optimal matrix pencil approximation problem in structural dynamic model updating abstract the problem of finding the least change adjustment to a given matrix pencil is considered in this paper desired matrix properties including satisfaction of characteristic equation symmetry positive semidefiniteness and sparsity are imposed as side constraints to form the optimal matrix pencil approximation problem such a problem is related to the frequently encountered engineering problem of a structural modification on the dynamic behavior of a structure alternating direction method is applied to this constrained minimization problem numerical results are included to illustrate the performance and application of the proposed method

an adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures we develop a finite element adaptive strategy with error control for the wave scattering by periodic structures the unbounded computational domain is truncated to a bounded one by an extension of the perfectly matched layer pml technique which attenuates both the outgoing and evanescent waves in the pml region pml parameters such as the thickness of the layer and the medium property are determined through sharp a posteriori error estimates numerical experiments are included to illustrate the competitive behavior of the proposed adaptive method

very simple methods for all pairs network flow analysis a very simple algorithm for the classical problem of computing the maximum network flow value between every pair of nodes in an undirected capacitated n node graph is presented as in the well known gomory hu method the method given here uses only n 1 maximum flow computations our algorithm is implemented by adding only five simple lines of code to any program that produces a minimum cut a program to produce an equivalent flow tree which is a compact representation of the flow values is obtained by adding only three simple lines of code to any program producing a minimum cut a very simple version of the gomory hu cut tree method that finds one minimum cut for every pair of nodes is also derived and it is shown that the seemingly fundamental operation of that method node contraction is not needed nor must crossing cuts be avoided as a result this version of the gomory hu method is implemented by adding less than ten simple lines of code to any program that produces a minimum cut the algor

on the o 1 t convergence rate of the parallel descent like method and parallel splitting augmented lagrangian method for solving a class of variational inequalities in this paper we extend parallel descent like method pdlm and parallel splitting augmented lagrangian method psalm for structured monotone variational inequalities whose operator is composed by three separable operators and prove their o 1 t convergence rate

a nonlinear programming algorithm for solving semidefinite programs via low rank factorization in this paper we present a nonlinear programming algorithm for solving semidefinite programs sdps in standard form the algorithm s distinguishing feature is a change of variables that replaces the symmetric positive semidefinite variable x of the sdp with a rectangular variable r according to the factorization x rr t the rank of the factorization i e the number of columns of r is chosen minimally so as to enhance computational speed while maintaining equivalence with the sdp fundamental results concerning the convergence of the algorithm are derived and encouraging computational results on some large scale test problems are also presented

a generalized proximal point algorithm and its convergence rate we propose a generalized proximal point algorithm ppa in the generic setting of finding a root of a maximal monotone operator in addition to the classical ppa a number of benchmark operator splitting methods in the pde and optimization literatures can be retrieved by this generalized ppa scheme we establish the convergence rate of this generalized ppa scheme under different conditions including estimating its worst case convergence rate measured by the iteration complexity under mild assumptions and deriving its linear convergence rate under certain stronger conditions throughout our discussion we pay particular attention to the special case where the operator is the sum of two maximal monotone operators and specify our theoretical results in the generic setting to this special case our result turns out to be a general and unified study on the convergence rate of a number of existing methods and subsumes some existing results in the literature

a proximal parallel splitting method for minimizing sum of convex functions with linear constraints in this paper we propose a proximal parallel decomposition algorithm for solving the optimization problems where the objective function is the sum of m separable functions i e they have no crossed variables and the constraint set is the intersection of cartesian products of some simple sets and a linear manifold the m subproblems are solved simultaneously per iterations which are sum of the decomposed subproblems of the augmented lagrange function and a quadratic term hence our algorithm is named as the proximal parallel splitting method we prove the global convergence of the proposed algorithm under some mild conditions that the underlying functions are convex and the solution set is nonempty to make the subproblems easier some linearized versions of the proposed algorithm are also presented together with their global convergence analysis finally some preliminary numerical results are reported to support the efficiency of the new algorithms

a modified alternating direction method for convex quadratically constrained quadratic semidefinite programs we propose a modified alternating direction method for solving convex quadratically constrained quadratic semidefinite optimization problems the method is a first order method therefore requires much less computational effort per iteration than the second order approaches such as the interior point methods or the smoothing newton methods in fact only a single inexact metric projection onto the positive semidefinite cone is required at each iteration we prove global convergence and provide numerical evidence to show the effectiveness of this method

adaptive galerkin methods with error control for a dynamical ginzburg landau model in superconductivity the time dependent ginzburg landau model which describes the phase transitions taking place in superconductors is a coupled system of nonlinear parabolic equations it is discretized semi implicitly in time and in space via continuous piecewise linear finite elements a posteriori error estimates are derived for the l infty l 2 norm by studying a dual problem of the linearization of the original system other than the dual of error equations numerical simulations are included which illustrate the reliability of the estimators and the flexibility of the proposed adaptive method

a modified forward backward splitting method for maximal monotone mappings we consider the forward backward splitting method for finding a zero of the sum of two maximal monotone mappings this method is known to converge when the inverse of the forward mapping is strongly monotone we propose a modification to this method in the spirit of the extragradient method for monotone variational inequalities under which the method converges assuming only the forward mapping is lipschitz continuous on some closed convex subset of its domain the modification entails an additional forward step and a projection step at each iteration applications of the modified method to decomposition in convex programming and monotone variational inequalities are discussed

an adaptive finite element method for the h ψ formulation of time dependent eddy current problems in this paper we develop an adaptive finite element method based on reliable and efficient a posteriori error estimates for the h formulation of eddy current problems with multiply connected conductors multiply connected domains are considered by making cuts the competitive performance of the method is demonstrated by an engineering benchmark problem team workshop problem 7 and a singular problem with analytic solution

beyond the flow decomposition barrier we introduce a new approach to the maximum flow problem this approach is based on assigning arc lengths based on the residual flow value and the residual arc capacities our approach leads to an o min n 2 3 m 1 2 m log n 2 m log u time bound for a network with n vertices m arcs and integral arc capacities in the range 1 u this is a fundamental improvement over the previous time bounds we also improve bounds for the gomory hu tree problem the parametric flow problem and the approximate s t cut problem

nonlinear proximal point algorithms using bregman functions with applications to convex programming a bregman function is a strictly convex differentiable function that induces a well behaved distance measure or d function on euclidean space this paper shows that for every bregman function there exists a nonlinear version of the proximal point algorithm and presents an accompanying convergence theory applying this generalization of the proximal point algorithm to convex programming one obtains the d function proximal minimization algorithm of censor and zenios and a wide variety of new multiplier methods these multiplier methods are different from those studied by kort and bertsekas and include nonquadratic variations on the proximal method of multipliers

consistency of trace norm minimization regularization by the sum of singular values also referred to as the trace norm is a popular technique for estimating low rank rectangular matrices in this paper we extend some of the consistency results of the lasso to provide necessary and sufficient conditions for rank consistency of trace norm minimization with the square loss we also provide an adaptive version that is rank consistent even when the necessary condition for the non adaptive version is not fulfilled

decomposable markov decision processes a fluid optimization approach decomposable markov decision processes mdps are problems where the stochastic system can be decomposed into multiple individual components although such mdps arise naturally in many practical applications they are often difficult to solve exactly due to the enormous size of the state space of the complete system which grows exponentially with the number of components in this paper we propose an approximate solution approach to decomposable mdps that is based on re solving a fluid linear optimization formulation of the problem at each decision epoch this formulation tractably approximates the problem by modeling transition behavior at the level of the individual components rather than the complete system we prove that our fluid formulation provides a tighter bound on the optimal value function than three state of the art formulations the approximate linear optimization formulation the classical lagrangian relaxation formulation and a novel alternate lagrangian relaxation that is based on relax

adaptive finite element frequency domain method for eddy current problems in the present paper adaptive finite element methods are studied for time harmonic eddy current problems in the case of three dimensional isotropic and linear materials we adopt the formulation based on the magnetic field and a magnetic scalar potential in this paper since it needs the least number of unknowns reliable and efficient a posteriori error estimates are obtained and the efficiency of the adaptive algorithm is demonstrated by numerical and engineering experiments

an adaptive finite element algorithm with reliable and efficient error control for linear parabolic problems an efficient and reliable a posteriori error estimate is derived for linear parabolic equations which does not depend on any regularity assumption on the underlying elliptic operator an adaptive algorithm with variable time step sizes and space meshes is proposed and studied which at each time step delays the mesh coarsening until the final iteration of the adaptive procedure allowing only mesh and time step size refinements before it is proved that at each time step the adaptive algorithm is able to reduce the error indicators and thus the error below any given tolerance within a finite number of iteration steps the key ingredient in the analysis is a new coarsening strategy numerical results are presented to show the competitive behavior of the proposed adaptive algorithm

distributed computation of pareto sets the needs of multidisciplinary engineering design have motivated the development of distributed solution approaches to computing efficient solutions to decomposable multiobjective optimization problems mops the decomposition is necessary due to the assumption that the overall mop is not solvable since access to its solution space is subproblem restricted the state of the art analyses for distributed solution approaches such as the alternating direction method of multipliers admm and the block coordinate descent bcd method were developed in the context of problem decomposition originating in a single objective setting and are not immediately applicable to mops applying certain scalarization techniques well suited for nonconvex mops the decomposable mop is reformulated into a single objective problem sop but the decomposability is not preserved and the sop is not suitable for the application of admm furthermore coupling between the subproblems makes bcd in its current form likewise inadequate

a new inexact alternating directions method for monotone variational inequalities the alternating directions method adm is an effective method for solving a class of variational inequalities vi when the proximal and penalty parameters in sub vi problems are properly selected in this paper we propose a new adm method which needs to solve two strongly monotone sub vi problems in each iteration approximately and allows the parameters to vary from iteration to iteration the convergence of the proposed adm method is proved under quite mild assumptions and flexible parameter conditions

applications of the method of partial inverses to convex programming decomposition a primal dual decomposition method is presented to solve the separable convex programming problem convergence to a solution and lagrange multiplier vector occurs from an arbitrary starting point the method is equivalent to the proximal point algorithm applied to a certain maximal monotone multifunction in the nonseparable case it specializes to a known method the proximal method of multipliers conditions are provided which guarantee linear convergence

a polynomial time algorithm to find the shortest cycle basis of a graph define the length of a basis of the cycle space of a graph to be the sum of the lengths of all cycles in the basis an algorithm is given that finds a cycle basis with the shortest possible length in o m 3 n operations where m is the number of edges and n is the number of vertices this is the first known polynomial time algorithm for this problem edges may be weighted or unweighted also the shortest cycle basis is shown to have at most 3 n 1 n 2 2 edges for the unweighted case o mn 2 algorithm to obtain a suboptimal cycle basis of length o n 2 for unweighted graphs is also given

improved pointwise iteration complexity of a regularized admm and of a regularized non euclidean hpe framework this paper describes a regularized variant of the alternating direction method of multipliers admm for solving linearly constrained convex programs it is shown that the pointwise iteration complexity of the new variant is better than the corresponding one for the standard admm method and that up to a logarithmic term is identical to the ergodic iteration complexity of the latter method our analysis is based on first presenting and establishing the pointwise iteration complexity of a regularized non euclidean hybrid proximal extragradient framework whose error condition at each iteration includes both a relative error and a summable error it is then shown that the new admm variant is a special instance of the latter framework where the sequence of summable errors is identically zero when the admm stepsize is less than one or a nontrivial sequence when the stepsize is in the interval 1 1 sqrt 5 2

alternating direction method for structure persevering finite element model updating problem the problem of finding the optimal approximation to the discrete stiffness matrix modeled by the finite element method is considered in this paper desired properties of the updated matrix including symmetry positive semidefiniteness and structure connectivity are imposed as side constraints besides these the optimal approximate matrix should be the least squares solution to the dynamics equation to the best of the author s knowledge the optimal matrix approximation problem containing all these constraints simultaneously has not been proposed in the literature earlier alternating direction method is first applied to this constrained minimization problem numerical examples are performed to illustrate the efficiency of the proposed method

general projective splitting methods for sums of maximal monotone operators we describe a general projective framework for finding a zero of the sum of n maximal monotone operators over a real hilbert space unlike prior methods for this problem we neither assume n 2 nor first reduce the problem to the case n 2 our analysis defines a closed convex extended solution set for which we can construct a separating hyperplane by individually evaluating the resolvent of each operator at the cost of a single computationally simple projection step this framework gives rise to a family of splitting methods of unprecedented flexibility numerous parameters including the proximal stepsize may vary by iteration and by operator the order of operator evaluation may vary by iteration and may be either serial or parallel the analysis essentially generalizes our prior results for the case n 2 we also include a relative error criterion for approximately evaluating resolvents which was not present in our earlier work

recovering low rank and sparse components of matrices from incomplete and noisy observations many problems can be characterized by the task of recovering the low rank and sparse components of a given matrix recently it was discovered that this nondeterministic polynomial time hard np hard task can be well accomplished both theoretically and numerically via heuristically solving a convex relaxation problem where the widely acknowledged nuclear norm and l 1 norm are utilized to induce low rank and sparsity this paper studies the recovery task in the general settings that only a fraction of entries of the matrix can be observed and the observation is corrupted by both impulsive and gaussian noise we show that the resulting model falls into the applicable scope of the classical augmented lagrangian method moreover the separable structure of the new model enables us to solve the involved subproblems more efficiently by splitting the augmented lagrangian function hence some splitting numerical algorithms are developed for solving the new recovery model some preliminary numerical experiments verify that these augmented lagrangian based splitting algorithms are easily implementable and surprisingly efficient for tackling the new recovery model

polar n complex and n bicomplex singular value decomposition and principal component pursuit informed by recent work on tensor singular value decomposition and circulant algebra matrices this paper presents a new theoretical bridge that unifies the hypercomplex and tensor based approaches to singular value decomposition and robust principal component analysis we begin our work by extending the principal component pursuit to olariu s polar n complex numbers as well as their bicomplex counterparts in doing so we have derived the polar n complex and n bicomplex proximity operators for both the ell 1 and trace norm regularizers which can be used by proximal optimization methods such as the alternating direction method of multipliers experimental results on two sets of audio data show that our algebraically informed formulation outperforms tensor robust principal component analysis we conclude with the message that an informed definition of the trace norm can bridge the gap between the hypercomplex and tensor based approaches our approach can be seen as a general methodology for generating other principal component pursuit algorithms with proper algebraic structures

the admm algorithm for distributed quadratic problems parameter selection and constraint preconditioning this paper presents optimal parameter selection and preconditioning of the alternating direction method of multipliers admm algorithm for a class of distributed quadratic problems which can be formulated as equality constrained quadratic programming problems the parameter selection focuses on the admm step size and relaxation parameter while the preconditioning corresponds to selecting the edge weights of the underlying communication graph we optimize these parameters to yield the smallest convergence factor of the iterates explicit expressions are derived for the step size and relaxation parameter as well as for the corresponding convergence factor numerical simulations justify our results and highlight the benefits of optimal parameter selection and preconditioning for the admm algorithm

alternating direction method with gaussian back substitution for separable convex programming we consider the linearly constrained separable convex minimization problem whose objective function is separable into m individual convex functions with nonoverlapping variables a douglas rachford alternating direction method of multipliers adm has been well studied in the literature for the special case of m 2 but the convergence of extending adm to the general case of m ge 3 is still open in this paper we show that the straightforward extension of adm is valid for the general case of m ge 3 if it is combined with a gaussian back substitution procedure the resulting adm with gaussian back substitution is a novel approach towards the extension of adm from m 2 to m ge 3 and its algorithmic framework is new in the literature for the adm with gaussian back substitution we prove its convergence via the analytic framework of contractive type methods and we show its numerical efficiency by some application problems

solving constrained total variation image restoration and reconstruction problems via alternating direction methods in this paper we study alternating direction methods for solving constrained total variation image restoration and reconstruction problems alternating direction methods can be implementable variants of the classical augmented lagrangian method for optimization problems with separable structures and linear constraints the proposed framework allows us to solve problems of image restoration impulse noise removal inpainting and image cartoon texture decomposition as the constrained model is employed we need only to input the noise level and the estimation of the regularization parameter is not required in these imaging problems experimental results for such imaging problems are presented to illustrate the effectiveness of the proposed method we show that the alternating direction method is very efficient for solving image restoration and reconstruction problems

fast alternating linearization methods for minimizing the sum of two convex functions we present in this paper alternating linearization algorithms based on an alternating direction augmented lagrangian approach for minimizing the sum of two convex functions our basic methods require at most o 1 epsilon iterations to obtain an epsilon optimal solution while our accelerated i e fast versions of them require at most o 1 sqrt epsilon iterations with little change in the computational effort required at each iteration for both types of methods we present one algorithm that requires both functions to be smooth with lipschitz continuous gradients and one algorithm that needs only one of the functions to be so algorithms in this paper are gauss seidel type methods in contrast to the ones proposed by goldfarb and ma in fast multiple splitting algorithms for convex optimization columbia university 2009 where the algorithms are jacobi type methods numerical results are reported to support our theoretical conclusions and demonstrate the practical potential of our algorithms

convergence rate analysis for the alternating direction method of multipliers with a substitution procedure for separable convex programming recently in he et al he bs tao m yuan xm 2012 alternating direction method with gaussian back substitution for separable convex programming siam j optim 22 2 313 340 we have showed the first possibility of combining the douglas rachford alternating direction method of multipliers admm with a gaussian back substitution procedure for solving a convex minimization model with a general separable structure this paper is a further study on this theme we first derive a general algorithmic framework to combine admm with either a forward or backward substitution procedure then we show that convergence of this framework can be easily proved from the contraction perspective and its local linear convergence rate is provable if certain error bound condition is assumed without such an error bound assumption we can estimate its worst case convergence rate measured by the iteration complexity

relative error approximate versions of douglas rachford splitting and special cases of the admm we derive a new approximate version of the alternating direction method of multipliers admm which uses a relative error criterion the new version is somewhat restrictive and allows only one of the two subproblems to be minimized approximately but nevertheless covers commonly encountered special cases the derivation exploits the long established relationship between the admm and both the proximal point algorithm ppa and douglas rachford dr splitting for maximal monotone operators along with a relative error of the ppa due to solodov and svaiter in the course of analysis we also derive a version of dr splitting in which one operator may be evaluated approximately using a relative error criterion we computationally evaluate our method on several classes of test problems and find that it significantly outperforms several alternatives on one problem class

fixed point continuation for ell_1 minimization methodology and convergence we present a framework for solving the large scale ell 1 regularized convex minimization problem min x 1 mu f x our approach is based on two powerful algorithmic ideas operator splitting and continuation operator splitting results in a fixed point algorithm for any given scalar mu continuation refers to approximately following the path traced by the optimal value of x as mu increases in this paper we study the structure of optimal solution sets prove finite convergence for important quantities and establish q linear convergence rates for the fixed point algorithm applied to problems with f x convex but not necessarily strictly convex the continuation framework motivated by our convergence results is demonstrated to facilitate the construction of practical algorithms

a customized douglas rachford splitting algorithm for separable convex minimization with linear constraints we consider applying the douglas rachford splitting method drsm to the convex minimization problem with linear constraints and a separable objective function the dual application of drsm has been well studied in the literature resulting in the well known alternating direction method of multipliers admm in this paper we show that the primal application of drsm in combination with an appropriate decomposition can yield an efficient structure exploiting algorithm for the model under consideration whose subproblems could be easier than those of admm both the exact and inexact versions of this customized drsm are studied and their numerical efficiency is demonstrated by some preliminary numerical results

solving coupled composite monotone inclusions by successive fejer approximations of their kuhn tucker set we propose a new class of primal dual fejer monotone algorithms for solving systems of composite monotone inclusions our construction is inspired by a framework used by eckstein and svaiter for the basic problem of finding a zero of the sum of two monotone operators at each iteration points in the graph of the monotone operators present in the model are used to construct a half space containing the kuhn tucker set associated with the system the primal dual update is then obtained via a relaxed projection of the current iterate onto this half space an important feature that distinguishes the resulting splitting algorithms from existing ones is that they do not require prior knowledge of bounds on the linear operators involved or the inversion of linear operators

linear rate convergence of the alternating direction method of multipliers for convex composite programming in this paper we aim to prove the linear rate convergence of the alternating direction method of multipliers admm for solving linearly constrained convex composite optimization problems under a mild calmness condition which holds automatically for convex composite piecewise linear quadratic programming we establish the global q linear rate of convergence for a general semi proximal admm with the dual step length being taken in 0 1 51 2 2 this semi proximal admm which covers the classic one has the advantage to resolve the potentially nonsolvability issue of the subproblems in the classic admm and possesses the abilities of handling the multi block cases efficiently we demonstrate the usefulness of the obtained results when applied to two and multi block convex quadratic semidefinite programming

on the o 1 t convergence rate of alternating direction method with logarithmic quadratic proximal regularization it was shown recently that the douglas rachford alternating direction method of multipliers can be combined with the logarithmic quadratic proximal regularization for solving a class of variational inequalities with separable structures this paper further shows a worst case o 1 t convergence rate for this algorithm where a general glowinski relaxation factor is used

a fast algorithm for sparse reconstruction based on shrinkage subspace optimization and continuation we propose a fast algorithm for solving the ell 1 regularized minimization problem min x in mathbb r n mu x 1 ax b 2 2 for recovering sparse solutions to an undetermined system of linear equations ax b the algorithm is divided into two stages that are performed repeatedly in the first stage a first order iterative shrinkage method yields an estimate of the subset of components of x likely to be nonzero in an optimal solution restricting the decision variables x to this subset and fixing their signs at their current values reduces the ell 1 norm x 1 to a linear function of x the resulting subspace problem which involves the minimization of a smaller and smooth quadratic function is solved in the second phase our code fpc as embeds this basic two stage algorithm in a continuation homotopy approach by assigning a decreasing sequence of values to mu this code exhibits state of the art performance in terms of both its speed and its ability to recover sparse signals

on the o 1 n convergence rate of the douglas rachford alternating direction method alternating direction methods adms have been well studied in the literature and they have found many efficient applications in various fields in this note we focus on the douglas rachford adm scheme proposed by glowinski and marrocco and we aim at providing a simple approach to estimating its convergence rate in terms of the iteration number the linearized version of this adm scheme which is known as the split inexact uzawa method in the image processing literature is also discussed

local linear convergence of the alternating direction method of multipliers on quadratic or linear programs we introduce a novel matrix recurrence yielding a new spectral analysis of the local transient convergence behavior of the alternating direction method of multipliers admm for the particular case of a quadratic program or a linear program we identify a particular combination of vector iterates whose convergence can be analyzed via a spectral analysis the theory predicts that admm should go through up to four convergence regimes such as constant step convergence or linear convergence ending with the latter when close enough to the optimal solution if the optimal solution is unique and satisfies strict complementarity

explicit convergence rate of a distributed alternating direction method of multipliers consider a set of n agents seeking to solve distributively the minimization problem inf x sum n 1 n f n x where the convex functions f n are local to the agents the popular alternating direction method of multipliers has the potential to handle distributed optimization problems of this kind we provide a general reformulation of the problem and obtain a class of distributed algorithms which encompass various network architectures the rate of convergence of our method is considered it is assumed that the infimum of the problem is reached at a point x star the functions f n are twice differentiable at this point and sum nabla 2 f n x star 0 in the positive definite ordering of symmetric matrices with these assumptions it is shown that the convergence to the consensus x star is linear and the exact rate is provided application examples where this rate can be optimized with respect to the admm free parameter rho are also given

the fermat weber location problem revisited the fermat weber location problem requires finding a point in n that minimizes the sum of weighted euclidean distances tom given points a one point iterative method was first introduced by weiszfeld in 1937 to solve this problem since then several research articles have been published on the method and generalizations thereof global convergence of weiszfeld s algorithm was proven in a seminal paper by kuhn in 1973 however since them given points are singular points of the iteration functions convergence is conditional on none of the iterates coinciding with one of the given points in addressing this problem kuhn concluded that whenever them given points are not collinear weiszfeld s algorithm will converge to the unique optimal solution except for a denumerable set of starting points as late as 1989 chandrasekaran and tamir demonstrated with counter examples that convergence may not occur for continuous sets of starting points when the given points are contained in an affine subspace of n we resolve this open question by proving that weiszfeld s algorithm converges to the unique optimal solution for all but a denumerable set of starting points if and only if the convex hull of the given points is of dimensionn

optimal parameter selection for the alternating direction method of multipliers admm quadratic problems the alternating direction method of multipliers admm has emerged as a powerful technique for large scale structured optimization despite many recent results on the convergence properties of admm a quantitative characterization of the impact of the algorithm parameters on the convergence times of the method is still lacking in this paper we find the optimal algorithm parameters that minimize the convergence factor of the admm iterates in the context of l 2 regularized minimization and constrained quadratic programming numerical examples show that our parameter selection rules significantly outperform existing alternatives in the literature

a characteristic galerkin method with adaptive error control for the continuous casting problem the continuous casting problem is a convection dominated nonlinearly degenerate diffusion problem it is discretized implicitly in time via the method of characteristics and in space via continuous piecewise linear finite elements a posteriori error estimates are derived for the l1l1 norm of temperature which exhibit a mild explicit dependence on velocity the analysis is based on special properties of a linear dual problem in non divergence form with vanishing diffusion and strong advection several simulations with realistic physical parameters illustrate the reliability of the estimators and the flexibility of the proposed adaptive method

solving semidefinite quadratic linear programs using sdpt3 this paper discusses computational experiments with linear optimization problems involving semidefinite quadratic and linear cone constraints sqlps many test problems of this type are solved using a new release of sdpt3 a matlab implementation of infeasible primal dual path following algorithms the software developed by the authors uses mehrotra type predictor corrector variants of interior point methods and two types of search directions the hkm and nt directions a discussion of implementation details is provided and computational results on problems from the sdplib and dimacs challenge collections are reported

linear convergence and metric selection for douglas rachford splitting and admm recently several convergence rate results for douglas rachford splitting and the alternating direction method of multipliers admm have been presented in the literature in this paper we show global linear convergence rate bounds for douglas rachford splitting and admm under strong convexity and smoothness assumptions we further show that the rate bounds are tight for the class of problems under consideration for all feasible algorithm parameters for problems that satisfy the assumptions we show how to select step size and metric for the algorithm that optimize the derived convergence rate bounds for problems with a similar structure that do not satisfy the assumptions we present heuristic step size and metric selection methods

efficient iterative solution of finite element discretized nonsmooth minimization problems abstract for the iterative solution of finite element discretized nonsmooth minimization problems the alternating direction method of multipliers admm is considered which is a flexible method to solve a large class of convex minimization problems particular features are its unconditional convergence with respect to the involved step size and its direct applicability in particular this article deals with the admm with variable step sizes and devises an adjustment rule for the step size relying on the monotonicity of the residual and discusses proper stopping criteria the proposed scheme is applied to finite element formulations of the obstacle problem and the rudin osher fatemi image denoising problem and the numerical experiments show significant improvements over established variants of the admm

scenarios and policy aggregation in optimization under uncertainty a common approach in coping with multiperiod optimization problems under uncertainty where statistical information is not really enough to support a stochastic programming model has been to set up and analyze a number of scenarios the aim then is to identify trends and essential features on which a robust decision policy can be based this paper develops for the first time a rigorous algorithmic procedure for determining such a policy in response to any weighting of the scenarios the scenarios are bundled at various levels to reflect the availability of information and iterative adjustments are made to the decision policy to adapt to this structure and remove the dependence on hindsight

a posteriori error indicators for maxwell s equations abstract in this paper we shall analyze a class of a posteriori error indicators for an electromagnetic scattering problem for maxwell s equations in the presence of a bounded inhomogeneous and anisotropic scatterer problems of this type arise when computing the interaction of electromagnetic radiation with biological tissue we briefly recall existence and uniqueness theory associated with this problem then we show how a posteriori error indicators can be derived using an adjoint equation approach the error indicators use both the jump in normal and tangential components of the field across faces in the mesh

an inexact lqp alternating direction method for solving a class of structured variational inequalities in this paper we present an inexact alternating direction method each iteration of the proposed method contains a prediction and a correction the predictor is obtained via solving the logarithmic quadratic proximal lqp system approximately under significantly relaxed accuracy criterion and the new iterate is computed directly by an explicit formula derived from the original lqp method under certain conditions the global convergence of the proposed method is proved

fast consensus by the alternating direction multipliers method the alternating direction multipliers method admm has been recently proposed as a practical and efficient algorithm for distributed computing we discuss its applicability to the average consensus problem in this paper by carefully relaxing admm augmentation coefficients we are able to analytically investigate its properties and to propose simple and strict analytical bounds these provide a clear indication on how to choose system parameters for optimized performance we prove both analytically and via simulations that the proposed approach exhibits convergence speed between the best in the literature classical and optimized solutions while providing the most powerful resilience to noise

properties of gomory hu co cycle bases let d v a be a directed graph with non negative arc weights we study the problem of computing certain special co cycle bases of d in particular a minimum weight weakly fundamental co cycle basis a co cycle in d corresponds to a cut in the underlying undirected graph and a 1 0 1 arc incidence vector is associated with each co cycle where the 1 coordinates are used for the arcs crossing the cut the weight of a co cycle c is the sum of the weights of those arcs a such that c a 1 the vector space over q generated by the arc incidence vectors of the co cycles is the co cycle space of d the co cycle space of d can also be defined as the orthogonal complement of the cycle space of d a set of linearly independent co cycles that span the co cycle space is a co cycle basis of d the problem of computing a co cycle basis c 1 c k such that the sum of weights of the co cycles in the basis is the least possible is the minimum co cycle basis problem a co cycle basis c 1 c k is weakly fundamental if for every i there is an arc a i such that c i a i 1 while c j a i 0 for j i the minimum cycle basis problem in directed and undirected graphs is a well studied problem and while polynomial time algorithms are known for these problems the problem of computing a minimum weight weakly fundamental cycle basis has recently been shown to be apx hard we show that the co cycle basis corresponding to the cuts of a gomory hu tree t of the underlying undirected graph of d is a minimum weight weakly fundamental co cycle basis of d this is in fact a minimum co cycle basis of d and it is also totally unimodular thus this is a special co cycle basis that simultaneously answers several questions in the domain of co cycle bases it is known that there is no such special cycle basis for general graphs

fast multiple splitting algorithms for convex optimization we present in this paper two different classes of general multiple splitting algorithms for solving finite dimensional convex optimization problems under the assumption that the function being minimized can be written as the sum of k convex functions each of which has a lipschitz continuous gradient we prove that the number of iterations needed by the first class of algorithms to obtain an epsilon optimal solution is o k 1 l epsilon where l is an upper bound on all of the lipschitz constants the algorithms in the second class are accelerated versions of those in the first class where the complexity result is improved to o sqrt k 1 l epsilon while the computational effort required at each iteration is almost unchanged to the best of our knowledge the complexity results presented in this paper are the first ones of this type that have been given for splitting and alternating direction type methods moreover all algorithms proposed in this paper are parallelizable which makes the

on the efficiency of adaptive finite element methods for elliptic problems with discontinuous coefficients the successful implementation of adaptive finite element methods based on a posteriori error estimates depends on several ingredients an a posteriori error indicator a refinement coarsening strategy and the choice of various parameters the objective of the paper is to examine the influence of these factors on the performance of adaptive finite element methods for a model problem the linear elliptic equation with strongly discontinuous coefficients we derive a new a posteriori error estimator which depends locally on the oscillations of the coefficients around singular points extensive numerical experiments are reported to support our theoretical results and to show the competitive behaviors of the proposed adaptive algorithm

alternating direction methods for latent variable gaussian graphical model selection chandrasekaran parrilo and willsky 2012 proposed a convex optimization problem for graphical model selection in the presence of unobserved variables this convex optimization problem aims to estimate an inverse covariance matrix that can be decomposed into a sparse matrix minus a low rank matrix from sample data solving this convex optimization problem is very challenging especially for large problems in this letter we propose two alternating direction methods for solving this problem the first method is to apply the classic alternating direction method of multipliers to solve the problem as a consensus problem the second method is a proximal gradient based alternating direction method of multipliers our methods take advantage of the special structure of the problem and thus can solve large problems very efficiently a global convergence result is established for the proposed methods numerical results on both synthetic data and gene expression data show that our methods usually solve problems with 1 million variables in 1 to 2 minutes and are usually 5 to 35i times faster than a state of the art newton cg proximal point algorithm

iterative reweighted linear least squares for exact penalty subproblems on product sets we present two matrix free methods for solving exact penalty subproblems on product sets that arise when solving large scale optimization problems the first approach is a novel iterative reweighting algorithm irwa which iteratively minimizes quadratic models of relaxed subproblems while automatically updating a relaxation vector the second approach is based on alternating direction augmented lagrangian adal technology applied to our setting the main computational costs of each algorithm are the repeated minimizations of convex quadratic functions which can be performed matrix free we prove that both algorithms are globally convergent under loose assumptions and that each requires at most o 1 varepsilon 2 iterations to reach varepsilon optimality of the objective function numerical experiments exhibit the ability of both algorithms to efficiently find inexact solutions however in certain cases these experiments indicate that irwa can be significantly more efficient than adal

on non ergodic convergence rate of douglas rachford alternating direction method of multipliers this note proposes a novel approach to derive a worst case o 1 k o 1 k convergence rate measured by the iteration complexity in a non ergodic sense for the douglas rachford alternating direction method of multipliers proposed by glowinski and marrocco

an improved prediction correction method for monotone variational inequalities with separable operators in this paper we study the variational inequality problems with a particular splitting structure in which the mapping f does not have an explicit form and only its function values can be employed in the numerical methods for solving such problems studies and applications of such problems can be found in fukushima 1992 3 glowinski 1984 4 glowinski 1989 5 and xu 2007 15 the paper he et al 2006 6 presents an efficient prediction correction method for such problems based on the predictor method in the latter this paper presents two classes of correction methods which are more convenient to be carried out than that in this reference numerical experiments show that the new method is effective

data oscillation and convergence of adaptive fem data oscillation is intrinsic information missed by the averaging process associated with finite element methods fem regardless of quadrature ensuring a reduction rate of data oscillation together with an error reduction based on a posteriori error estimators we construct a simple and efficient adaptive fem for elliptic partial differential equations pdes with linear rate of convergence without any preliminary mesh adaptation nor explicit knowledge of constants any prescribed error tolerance is thus achieved in a finite number of steps a number of numerical experiments in two and three dimensions yield quasi optimal meshes along with a competitive performance

distributed estimation of laplacian eigenvalues via constrained consensus optimization problems from the recent literature we know that some consecutive measurements of the consensus protocol can be used to compute the exact average of the initial condition in this paper we show that these measurements can also be used for estimating the laplacian eigenvalues of the graph representing the network as recently shown in the literature by solving the factorization of the averaging matrix the laplacian eigenvalues can be inferred herein the problem is posed as a constrained consensus problem formulated two fold the first formulation direct approach yields a non convex optimization problem solved in a distributed way by means of the method of lagrange multipliers the second formulation indirect approach is obtained after an adequate re parameterization the problem is then convex and is solved by using the distributed subgradient algorithm and the alternating direction method of multipliers admm the proposed algorithms allow estimating the actual laplacian eigenvalues with high accuracy

a primal dual splitting algorithm for finding zeros of sums of maximal monotone operators we consider the primal problem of finding the zeros of the sum of a maximal monotone operator and the composition of another maximal monotone operator with a linear continuous operator by formulat

on the global linear convergence of the admm with multiblock variables the alternating direction method of multipliers admm has been widely used for solving structured convex optimization problems in particular the admm can solve convex programs that minimize the sum of n convex functions whose variables are linked by some linear constraints while the convergence of the admm for n 2 was well established in the literature it remained an open problem for a long time whether the admm for n ge 3 is still convergent recently it was shown in chen et al math program 2014 doi 10 1007 s10107 014 0826 5 that without additional conditions the admm for n ge 3 generally fails to converge in this paper we show that under some easily verifiable and reasonable conditions the global linear convergence of the admm when n geq 3 can still be ensured which is important since the admm is a popular method for solving large scale multiblock optimization models and is known to perform very well in practice even when n ge 3 our study aims to offer an explanation for

proximal decomposition via alternating linearization a new approximate proximal point method for minimizing the sum of two convex functions is introduced it replaces the original problem by a sequence of regularized subproblems in which the functions are alternately represented by linear models the method updates the linear models and the prox center as well as the prox coefficient it is monotone in terms of the objective values and converges to a solution of the problem if any a dual version of the method is derived and analyzed applications of the methods to multistage stochastic programming problems are discussed and preliminary numerical experience is presented

a proximal based decomposition method for convex minimization problems this paper presents a decomposition method for solving convex minimization problems at each iteration the algorithm computes two proximal steps in the dual variables and one proximal step in the primal variables we derive this algorithm from rockafellar s proximal method of multipliers which involves an augmented lagrangian with an additional quadratic proximal term the algorithm preserves the good features of the proximal method of multipliers with the additional advantage that it leads to a decoupling of the constraints and is thus suitable for parallel implementation we allow for computing approximately the proximal minimization steps and we prove that under mild assumptions on the problem s data the method is globally convergent and at a linear rate the method is compared with alternating direction type methods and applied to the particular case of minimizing a convex function over a finite intersection of closed convex sets

on the linear convergence of the alternating direction method of multipliers we analyze the convergence rate of the alternating direction method of multipliers admm for minimizing the sum of two or more nonsmooth convex separable functions subject to linear constraints previous analysis of the admm typically assumes that the objective function is the sum of only two convex functions defined on two separable blocks of variables even though the algorithm works well in numerical experiments for three or more blocks moreover there has been no rate of convergence analysis for the admm without strong convexity in the objective function in this paper we establish the global r linear convergence of the admm for minimizing the sum of any number of convex separable functions assuming that a certain error bound condition holds true and the dual stepsize is sufficiently small such an error bound condition is satisfied for example when the feasible set is a compact polyhedron and the objective function consists of a smooth strictly convex function composed with a linear mapping and a nonsmooth ell 1 l1 regularizer this result implies the linear convergence of the admm for contemporary applications such as lasso without assuming strong convexity of the objective function

a parallel algorithm for a class of convex programs a parallel algorithm is proposed in this paper for solving the problem min q x x in c 1 cap cdots cap c m where q is an uniformly convex function and c i are closed convex sets in r n in each iteration of the method we solve in parallel m independent subproblems each minimizing a definite quadratic function over an individual set c i the method has attractive convergence properties and can be implemented as parallel algorithms for tackling definite quadratic programs linear programs systems of linear equations and systems of generalized nonlinear inequalities

a variable penalty alternating directions method for convex optimization we study a generalized version of the method of alternating directions as applied to the minimization of the sum of two convex functions subject to linear constraints the method consists of solving consecutively in each iteration two optimization problems which contain in the objective function both lagrangian and proximal terms the minimizers determine the new proximal terms and a simple update of the lagrangian terms follows we prove a convergence theorem which extends existing results by relaxing the assumption of uniqueness of minimizers another novelty is that we allow penalty matrices and these may vary per iteration this can be beneficial in applications since it allows additional tuning of the method to the problem and can lead to faster convergence relative to fixed penalties as an application we derive a decomposition scheme for block angular optimization and present computational results on a class of dual block angular problems

an adaptive multilevel method for time harmonic maxwell equations with singularities we develop an adaptive edge finite element method based on reliable and efficient residual based a posteriori error estimates for low frequency time harmonic maxwell equations with singularities the resulting discrete problem is solved by the multigrid preconditioned minres minimum residual iteration algorithm we demonstrate the efficiency and robustness of the proposed method by extensive numerical experiments for cavity problems with singular solutions which include in particular scattering over screens

on the convergence of the exponential multiplier method for convex programming in this paper we analyze the exponential method of multipliers for convex constrained minimization problems which operates like the usual augmented lagrangian method except that it uses an exponential penalty function in place of the usual quadratic we also analyze a dual counterpart the entropy minimization algorithm which operates like the proximal minimization algorithm except that it uses a logarithmic entropy proximal term in place of a quadratic we strengthen substantially the available convergence results for these methods and we derive the convergence rate of these methods when applied to linear programs

comparison of several fast algorithms for projection onto an ellipsoid abstract projecting a point onto an ellipsoid is one of the fundamental problems in convex analysis and numerical algorithms recently several fast algorithms were proposed for solving this problem such as lin han algorithm maximum 2 dimensional inside ball algorithm sequential 2 dimensional projection algorithm and hybrid projection algorithms of dai in this paper we rewrite the problem as a constrained convex optimization problem with separable objective functions which enables the use of the alternating direction method of multipliers admm furthermore since the efficiency of admm depends on the penalty parameter we choose it in a self adaptive manner resulting in the self adaptive admm s admm all these methods converge with a global linear rate we compare them theoretically and numerically and find that s admm is the most efficient one we also illustrate the flexibility of admm by applying it to the more general problem of projecting a point onto the intersection of several ellipsoids dantzig selector and image restoration and reconstruction

globally convergent nonlinear 3d inverse model for smart materials with hessian based optimization abstract a globally convergent and fully coupled 3d inverse model for smart materials is presented in practice stress and field electric magnetic or temperature are applied to smart materials whereas strain and flux density electric magnetic or temperature are measured we refer to constitutive models that follow this scheme as direct models in certain design and control situations however inverse models are necessary in which the field and stress are found from specified flux density and strain this inversion typically involves an iterative procedure which may be prone to convergence issues an inverse model approach is proposed for arbitrary smart materials the inversion requirement is a continuous and second order differentiable direct model for any chosen smart material the approach is globally convergent which makes it ideal for use in finite element frameworks the premise of the proposed iterative system model is to constitute a recursive correction formula based on second order approximations of a novel scalar error function which offers a faster convergence rate a continuation approach is then used to achieve global convergence for arbitrary input parameters magnetostrictive galfenol is chosen to illustrate the effectiveness of the inverse model and compact analytical derivations of the jacobian and hessian matrices are presented the convergence rate of the proposed approach is superior to that of an existing inverse model finally the inverse model s robustness is demonstrated through integration of the model into a finite element framework to simulate a magnetostrictive composite plate actuator in full 3d