order multigrid method for the lowest order raviart thomas mixed triangular finite element is developed the algorithm and the convergence analysis are based on the equivalence between raviart thomas mixed methods and certain nonconforming methods both the dirichlet and singular neumann boundary value problems for second order elliptic equations are discussed

an approximation scheme for distributionally robust nonlinear optimization we consider distributionally robust optimization problems drops with nonlinear and nonconcave dependence on uncertain parameters the drop can be written as a nonsmooth nonlinear program with a

strong duality in nonconvex quadratic optimization with two quadratic constraints we consider the problem of minimizing an indefinite quadratic function subject to two quadratic inequality constraints when the problem is defined over the complex plane we show that strong duality holds and obtain necessary and sufficient optimality conditions we then develop a connection between the image of the real and complex spaces under a quadratic mapping which together with the results in the complex case lead to a condition that ensures strong duality in the real setting preliminary numerical simulations suggest that for random instances of the extended trust region subproblem the sufficient condition is satisfied with a high probability furthermore we show that the sufficient condition is always satisfied in two classes of nonconvex quadratic problems finally we discuss an application of our results to robust least squares problems

quadratically constrained attitude control via semidefinite programming we consider the attitude control problem augmented with an arbitrary number of nonconvex quadratic constraints by utilizing the implicit magnitude constraint of the state vector i e quaternion we are able to reduce this problem to a convex quadratically constrained quadratic program or a semidefinite program this result leads to an efficient solution strategy for general quadratically constrained attitude control problems an example demonstrates the proposed methodology

an efficient algorithm for globally minimizing a quadratic function under convex quadratic constraints in this paper we investigate two approaches to minimizing a quadratic form subject to the intersection of finitely many ellipsoids the first approach is the d c difference of convex functions optimization algorithm abbr dca whose main tools are the proximal point algorithm and or the projection subgradient method in convex minimization the second is a branch and bound scheme using lagrangian duality for bounding and ellipsoidal bisection in branching the dca was first introduced by pham dinh in 1986 for a general d c program and later developed by our various work is a local method but from a good starting point it provides often a global solution this motivates us to combine the dca and our branch and bound algorithm in order to obtain a good initial point for the dca and to prove the globality of the dca in both approaches we attempt to use the ellipsoidal constrained quadratic programs as the main subproblems the idea is based upon the fact that these programs can be efficiently solved by some available polynomial and nonpolynomial time algorithms among them the dca with restarting procedure recently proposed by pham dinh and le thi has been shown to be the most robust and fast for large scale problems several numerical experiments with dimension up to 200 are given which show the effectiveness and the robustness of the dca and the combined dca branch and bound algorithm

a cal v cycle multigrid approach for mortar finite elements mortar methods based on dual lagrange multipliers provide a flexible tool for the numerical approximation of partial differential equations the associated finite element spaces are in general nonconforming and nonnested optimal multigrid results have previously been established for cal w cycle and the variable cal v cycle multigrid methods in this paper we introduce a new multigrid method based on a nested sequence of modified mortar spaces for which we can establish that the cal v cycle with one smoothing step has contraction numbers uniformly bounded away from one to obtain nested mortar spaces we apply a product form of certain corrections at the interfaces numerical results demonstrate the efficiency of the resulting multigrid solver

on a subproblem of trust region algorithms for constrained optimization we study a subproblem that arises in some trust region algorithms for equality constrained optimization it is the minimization of a general quadratic function with two special quadratic constraints properties of such subproblems are given it is proved that the hessian of the lagrangian has at most one negative eigenvalue and an example is presented to show that the hessian may have a negative eigenvalue when one constraint is inactive at the solution

new convergence estimates for multigrid algorithms in this paper new convergence estimates are proved for both symmetric and nonsymmetric multigrid algorithms applied to symmetric positive definite problems our theory relates the convergence of multigrid algorithms to a regularity and approximation parameter cap alpha epsilon 0 1 and the number of relaxations m we show that for the symmetric and nonsymmetric nu cycles the multigrid iteration converges for any positive m at a rate which deteriorates no worse than 1 cj sup 1 sup cap alpha sup sup cap alpha where j is the number of grid levels we then define a generalized nu cycle algorithm which involves exponentially increasing for example doubling the number of smoothings on successively coarser grids we show that the resulting symmetric and nonsymmetric multigrid iterations converge for any cap alpha with rates that are independent of the mesh size the theory is presented in an abstract setting which can be applied to finite element multigrid and finite difference multigrid methods

an exact algorithm for nonconvex quadratic integer minimization using ellipsoidal relaxations we propose a branch and bound algorithm for minimizing a not necessarily convex quadratic function over integer variables the algorithm is based on lower bounds computed as continuous minima of the objective function over appropriate ellipsoids in the nonconvex case we use ellipsoids enclosing the feasible region of the problem in spite of the nonconvexity these minima can be computed quickly the corresponding optimization problems are equivalent to trust region subproblems we present several ideas that allow us to accelerate the solution of the continuous relaxation within a branch and bound scheme and examine the performance of the overall algorithm by computational experiments good computational performance is shown especially for ternary instances

convergence estimates for multigrid algorithms without regularity assumptions a new technique for proving rate of convergence estimates of multi grid algorithms for symmetric positive definite problems will be given in this paper the standard multigrid theory requires a regularity and approxima tion assumption in contrast the new theory requires only an easily verified approximation assumption this leads to convergence results for multigrid re finement applications problems with irregular coefficients and problems whose coefficients have large jumps in addition the new theory shows why it suffices to smooth only in the regions where new nodes are being added in multigrid refinement applications

a linear time algorithm for trust region problems we consider the fundamental problem of minimizing a general quadratic function over an ellipsoidal domain also known as the trust region sub problem we give the first provable linear time in the number of non zero entries of the input algorithm for approximately solving this problem specifically our algorithm returns an epsilon approximate solution in runtime of order n sqrt epsilon where n is the number of non zero entries in the input this matches the runtime of nesterov s accelerated gradient descent suitable for the special case in which the quadratic objective is convex and the runtime of the lanczos method which is applicable when the problem is purely quadratic

an unconstrained minimization method for solving low rank sdp relaxations of the maxcut problem in this paper we consider low rank semidefinite programming lrsdp relaxations of combinatorial quadratic problems that are equivalent to the maxcut problem using the gramian representation of a positive semidefinite matrix the lrsdp problem can be formulated as the nonconvex nonlinear programming problem of minimizing a quadratic function with quadratic equality constraints for the solution of this problem we propose a continuously differentiable exact merit function that exploits the special structure of the constraints and we use this function to define an efficient and globally convergent algorithm finally we test our code on an extended set of instances of the maxcut problem and we report comparisons with other existing codes

alternative characterization of time integration schemes this work presents an alternative definition of the numerical angular frequency and numerical damping ratio within the framework of direct time integration of second order differential equations the classical approach based on the amplification matrix concept deals with the ability of the numerical schemes to deliver an accurate response in time for problems with free vibration behaviour it is well suited to the study of the schemes both explicit and implicit used in transient dynamic analysis the alternative approach emphasizes the capability of the scheme to deliver an accurate resonant response for problems with forced vibration behaviour such frequency viewpoint is helpful in the study of the implicit schemes used in civil engineering applications

indefinite trust region subproblems and nonsymmetric eigenvalue perturbations this paper extends the theory of trust region subproblems in two ways i it allows indefinite inner products in the quadratic constraint and ii it uses a two sided upper and lower bound quadratic constraint characterizations of optimality are presented that have no gap between necessity and sufficiency conditions for the existence of solutions are given in terms of the definiteness of a matrix pencil a simple dual program is introduced that involves the maximization of a strictly concave function on an interval this dual program simplifies the theory and algorithms for trust region subproblems it also illustrates that the trust region subproblems are implicit convex programming problems and thus explains why they are so tractable the duality theory also provides connections to eigenvalue perturbation theory trust region subproblems with zero linear term in the objective function correspond to eigenvalue problems and adding a linear term in the objective function is seen to correspond to a p

a two level algorithm for the weak galerkin discretization of diffusion problems this paper analyzes a two level algorithm for the weak galerkin wg finite element methods based on local raviart thomas rt and brezzi douglas marini bdm mixed elements for two and three dimensional diffusion problems with dirichlet condition we first show the condition numbers of the stiffness matrices arising from the wg methods are of o h 2 we use an extended version of the xu zikatanov xz identity to derive the convergence of the algorithm without any regularity assumption finally we provide some numerical results

multigrid methods for a quad curl problem based on c0 interior penalty method abstract in this paper we study the multigrid algorithm for a quad curl problem based on c 0 interior penalty method on bounded polygonal domains it is shown that the contraction numbers of w cycle multigrid algorithm for the c 0 interior penalty method converge uniformly with rate of m where m is the number of pre smoothing and post smoothing steps is the index of elliptic regularity the performance of the multigrid fast solver is illustrated by numerical results

globally solving the trust region subproblem using simple first order methods we consider the trust region subproblem which is given by a minimization of a quadratic not necessarily convex function over the euclidean ball based on the well known second order necessary and

convex hull of two quadratic or a conic quadratic and a quadratic inequality in this paper we consider an aggregation technique introduced by yildiran j math control inf 26 417 450 2009 to study the convex hull of regions defined by two quadratic inequalities or by a conic quadratic and a quadratic inequality yildiran 2009 shows how to characterize the convex hull of open sets defined by two strict quadratic inequalities using linear matrix inequalities we show how this aggregation technique can be easily extended to yield valid conic quadratic inequalities for the convex hull of open sets defined by two strict quadratic inequalities or by a strict conic quadratic and a strict quadratic inequality we also show that for sets defined by a strict conic quadratic and a strict quadratic inequality under one additional containment assumption these valid inequalities characterize the convex hull exactly we also show that under certain topological assumptions the results from the open setting can be extended to characterize the closed convex hull of sets defined with non strict conic and quadratic inequalities

s lemma with equality and its applications let f x x tax 2a tx c f x xtax 2atx c and h x x tbx 2b tx d h x xtbx 2btx d be two quadratic functions having symmetric matrices a a and b b the s lemma with equality asks when the unsolvability of the system f x 0 h x 0 f x 0 h x 0 implies the existence of a real number mu such that f x mu h x ge 0 forall x in mathbb r n f x h x 0 x rn the problem is much harder than the inequality version which asserts that under slater condition f x 0 h x le 0 f x 0 h x 0 is unsolvable if and only if f x mu h x ge 0 forall x in mathbb r n f x h x 0 x rn for some mu ge 0 0 in this paper we show that the s lemma with equality does not hold only when the matrix a a has exactly one negative eigenvalue and h x h x is a non constant linear function b 0 b not 0 b 0 b 0 as an application we can globally solve inf f x h x 0 inf f x h x 0 as well as the two sided generalized trust region subproblem inf f x l le h x le u inf f x l h x u without any condition moreover the convexity of the joint numerical range f x h 1 x ldots h p x x in mathbb r n f x h1 x hp x x rn where f f is a possibly non convex quadratic function and h 1 x ldots h p x h1 x hp x are affine functions can be characterized using the newly developed s lemma with equality

multilevel schwarz methods we consider the solution of the algebraic system of equations which result from the discretization of second order elliptic equations a class of multilevel algorithms are studied using the additive schwarz framework we establish that the condition number of the iteration operators are bounded independent of mesh sizes and the number of levels this is an improvement on dryja and widlund s result on a multilevel additive schwarz algorithm as well as bramble pasciak and xu s result on the bpx algorithm some multiplicative variants of the multilevel methods are also considered we establish that the energy norms of the corresponding iteration operators are bounded by a constant less than one which is independent of the number of levels for a proper ordering the iteration operators correspond to the error propagation operators of certain v cycle multigrid methods using gauss seidel and damped jacobi methods as smoothers respectively

on the multigrid f cycle in a recent paper a bound was derived on the convergence of the multigrid v cycle for the case when the solution is in the sobolev space h 1 a but not in h 1 a a 62 a showing that the convergence factor approaches one only as 1 o k a 1 a for a large number of levels k we now extend the technique to obtain the asymptotically better bound 1 o k 1 a 2 a on the multigrid f cycle we also show that in many cases for practical values of k one gets the same bound for the f cycle as for the v cycle with a 1

convergence of the multigrid v cycle algorithm for second order boundary value problems without full elliptic regularity the multigrid v cycle algorithm using the richardson relaxation scheme as the smoother is studied in this paper for second order elliptic boundary value problems the contraction number of the v cycle algorithm is shown to improve uniformly with the increase of the number of smoothing steps without assuming full elliptic regularity as a consequence the v cycle convergence result of braess and hackbusch is generalized to problems without full elliptic regularity

randomized algorithms for optimal solutions of double sided qcqp with applications in signal processing quadratically constrained quadratic programming qcqp with double sided constraints has plenty of applications in signal processing as have been addressed in recent years qcqp problems are hard to solve in general and they are typically approached by solving a semidefinite programming sdp relaxation followed by a postprocessing procedure existing postprocessing schemes include gaussian randomization to generate an approximate solution rank reduction procedure the so called purification and some specific rank one matrix decomposition techniques to yield a globally optimal solution in this paper we propose several randomized postprocessing methods to output not an approximate solution but a globally optimal solution for some solvable instances of the double sided qcqp i e instances with a small number of constraints we illustrate their applicability in robust receive beamforming radar optimal code design and broadcast beamforming for multiuser communications as a byproduct we derive an alternative shorter proof for the sturm zhang rank one matrix decomposition theorem

the generalized trust region subproblem solution complexity and convex hull results we consider the generalized trust region subproblem gtrs of minimizing a nonconvex quadratic objective over a nonconvex quadratic constraint a lifting of this problem recasts the gtrs as minimizing a linear objective subject to two nonconvex quadratic constraints our first main contribution is structural we give an explicit description of the convex hull of this nonconvex set in terms of the generalized eigenvalues of an associated matrix pencil this result may be of interest in building relaxations for nonconvex quadratic programs moreover this result allows us to reformulate the gtrs as the minimization of two convex quadratic functions in the original space our next set of contributions is algorithmic we present an algorithm for solving the gtrs up to an epsilon additive error based on this reformulation we carefully handle numerical issues that arise from inexact generalized eigenvalue and eigenvector computations and establish explicit running time guarantees for these algorithms notably our algorithms run in linear in the size of the input time furthermore our algorithm for computing an epsilon optimal solution has a slightly improved running time dependence on epsilon over the state of the art algorithm our analysis shows that the dominant cost in solving the gtrs lies in solving a generalized eigenvalue problem establishing a natural connection between these problems finally generalizations of our convex hull results allow us to apply our algorithms and their theoretical guarantees directly to equality interval and hollow constrained variants of the gtrs this gives the first linear time algorithm in the literature for these variants of the gtrs

a semidefinite framework for trust region subproblems with applications to large scale minimization primal dual pairs of semidefinite programs provide a general framework for the theory and algorithms for the trust region subproblem trs this latter problem consists in minimizing a general quadratic function subject to a convex quadratic constraint and therefore it is a generalization of the minimum eigenvalue problem the importance of trs is due to the fact that it provides the step in trust region minimization algorithms the semidefinite framework is studied as an interesting instance of semidefinite programming as well as a tool for viewing known algorithms and deriving new algorithms for trs in particular a dual simplex type method is studied that solves trs as a parametric eigenvalue problem this method uses the lanczos algorithm for the smallest eigenvalue as a black box therefore the essential cost of the algorithm is the matrix vector multiplication and thus sparsity can be exploited a primal simplex type method provides steps for the so called hard case extensive numerical tests for large sparse problems are discussed these tests show that the cost of the algorithm is 1 n times the cost of finding a minimum eigenvalue using the lanczos algorithm where 0 n 1 is a fraction which decreases as the dimension increases

finite element analysis for the axisymmetric laplace operator on polygonal domains abstract let l r 2 r r 2 z 2 we consider the equation l u f on a bounded polygonal domain with suitable boundary conditions derived from the three dimensional axisymmetric poisson s equation we establish the well posedness regularity and fredholm results in weighted sobolev spaces for possible singular solutions caused by the singular coefficient of the operator l as r 0 and by non smooth points on the boundary of the domain in particular our estimates show that there is no loss of regularity of the solution in these weighted sobolev spaces besides by analyzing the convergence property of the finite element solution we provide a construction of improved graded meshes such that the quasi optimal convergence rate can be recovered on piecewise linear functions for singular solutions the introduction of a new projection operator from the weighted space to the finite element subspace certain scaling arguments and a calculation of the index of the fredholm operator together with our regularity results are the ingredients of the finite element estimates

multigrid methods for saddle point problems oseen system abstract we develop and analyze multigrid methods for the oseen system in fluid flow we show that the w cycle algorithm is a uniform contraction if the number of smoothing steps is sufficiently large numerical results that illustrate the performance of the methods are also presented

a note on convergence of v cycle nonconforming multigrid methods1this research was supported by the doctoral program foundation of china universities and by nsf and china postdoctoral science foundation 1 in this paper the nonconforming multigrid methods are studied i e the fine mesh spaces and coarse mesh spaces also use the same nonconforming finite element contrary to the observation by chen convergence of multigrid method for nonconforming method pre print 1996 even if quadratic forms are not changed on coarse meshes v cycle nonconforming multigrid methods still can converge the key of our methods is the choice of a good intergrid operator moreover optimal convergent order of our v cycle nonconforming multigrid methods is obtained

some estimates for a weighted l 2 projection this paper is devoted to the error estimates for some weighted l2 projections nearly optimal estimates are obtained these estimates can be applied to the analysis of the usual multigrid method multilevel preconditioner and domain decomposition method for solving elliptic boundary problems whose coefficients have large jump discontinuities

second order global optimality conditions for convex composite optimization in recent years second order sufficient conditions of an isolated local minimizer for convex composite optimization problems have been established in this paper second order optimality conditions are obtained of aglobal minimizer for convex composite problems with a non finite valued convex function and a twice strictly differentiable function by introducing a generalized representation condition this result is applied to a minimization problem with a closed convex set constraint which is shown to satisfy the basic constraint qualification in particular second order necessary and sufficient conditions of a solution for a variational inequality problem with convex composite inequality constraints are obtained 1998 the mathematical programming society inc published by elsevier science b v

a simple duality proof in convex quadratic programming with a quadratic constraint and some applications abstract in this paper a simple derivation of duality is presented for convex quadratic programs with a convex quadratic constraint this problem arises in a number of applications including trust region subproblems of nonlinear programming regularized solution of ill posed least squares problems and ridge regression problems in statistical analysis in general the dual problem is a concave maximization problem with a linear equality constraint we apply the duality result to 1 the trust region subproblem 2 the smoothing of empirical functions and 3 to piecewise quadratic trust region subproblems arising in nonlinear robust huber m estimation problems in statistics the results are obtained from a straightforward application of lagrange duality

convergence of nonconforming v cycle and f cycle multigrid algorithms for second order elliptic boundary value problems the convergence of v cycle and f cycle multigrid algorithms with a sufficiently large number of smoothing steps is established for nonconforming finite element methods for second order elliptic boundary value problems

sharp estimates for multigrid rates of convergence with general smoothing and acceleration in this paper we prove the convergence of the multilevel iterative method for solving linear equations that arise from elliptic partial differential equations our theory is presented entirely in terms of the generalized condition number kappa of the matrix a and the smoothing matrix b this leads to a completely algebraic analysis of the method as an iterative technique for solving linear equations the properties of the elliptic equation and discretization procedure enter only when we seek to estimate kappa just as in the case of most standard iterative methods here we consider the fundamental two level iteration and the v and w cycles of the j level iteration j 2 we prove that the v and w cycles converge even when only one smoothing iteration is used we present several examples of the computation of kappa using both fourier analysis and standard finite element techniques we compare the predictions of our theorems with the actual rate of convergence our analysis also shows that

convergence estimates for product iterative methods with applications to domain decomposition in this paper we consider iterative methods for the solution of symmetric positive definite problems on a space which are defined in terms of products of operators defined with respect to a number of subspaces the simplest algorithm of this sort has an error reducing operator which is the product of orthogonal projections onto the complement of the subspaces new normreduction estimates for these iterative techniques will be presented in an abstract setting applications are given for overlapping schwarz algorithms with many subregions for finite element approximation of second order elliptic problems

poincare friedrichs inequalities for piecewise h 1 functions poincare friedrichs inequalities for piecewise h1 functions are established they can be applied to classical nonconforming finite element methods mortar methods and discontinuous galerkin methods

doubly constrained robust capon beamformer with ellipsoidal uncertainty sets the doubly constrained robust dcr capon beamformer with a spherical uncertainty set was introduced and studied by stoica and wang here we consider the generalized dcr problem gdcr in which the uncertainty set is an ellipsoid rather than a sphere although as noted previously by stoica and wang this problem is nonconvex and appears to be intractable we show that it can be solved efficiently in fact we prove that the gdcr beamformer can be obtained as a solution to a convex optimization problem to this end we first derive a strong duality result for nonconvex quadratic programs with two quadratic constraints over the complex domain specializing the results to our context leads to a semidefinite programming formulation of the gdcr beamformer

an optimal order multigrid method for p1 nonconforming finite elements etude d une methode multigrille d ordre optimal pour la resolution des problemes aux limites elliptiques du second ordre a l aide d elements finis non conformes

a survey of the s lemma in this survey we review the many faces of the s lemma a result about the correctness of the s procedure the basic idea of this widely used method came from control theory but it has important consequences in quadratic and semidefinite optimization convex geometry and linear algebra as well these were all active research areas but as there was little interaction between researchers in these different areas their results remained mainly isolated here we give a unified analysis of the theory by providing three different proofs for the s lemma and revealing hidden connections with various areas of mathematics we prove some new duality results and present applications from control theory error estimation and computational geometry

multigrid algorithms for c 0 interior penalty methods multigrid algorithms for c 0 interior penalty methods for fourth order elliptic boundary value problems on polygonal domains are studied in this paper it is shown that v cycle f cycle and w cycle algorithms are contractions if the number of smoothing steps is sufficiently large the contraction numbers of these algorithms are bounded by cm alpha where m is the number of presmoothing and postsmoothing steps alpha is the index of elliptic regularity and the positive constant c is mesh independent these estimates are established for a smoothing scheme that uses a poisson solve as a preconditioner which can be easily implemented because the c 0 finite element spaces are standard spaces for second order problems furthermore the variable v cycle algorithm is also shown to be an optimal preconditioner

on the solution of the tikhonov regularization of the total least squares problem total least squares tls is a method for treating an overdetermined system of linear equations bf a bf x approx bf b where both the matrix bf a and the vector bf b are contaminated by noise tikhonov regularization of the tls trtls leads to an optimization problem of minimizing the sum of fractional quadratic and quadratic functions as such the problem is nonconvex we show how to reduce the problem to a single variable minimization of a function mathcal g over a closed interval computing a value and a derivative of mathcal g consists of solving a single trust region subproblem for the special case of regularization with a squared euclidean norm we show that mathcal g is unimodal and provide an alternative algorithm which requires only one spectral decomposition a numerical example is given to illustrate the effectiveness of our method

v cycle convergence with unsymmetric smoothers and application to an anisotropic model problem we generalize the abstract v cycle convergence proof of j xu siam rev 34 1992 pp 581 613 and h yserentant acta numerica 1993 pp 285 326 to include unsymmetric smoothing operations as well applying our theory to an anisotropic model problem we obtain optimal results i e robust convergence even when using only one smoothing step of a line jacobi a line gauss seidel or ilu smoothing

new estimates for multilevel algorithms including the v cycle the purpose of this paper is to provide new estimates for certain multilevel algorithms in particular we are concerned with the simple additive multilevel algorithm discussed recently together with j xu and the standard v cycle algorithm with one smoothing step per grid we shall prove that these algorithms have a uniform reduction per iteration independent of the mesh sizes and number of levels even on nonconvex domains which do not provide full elliptic regularity for example the theory applies to the standard multigrid vcycle on the l shaped domain or a domain with a crack and yields a uniform convergence rate we also prove uniform convergence rates for the multigrid v cycle for problems with nonuniformly refined meshes finally we give a new multigrid approach for problems on domains with curved boundaries and prove a uniform rate of convergence for the corresponding multigrid v cycle algorithms

nonconforming finite elements and the cascadic multi grid method we derive sufficient conditions under which the cascadic multi grid method applied to nonconforming finite element discretizations yields an optimal solver key ingredients are optimal error estimates of such discretizations which we therefore study in detail we derive a new efficient modified morley finite element method optimal cascadic multi grid methods are obtained for problems of second and using a new smoother of fourth order as well as for the stokes problem

on the abstract theory of additive and multiplicative schwarz algorithms in recent years it has been shown that many modern iterative algorithms multigrid schemes multilevel preconditioners domain decomposition methods etc for solving problems resulting from the discretization of pdes can be interpreted as additive jacobi like or multiplicative gauss seidel like subspace correction methods the key to their analysis is the study of certain metric properties of the underlying splitting of the discretization space v into a sum of subspaces v j and the splitting of the variational problem on v into auxiliary problems on these subspaces in this paper we propose a modification of the abstract convergence theory of the additive and multiplicative schwarz methods that makes the relation to traditional iteration methods more explicit the analysis of the additive and multiplicative schwarz iterations can be carried out in almost the same spirit as in the traditional block matrix situation making convergence proofs of multilevel and domain decomposition methods clearer or at least more classical in addition we present a new bound for the convergence rate of the appropriately scaled multiplicative schwarz method directly in terms of the condition number of the corresponding additive schwarz operator these results may be viewed as an appendix to the recent surveys x ys

multigrid algorithms for symmetric discontinuous galerkin methods on graded meshes we study a class of symmetric discontinuous galerkin methods on graded meshes optimal order error estimates are derived in both the energy norm and the l 2 norm and we establish the uniform convergence of v cycle f cycle and w cycle multigrid algorithms for the resulting discrete problems numerical results that confirm the theoretical results are also presented

a second order cone based approach for solving the trust region subproblem and its variants we study the trust region subproblem trs of minimizing a nonconvex quadratic function over the unit ball with additional conic constraints despite having a nonconvex objective it is known that the classical trs and a number of its variants are polynomial time solvable in this paper we follow a second order cone soc based approach to derive an exact convex reformulation of the trs under a structural condition on the conic constraint our structural condition is immediately satisfied when there are no additional conic constraints and it generalizes several such conditions studied in the literature as a result our study highlights an explicit connection between the classical nonconvex trs and smooth convex quadratic minimization which allows for the application of cheap iterative methods such as nesterov s accelerated gradient descent to the trs furthermore under slightly stronger conditions we give a low complexity characterization of the convex hull of the epigraph of the nonconvex quadrati

quadratic matrix programming we introduce and study a special class of nonconvex quadratic problems in which the objective and constraint functions have the form f boldmath x tr boldmath x t boldmath a boldmath x 2 tr boldmath b t boldmath x c boldmath x in real r n times r the latter formulation is termed quadratic matrix programming qmp of order r we construct a specially devised semidefinite relaxation sdr and dual for the qmp problem and show that under some mild conditions strong duality holds for qmp problems with at most r constraints using a result on the equivalence of two characterizations of the nonnegativity property of quadratic functions of the above form we are able to compare the constructed sdr and dual problems to other known sdrs and dual formulations of the problem an application to robust least squares problems is discussed

hidden convexity in some nonconvex quadratically constrained quadratic programming we consider the problem of minimizing an indefinite quadratic objective function subject to twosided indefinite quadratic constraints under a suitable simultaneous diagonalization assumption which trivially holds for trust region type problems we prove that the original problem is equivalent to a convex minimization problem with simple linear constraints we then consider a special problem of minimizing a concave quadratic function subject to finitely many convex quadratic constraints which is also shown to be equivalent to a minimax convex problem in both cases we derive the explicit nonlinear transformations which allow for recovering the optimal solution of the nonconvex problems via their equivalent convex counterparts special cases and applications are also discussed we outline interior point polynomial time algorithms for the solution of the equivalent convex programs

socp reformulation for the generalized trust region subproblem via a canonical form of two symmetric matrices we investigate in this paper the generalized trust region subproblem gtrs of minimizing a general quadratic objective function subject to a general quadratic inequality constraint by applying a simultaneous block diagonalization approach we obtain a congruent canonical form for the symmetric matrices in both the objective and constraint functions by exploiting the block separability of the canonical form we show that all gtrss with an optimal value bounded from below are second order cone programming socp representable our result generalizes the recent work of ben tal and den hertog math program 143 1 2 1 29 2014 which establishes the socp representability of the gtrs under the assumption of the simultaneous diagonalizability of the two matrices in the objective and constraint functions we then derive a closed form solution for the gtrs when the two matrices are not simultaneously diagonalizable we further extend our method to two variants of the gtrs in which the inequality constraint is replaced by either an equality constraint or an interval constraint

a generalized bpx multigrid framework covering nonnested v cycle methods more than a decade ago bramble pasciak and xu developed a framework in analyzing the multigrid methods with nonnested spaces or non inherited quadratic forms it was subsequently known as the bpx multigrid framework which was widely used in the analysis of multigrid and domain decomposition methods however the framework has an apparent limit in the analysis of nonnested v cycle methods and it produces a variable v cycle or nonuniform convergence rate v cycle methods or other nonoptimal results in analysis thus far this paper completes a long time effort in extending the bpx multigrid framework so that it truly covers the nonnested v cycle we will apply the extended bpx framework to the analysis of many v cycle nonnested multigrid methods some of them were proven previously only for two level and w cycle iterations some numerical results are presented to support the theoretical analysis of this paper

a new convergence proof for the multigrid method including the v cycle for a positive definite finite element equation we describe a multigrid iteration and prove convergence under natural assumptions on the discretization and the elliptic problem hitherto existing convergence proofs require a sufficiently large number of smoothing iterations and exclude the v cycle the presented proof applies to procedures with any number of smoothing iterations and to the v cycle

an optimal order process for solving finite element equations a k level iterative procedure for solving the algebraic equations which arise from the finite element approximation of elliptic boundary value problems is presented and analyzed the work estimate for this procedure is proportional to the number of unknowns an optimal order result general geometry is permitted for the underlying domain but the shape plays a role in the rate of convergence through elliptic regularity finally a short discussion of the use of this method for parabolic problems is presented

assessment of explicit and semi explicit classes of model based algorithms for direct integration in structural dynamics summary the model based algorithms available in the literature are primarily developed for the direct integration of the equations of motion for hybrid simulation in earthquake engineering an experimental method where the system response is simulated by dividing it into a physical and an analytical domain the term model based indicates that the algorithmic parameters are functions of the complete model of the system to enable unconditional stability to be achieved within the framework of an explicit formulation these two features make the model based algorithms also a potential candidate for computations in structural dynamics based on the algorithmic difference equations these algorithms can be classified as either explicit or semi explicit where the former refers to the algorithms with explicit difference equations for both displacement and velocity while the latter for displacement only the algorithms pertaining to each class are reviewed and a new family of second order unconditionally stable parametrically dissipative semi explicit algorithms is presented numerical characteristics of these two classes of algorithms are assessed under linear and nonlinear structural behavior representative numerical examples are presented to complement the analytical findings the analysis and numerical examples demonstrate the advantages and limitations of these two classes of model based algorithms for applications in structural dynamics this article is protected by copyright all rights reserved

on the complexity of quadratic programming with two quadratic constraints the complexity of quadratic programming problems with two quadratic constraints is an open problem in this paper we show that when one constraint is a ball constraint and the hessian of the quadratic function defining the other constraint is positive definite then under quite general conditions the problem can be solved in polynomial time in the real number model of computation through an approach based on the analysis of the dual space of the lagrange multipliers however the degree of the polynomial is rather large thus making the result mostly of theoretical interest

a convex optimization approach for minimizing the ratio of indefinite quadratic functions over an ellipsoid we consider the nonconvex problem rq of minimizing the ratio of two nonconvex quadratic functions over a possibly degenerate ellipsoid this formulation is motivated by the so called regularized total least squares problem rtls which is a special case of the problem s class we study we prove that under a certain mild assumption on the problem s data problem rq admits an exact semidefinite programming relaxation we then study a simple iterative procedure which is proven to converge superlinearly to a global solution of rq and show that the dependency of the number of iterations on the optimality tolerance varepsilon grows as o sqrt ln varepsilon 1

the cascadic multigrid method for elliptic problems the paper deals with certain adaptive multilevel methods at the confluence of nested multigrid methods and iterative methods based on the cascade principle of 10 from the multigrid point of view no correction cycles are needed from the cascade principle view a basic iteration method without any preconditioner is used at successive refinement levels for a prescribed error tolerance on the final level more iterations must be spent on coarser grids in order to allow for less iterations on finer grids a first candidate of such a cascadic multigrid method was the recently suggested cascadic conjugate gradient method of 9 in short ccg method whichused the cg method as basic iteration method on each level in 18 it has been proven that the ccg method is accurate with optimal complexity for elliptic problems in 2d and quasi uniform triangulations the present paper simplifies that theory and extends it to more general basic iteration methods like the traditional multigrid smoothers moreover an adaptive control strategy for the number of iterations on successive refinement levels for possibly highly non uniform grids is worked out on the basis of a posteriori estimates numerical tests confirm the efficiency and robustness of the cascadic multigrid method

a geometric characterization of strong duality in nonconvex quadratic programming with linear and nonconvex quadratic constraints we first establish a relaxed version of dines theorem associated to quadratic minimization problems with finitely many linear equality and a single nonconvex quadratic inequality constraints the case of unbounded optimal valued is also discussed then we characterize geometrically the strong duality and some relationships with the conditions employed in finsler theorem are established furthermore necessary and sufficient optimality conditions with or without the slater assumption are derived our results can be used to situations where none of the results appearing elsewhere are applicable in addition a revisited theorem due to frank and wolfe along with that due to eaves is established for asymptotically linear sets

mixed finite element solution of time dependent problems a mixed formulation of the finite element method is used to establish a higher order incremental method for the solution of second order hyperbolic problems the displacement the velocity and optionally the acceleration fields are approximated independently in time using hierarchical bases the time approximation criterion preserves hyperbolicity in the sense that it replaces the solution of hyperbolic problems by the solution of uncoupled helmholtz type elliptic problems which can be subsequently solved using the alternative methods currently in use for discretization of the space dimension the development of the time integration procedure the characterization of its performance in terms of stability accuracy and convergence are illustrated using a polynomial time basis in order to stress the fact that the procedure can be implemented using alternative time bases a wavelet system is used in the solution of nonlinear parabolic and hyperbolic problems the method is well suited to parallel processing and to large time stepping the extension of its application to the solution of other than linear second order hyperbolic problems is discussed

multigrid methods for nonconforming finite element methods the present paper analyzes a multigrid algorithm for the crouzeix raviart discretization of the poisson and stokes equations in two and three dimensions the central point is the construction of easily computable l 2 projections based on suitable quadrature rules for the transfer from coarse to fine grids and vice versa