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

lipschitz continuity of solutions of linear inequalities programs and complementarity problems it is shown that solutions of linear inequalities linear programs and certain linear complementarity problems e g those with p matrices or z matrices but not semidefinite matrices are lipschitz continuous with respect to changes in the right hand side data of the problem solutions of linear programs are not lipschitz continuous with respect to the coefficients of the objective function the lipschitz constant given here is a generalization of the role played by the norm of the inverse of a nonsingular matrix in bounding the perturbation of the solution of a system of equations in terms of a right hand side perturbation

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

stochastic differential equations with reflecting boundary conditions on resout des equations differentielles stochastiques a conditions aux limites reflechissantes par une approche directe basee sur le probleme de skorokhod

reflected brownian motion in a cone with radially homogeneous reflection field this work is concerned with the existence and uniqueness of a strong markov process that has continuous sample paths and the following ad ditional properties i the state space is a cone in dimensions d 3 and the process behaves in the interior of the cone like ordinary brownian motion ii the process reflects instantaneously at the boundary of the cone the direction of reflection being fixed on each radial line emanating from the vertex of the cone iii the amount of time that the process spends at the vertex of the cone is zero i e the set of times for which the process is at the vertex has zero lebesgue measure the question of existence and uniqueness is cast in precise mathematical terms as a submartingale problem in the style used by stroock and varadhan for diffusions on smooth domains with smooth boundary conditions the question is resolved in terms of a real parameter a which in general depends in a rather complicated way on the geometric data of the problem i e on the cone and the directions of reflection however a criterion is given for determining whether a 0 it is shown that there is a unique continuous strong markov process satisfying i iii above if and only if a 2 there is a unique continuous strong markov process satisfying i and ii above it reaches the vertex of the cone almost surely and remains there these results are illustrated in concrete terms for some special cases the process considered here serves as a model for comparison with a re flected brownian motion in a cone having a nonradially homogeneous reflection field this is discussed in a subsequent work by kwon

separation of nonconvex sets with general augmenting functions in this paper we consider two geometric optimization problems that are dual to each other and characterize conditions under which the optimal values of the two problems are equal this characterization relies on establishing separation results for nonconvex sets using general concave surfaces defined in terms of convex augmenting functions we prove separation results for augmenting functions that are bounded from below unbounded augmenting functions and asymptotic augmenting functions

scaling and nonscaling finite size effects in the gaussian and the mean spherical model with free boundary conditions we calculate finite size effects of the gaussian model in a lxl d 1 box geometry with free boundary conditions in one direction and periodic boundary conditions in d 1 directions for 2 infinity finite size scaling is found to be valid for d 3 but logarithmic deviations from finite size scaling are found for the free energy and energy density at the gaussian upper borderline dimension d 3 the logarithms are related to the vanishing critical exponent 1 alpha nu d 3 2 of the gaussian surface energy density the latter has a cusplike singularity in d 3 dimensions we show that these properties are the origin of nonscaling finite size effects in the mean spherical model with free boundary conditions in dgreater than or equal to3 dimensions at bulk t c in d 3 dimensions we find an unexpected nonlogarithmic violation of finite size scaling for the susceptibility chisimilar tol 3 of the mean spherical model in film geometry whereas only a logarithmic deviation chisimilar tol 2 ln l exists for box geometry the result for film geometry is explained by the existence of the lower borderline dimension d l 3 as implied by the mermin wagner theorem that coincides with the gaussian upper borderline dimension d 3 for 3 4 we find a power law violation of scaling chisimilar tol d 1 at bulk t c for box geometry and a nonscaling temperature dependence chi surface similar toxi d of the surface susceptibility above t c for 2 3 dimensions we show the validity of universal finite size scaling for the susceptibility of the mean spherical model with free boundary conditions for both box and film geometry and calculate the corresponding universal scaling functions for tgreater than or equal tot c

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

uniqueness for diffusions with piecewise constant coefficients letl be a second order partial differential operator inrd letrd be the finite union of disjoint polyhedra suppose that the diffusion matrix is everywhere non singular and constant on each polyhedron and that the drift coefficient is bounded and measurable we show that the martingale problem associated withl is well posed

brownian motion in a wedge with variable reflection existence and uniqueness existence and uniqueness in law of reflecting brownian motion in a wedge is proved the direction of reflection along the sides of the wedge varies in a reasonable fashion except perhaps at the corner

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

systems with logarithmic specific heat finite size scaling a detailed study is presented of the finite size scaling in systems with vanishing critical exponent alpha which usually have logarithmic specific heat singularity the appropriate form of the finite size hyperuniversality is established recent results on complex temperature plane zeros of the partition function are extended to the alpha 0 case

perturbation analysis of a condition number for convex inequality systems and global error bounds for analytic systems in this paper several types of perturbations on a convex inequality system are considered and conditions are obtained for the system to be well conditioned under these types of perturbations where the well conditionedness of a convex inequality system is defined in terms of the uniform boundedness of condition numbers under a set of perturbations it is shown that certain types of perturbations can be used to characterize the well conditionedness of a convex inequality system in which either the system has a bounded solution set and satisfies the slater condition or an associated convex inequality system which defines the recession cone of the solution set for the system satisfies the slater condition finally sufficient conditions are given for the existence of a global error bound for an analytic system it is shown that such a global error bound always holds for any inequality system defined by finitely many convex analytic functions when the zero vector is in the relative interior of the domain of an associated convex conjugate function

brownian motion in a wedge with oblique reflection this work is concerned with the existence and uniqueness of a strong markov process that has continuous sample paths and the following additional properties i the state space is an infinite two dimensional wedge and the process behaves in the interior of the wedge like an ordinary brownian motion ii the process reflects instantaneously at the boundary of the wedge the angle of reflection being constant along each side iii the amount of time that the process spends at the comer of the wedge is zero i e the set of times for which the process is at the comer has lebesgue measure zero hereafter let be the angle of the wedge 0 2 let 1 and 2 be the angles of reflection on the two sides of the wedge measured from the inward normals the positive angles being toward the corner 1 2 and set 1 2 the question of existence and uniqueness is recast as a submartingale problem in the style used by stroock and varadhan diffusion processes with boundary conditions comm pure appl math 24 1971 pp 147 225 for diffusions on smooth domains with smooth boundary conditions it is shown that no solution exists if 2 in this case there is a unique continuous strong markov process satisfying i ii above it reaches the corner of the wedge almost surely and it remains there if 2 however then there is a unique continuous strong markov process statisfying i iii it is shown that starting away from the corner this process does not reach the corner of the wedge if 0 and does reach the corner if 0 2 the general theory of multi dimensional diffusions does not apply to the above problem because in general the boundary of the state space is not smooth and there is a discontinuity in the direction of reflection at the corner for some values of the process arises from diffusion approximations to storage systems and queueing networks i the state space is an infinite two dimensional wedge and the process behaves in the interior of the wedge like an ordinary brownian motion ii the process reflects instantaneously at the boundary of the wedge and the angle of reflection being constant along each side iii the amount of time that the process spends at the corner of the wedge is zero i e the set of times for which the process is at the corner has lebesgue measure zero

on the o 1 t convergence rate of ye yuan s modified alternating direction method of multipliers the alternating direction method of multipliers admm is known to be a classic and efficient method for constrained optimization problem with two blocks of variables and its empirical efficiency has been well illustrated in various fields specially for improving its speed performance ye and yuan suggested to do an additional extension with an optimal step size on the variables after each iteration of the primary admm indeed the numerical experiments indicate that this modified admm improves the speed performance of the admm by around 40 without changing the algorithmic framework much recently the o 1 t convergence rate of the primary admm is established inspired by its idea in this paper we show that this improved admm also has o 1 t convergence rate the reason that larger c yields better speed performance is also investigated and explained

total nuclear variation and jacobian extensions of total variation for vector fields we explore a class of vectorial total variation vtv measures formed as the spatial sum of a pixel wise matrix norm of the jacobian of a vector field we give a theoretical treatment that indicates that while color smearing and affine coupling bias often reported as gray scale bias are typically cited as drawbacks for vtv these are actually fundamental to smoothing vector direction i e smoothing hue and saturation in color images additionally we show that encouraging different vector channels to share a common gradient direction is equivalent to minimizing jacobian rank we thus propose total nuclear variation tnv and since nuclear norm is the convex envelope of matrix rank we argue that tnv is the optimal convex regularizer for enforcing shared directions we also propose extended jacobians which use larger neighborhoods than the conventional finite difference operator and we discuss efficient vtv optimization algorithms in simple color image denoising experiments tnv outperformed other common vtv regularizers and was further improved by using extended jacobians tnv was also competitive with the method of non local means often outperforming it by 0 25 to 2 db when using extended jacobians

finite size dependence of the helicity modulus within the mean spherical model the validity of the finite size scaling prediction about the existence of logarithmic corrections in the helicity modulus of three dimensional o n symmetric order parameter systems in confined geometries is studied for the three dimensional mean spherical model of geometryl 3 s d d 0 d 3 for a fully finite geometry the general case ofd p 0 periodic d a 0 antiperiodic d 0 0 free andd 1 0 fixed d p da d0 d1 d d 3 boundary conditions is considered whereas for film d 2 and cylinder d 1 geometries only the case of antiperiodic and or periodic boundary conditions is investigated the corresponding expressions for the finite size scaling function of the helicity modulus and its asymptotics in the vicinity below and above the bulk critical temperaturet c and the shifted critical temperaturet c l are derived the obtained results are not in agreement with the hypothesis of the existence of a log l correction term to the finite size behavior of the helicity modulus in the finite size critical region if d 3 in the case of film and cylinder geometries there are no logarithmic corrections in the case of a fully finite geometry a universal logarithmic correction term d 0 d 1 4 2da 1 2 lnl l is obtained only for t c t l lnl

interior gradient and epsilon subgradient descent methods for constrained convex minimization we extend epsilon subgradient descent methods for unconstrained nonsmooth convex minimization to constrained problems over polyhedral sets in particular over r p this is done by replacing the usual squared quadratic regularization term used in subgradient schemes by the logarithmic quadratic distancelike function recently introduced by the authors we then obtain interior e subgradient descent methods which allow us to provide a natural extension of bundle methods and polyak s subgradient projection methods for nonsmooth convex minimization furthermore similar extensions are considered for smooth constrained minimization to produce interior gradient descent methods global convergence as well as an improved global efficiency estimate are established for these methods within a unifying principle and minimal assumptions on the problem s data

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

the adjoint process of killed reflected brownian motion in a cone and applications let xt be reflected brownian motion rbm in a cone with radially homogeneous reflection killed upon reaching the vertex of the cone we determine the adjoint process and use it to find the martin boundary of the killed rbm together with all the corresponding positive harmonic functions then we can identify and prove uniqueness up to positive scalar multiples of the invariant measure for killed rbm and rbm without killing along the way we prove the strong feller property of the resolvent of rbm no killing 1 introduction denote by s d 1 d 3 the unit sphere in d let s d 1 be a domain such that s d 1 is nonempty and the boundary

lagrangian regularization approach to constrained optimization problems abstract this paper proposes a new regularization approach referred to as the lagrangian regularization approach which aims to circumvent the nondifferentiability of a positively homogeneous function r m in a conceptual unconstrained reformulation for constrained optimization problems with the appropriate choices of regularizing function we obtain a family of smooth functions that include as special cases the existing penalty and barrier functions as such our approach can be used as an instrumental tool to resolve the nondifferentiability of r m as well as a unified way to construct penalty functions for convex programming cases we present its global convergence analysis

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

failure of universal finite size scaling above the upper critical dimension abstract previous theories have predicted that o n symmetric systems in a finite cubic geometry with periodic boundary conditions have universal finite size scaling functions near criticality in d 4 dimensions on the basis of exact results for the o n symmetric 4 model in the large n limit we show that universal finite size scaling does not hold in the predicted form because of significant cut off and lattice effects for d 4 it is shown that finite size scaling is valid with two reference lengths which turn out to be identical with the amplitudes of the bulk correlation length for the 4 field theory the finite size scaling functions are shown to be non universal i e to depend explicitly on the cut off and on the bare four point coupling constant whereas for a 4 lattice model the finite size scaling functions have a different form that is independent of the lattice spacing and the four point coupling constant

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

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

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

nonlinear rescaling vs smoothing technique in convex optimization we introduce an alternative to the smoothing technique approach for constrained optimization as it turns out for any given smoothing function there exists a modification with particular properties we use the modification for nonlinear rescaling nr the constraints of a given constrained optimization problem into an equivalent set of constraints the constraints transformation is scaled by a vector of positive parameters the lagrangian for the equivalent problems is to the correspondent smoothing penalty functions as augmented lagrangian to the classical penalty function or mbfs to the barrier functions moreover the lagrangians for the equivalent problems combine the best properties of quadratic and nonquadratic augmented lagrangians and at the same time are free from their main drawbacks sequential unconstrained minimization of the lagrangian for the equivalent problem in primal space followed by both lagrange multipliers and scaling parameters update leads to a new class of nr multipliers methods which are equivalent to the interior quadratic prox methods for the dual problem we proved convergence and estimate the rate of convergence of the nr multipliers method under very mild assumptions on the input data we also estimate the rate of convergence under various assumptions on the input data in particular under the standard second order optimality conditions the nr method converges with q linear rate without unbounded increase of the scaling parameters which correspond to the active constraints we also established global quadratic convergence of the nr methods for linear programming with unique dual solution we provide numerical results which strongly support the theory

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

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

convergence analysis of a proximal like minimization algorithm using bregman functions an alternative convergence proof of a proximal like minimization algorithm using bregman functions recently proposed by censor and zenios is presented the analysis allows the establishment of a global convergence rate of the algorithm expressed in terms of function values

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

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

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

a generalization of the proximal point algorithm the problem considered in this paper is to find a solution to the generalized equation 0 in t x y where t is a maximal monotone operator on the product h 1 times h 2 of two hilbert spaces h 1 and h 2 we give a generalization of the proximal map and the proximal point algorithm in which the proposed iterative procedure is based on just one variable applying to convex programming problems instead of adding a quadratic term for all variables as in the proximal point algorithm a quadratic term for a subset of variables is added this paper proves that under a mild assumption our algorithm has the same convergence properties as the regular proximal point algorithm

stochastic differential equations for multi dimensional domain with reflecting boundary in this paper we prove that there exists a unique solution of the skorohod equation for a domain inr d with a reflecting boundary condition we remove the admissibility condition of the domain which is assumed in the work 4 of lions and sznitman we first consider a deterministic case and then discuss a stochastic case

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

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

a new partial splitting augmented lagrangian method for minimizing the sum of three convex functions in this paper we propose a new partial splitting augmented lagrangian method for solving the separable constrained convex programming problem where the objective function is the sum of three separable convex functions and the constraint set is also separable into three parts the proposed algorithm combines the alternating direction method adm and parallel splitting augmented lagrangian method psalm where two operators are handled by a parallel method while the third operator and the former two are dealt with by an alternating manner under mild conditions we prove the global convergence of the new method we also report some preliminary numerical results on constrained matrix optimization problem illustrating the advantage of the new algorithm over the most recently padalm of peng and wu 2010 12

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

an inexact alternating direction method for solving a class of structured variational inequalities the alternating direction method is mainly adopted to solve large scale variational inequality problems with separable structure the method is effective because it solves the original high dimensional variational inequality problem by solving a series of much easier low dimensional subproblems in this paper we present an inexact alternating directions method compared with the quadratic proximal alternating direction methods the proposed method solves a series of related systems of nonlinear equations instead of a series of sub vis the inexact criteria are more relaxed than the ones used by he et al 7 the generated sequence is fejer monotone with respect to the solution set and the convergence is proved under suitable conditions

finite size effects in fluid interfaces abstract finite size effects in interfacial tensions of fluid interfaces are studied at the level of the gaussian model of capillary waves it had been suggested that such effects might play a significant role in surface tension measurements based on capillary rise between plates spaced a small distance d apart the finite size correction varying as 1 d a reconsideration of the thermodynamics of finite size effects followed by careful calculations for the gaussian model leads to a finite size correction varying as 1 d 2 with a universal coefficient the effect is too small to be noticeable in current experiments the calculations have been carried out for rectangular interfaces of arbitrary dimensionality and general shapes subject to many types of boundary conditions they also have implications for finite size effects in interfacial free energies in two and three dimensional ising models which are discussed in connection with recent exact and monte carlo calculations

nuclear norm based matrix regression with applications to face recognition with occlusion and illumination changes recently regression analysis has become a popular tool for face recognition most existing regression methods use the one dimensional pixel based error model which characterizes the representation error individually pixel by pixel and thus neglects the two dimensional structure of the error image we observe that occlusion and illumination changes generally lead approximately to a low rank error image in order to make use of this low rank structural information this paper presents a two dimensional image matrix based error model namely nuclear norm based matrix regression nmr for face representation and classification nmr uses the minimal nuclear norm of representation error image as a criterion and the alternating direction method of multipliers admm to calculate the regression coefficients we further develop a fast admm algorithm to solve the approximate nmr model and show it has a quadratic rate of convergence we experiment using five popular face image databases the extended yale b ar eurecom multi pie and frgc experimental results demonstrate the performance advantage of nmr over the state of the art regression based methods for face recognition in the presence of occlusion and illumination variations

finite size effects in the spherical model of finite thickness a detailed analysis of the finite size effects on the bulk critical behaviour of the d dimensional mean spherical model confined to a film geometry with finite thickness l is reported along the finite direction different kinds of boundary conditions are applied periodic p antiperiodic a and free surfaces with dirichlet d neumann n and a combination of neumann and dirichlet nd on both surfaces a systematic method for the evaluation of the finite size corrections to the free energy for the different types of boundary conditions is proposed the free energy density and the equation for the spherical field are computed for arbitrary d it is found for 2 d 4 that the singular part of the free energy has the required finite size scaling form at the bulk critical temperature only for p and a for the remaining boundary conditions the standard finite size scaling hypothesis is not valid at d 3 the critical amplitude of the singular part of the free energy related to the so called casimir amplitude is estimated we obtain p 2 3 5 0 153 051 a 0 274 543 and nd 0 019 22 implying a fluctuation induced attraction between the surfaces for p and repulsion in the other two cases for d and n we find a logarithmic dependence on l

finite size effects in film geometry with nonperiodic boundary conditions gaussian model and renormalization group theory at fixed dimension finite size effects are investigated in the gaussian model with isotropic and anisotropic short range interactions in film geometry with nonperiodic boundary conditions bc above at and below the bulk critical temperature tc we have obtained exact results for the free energy and the casimir force for antiperiodic neumann dirichlet and neumann dirichlet mixed bc in 1 d 4 dimensions for the casimir force finite size scaling is found to be valid for all bc for the free energy finite size scaling is valid in 1 d 3 and 3 d 4 dimensions for antiperiodic neumann and dirichlet bc but logarithmic deviations from finite size scaling exist in d 3 dimensions for neumann and dirichlet bc this is explained in terms of the borderline dimension d 3 where the critical exponent 1 d 3 2 of the gaussian surface energy density vanishes for neumann dirichlet bc finite size scaling is strongly violated above tc for 1 d 4 because of a cancelation of the leading scaling terms for antiperiodic dirichlet and neumann dirichlet bc a finite film critical temperature tc film l tc exists at finite film thickness l our results include an exact description of the dimensional crossover between the d dimensional finite size critical behavior near bulk tc and the d 1 dimensional critical behavior near tc film l this dimensional crossover is illustrated for the critical behavior of the specific heat particular attention is paid to an appropriate representation of the free energy in the region tc film l t tc for 2 d 4 the gaussian results are renormalized and reformulated as one loop contributions of the 4 field theory at fixed dimension d and are then compared with the e 4 d expansion results at e 1 as well as with d 3 monte carlo data for d 2 the gaussian results for the casimir force scaling function are compared with those for the ising model with periodic antiperiodic and free bc unexpected exact relations are found between the gaussian and ising scaling functions for both the d dimensional gaussian model and the two dimensional ising model it is shown that anisotropic couplings imply nonuniversal scaling functions of the casimir force that depend explicitly on microscopic couplings our gaussian results provide the basis for the investigation of finite size effects of the mean spherical model in film geometry with nonperiodic bc above at and below the bulk critical temperature

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

a descent lqp alternating direction method for solving variational inequality problems with separable structure based on the logarithmic quadratic proximal method a descent alternating direction method is introduced for structured variational inequalities in the paper the proposal method generates the new iteration by searching the optimal step size along the integrated descent direction from two descent directions global convergence of the proposal method is proved under certain assumptions to illustrate the proposal method and demonstrate its efficiency some applications and their numerical results are also provided

the submartingal problem for brownian motion in a cone with non constant oblique reflection this work is concerned with the existence and uniqueness of a strong markov process that has continuous sample paths and the following additional properties 1 the state space is a cone ind dimensions d 3 and the process behaves in the interior of the cone like ordinary brownian motion 2 the process reflects instantaneously at the boundary of the cone the direction of reflection is continuous except at the vertex and has a limit which is fixed on each radial line emanating from the vertex 3 the amount of time that the process spends at the vertex of the cone is zero i e the set of times for which the process is at the vertex has zero lebesgue measure

brownian motion in a wedge with variable skew reflection does planar brownian motion confined to a wedge by skew reflection on the sides approach the vertex of the wedge this question has been answered by varadhan and williams in the case where the direction of reflection is constant on each of the sides but here we address the question when the direction reflected is allowed to vary a necessary condition and a sufficient condition are obtained for the vertex to be reached the conditions are of a geometric nature and the gap between them is quite small

reflected brownian motion in a cone semimartingale property in this paper the object of study is reflected brownian motion in a cone ind dimensions d 3 with nonconstant oblique reflection on each radial line emanating from the vertex of the cone the basic question considered here is when is this process a semimartingale conditions for the existence and uniqueness of the process for which the vertex is an instantaneous state were given by kwon which is resolved in terms of a real parameter depending on the cone and the direction of reflection it is shown that starting from any point of the cone the process is a semimartingale if 1 0 and not a semimartingale if 2

interior proximal and multiplier methods based on second order homogeneous kernels we study a class of interior proximal algorithms and nonquadratic multiplier methods for solving convex programs where the usual proximal quadratic term is replaced by an homogeneous functional of order two defined in terms of a convex function we prove under mild assumptions several new convergence results in both the primal and dual framework allowing also for approximate minimization in particular we introduce a new class of interior proximal methods which is globally convergent and allows for generating c multiplier methods with bounded hessians which exhibit strong convergence properties we also consider linearly constrained convex problems and establish global quadratic convergence rates results for linear programs we then study in detail a particular realization of these algorithms leading to a new class of logarithmic quadratic interior point algorithms which are shown to enjoy several attractive properties for solving constrained convex optimization problems

on full jacobian decomposition of the augmented lagrangian method for separable convex programming the augmented lagrangian method alm is a benchmark for solving a convex minimization model with linear constraints we consider the special case where the objective is the sum of m functions without coupled variables for solving this separable convex minimization model it is usually required to decompose the alm subproblem at each iteration into m smaller subproblems each of which only involves one function in the original objective easier subproblems capable of taking full advantage of the functions properties individually could thus be generated in this paper we focus on the case where full jacobian decomposition is applied to alm subproblems i e all the decomposed alm subproblems are eligible for parallel computation at each iteration for the first time we show by an example that the alm with full jacobian decomposition could be divergent to guarantee the convergence we suggest combining a relaxation step and the output of the alm with full jacobian decomposition a novel analysis

message passing for graph structured linear programs proximal methods and rounding schemes the problem of computing a maximum a posteriori map configuration is a central computational challenge associated with markov random fields there has been some focus on tree based linear programming lp relaxations for the map problem this paper develops a family of super linearly convergent algorithms for solving these lps based on proximal minimization schemes using bregman divergences as with standard message passing on graphs the algorithms are distributed and exploit the underlying graphical structure and so scale well to large problems our algorithms have a double loop character with the outer loop corresponding to the proximal sequence and an inner loop of cyclic bregman projections used to compute each proximal update we establish convergence guarantees for our algorithms and illustrate their performance via some simulations we also develop two classes of rounding schemes deterministic and randomized for obtaining integral configurations from the lp solutions our deterministic rounding schemes use a re parameterization property of our algorithms so that when the lp solution is integral the map solution can be obtained even before the lp solver converges to the optimum we also propose graph structured randomized rounding schemes applicable to iterative lp solving algorithms in general we analyze the performance of and report simulations comparing these rounding schemes

diversity of critical behavior within a universality class we study spatial anisotropy effects on the bulk and finite size critical behavior of the mathrm o n symmetric anisotropic ensuremath varphi 4 lattice model with periodic boundary conditions in a d dimensional hypercubic geometry above at and below t c the absence of two scale factor universality is discussed for the bulk order parameter correlation function the bulk scattering intensity and for several universal bulk amplitude relations the anisotropy parameters are observable by scattering experiments at t c for the confined system renormalization group theory within the minimal subtraction scheme at fixed dimension d for 2ldl4 is employed in contrast to the ensuremath epsilon 4 ensuremath d expansion the fixed d finite size approach keeps the exponential form of the order parameter distribution function unexpanded for the case of cubic symmetry and for n 1 our perturbation approach yields excellent agreement with the monte carlo mc data for the finite size amplitude of the free energy of the three dimensional ising model at t c by mon phys rev lett 54 2671 1985 the ensuremath epsilon expansion result is in less good agreement below t c a minimum of the scaling function of the excess free energy is found we predict a measurable dependence of this minimum on the anisotropy parameters the relative anisotropy effect on the free energy is predicted to be significantly larger than that on the binder cumulant our theory agrees quantitatively with the nonmonotonic dependence of the binder cumulant on the ferromagnetic next nearest neighbor nnn coupling of the two dimensional ising model found by mc simulations of selke and shchur j phys a 38 l739 2005 our theory also predicts a nonmonotonic dependence for small values of the antiferromagnetic nnn coupling and the existence of a lifshitz point at a larger value of this coupling the nonuniversal anisotropy effects in the finite size scaling regime are predicted to satisfy a kind of restricted universality the tails of the large l behavior at t ensuremath ne t c violate both finite size scaling and universality even for isotropic systems as they depend on the bare four point coupling of the ensuremath varphi 4 theory on the cutoff procedure and on subleading long range interactions

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

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