proximal decomposition on the graph of a maximal monotone operator we present an algorithm to solve find x y in a times a bot such that y in tx where a is a subspace and t is a maximal monotone operator the algorithm is based on the proximal decomposition on the graph of a monotone operator and we show how to recover spingarn s decomposition method we give a proof of convergence that does not use the concept of partial inverse and show how to choose a scaling factor to accelerate the convergence in the strongly monotone case numerical results performed on quadratic problems confirm the robust behaviour of the algorithm

a bundle bregman proximal method for convex nondifferentiable minimization by taking xk to be an approximate minimizer of where is a piecewise linear model of f constructed from accumulated subgradient linearizations of f dh is the d function of a generalized bregman function h and tk 0 convergence under implementable criteria is established by extending our recent framework of bregman proximal minimization which is of independent interest e g for nonquadratic multiplier methods for constrained minimization in particular we provide new insights into the convergence properties of bundle methods based on h 2

adaptive surface related multiple elimination the major amount of multiple energy in seismic data is related to the large reflectivity of the surface a method is proposed for the elimination of all surface related multiples by means of a process that removes the influence of the surface reflectivity from the data an important property of the proposed multiple elimination process is that no knowledge of the subsurface is required on the other hand the source signature and the surface reflectivity do need to be provided as a consequence the proposed process has been implemented adaptively meaning that multiple elimination is designed as an inversion process where the source and surface reflectivity properties are estimated and where the multiple free data equals the inversion residue results on simulated data and field data show that the proposed multiple elimination process should be considered as one of the key inversion steps in stepwise seismic inversion

large scale molecular optimization from distance matrices by a d c optimization approach a so called dca method based on a d c difference of convex functions optimization approach algorithm for solving large scale distance geometry problems is developed different formulations of equivalent d c programs in the l 1 approach are stated via the lagrangian duality without gap relative to d c programming and new nonstandard nonsmooth reformulations in the l infty approach resp the l 1 l infty approach are introduced substantial subdifferential calculations permit us to compute sequences of iterations in the dca quite simply the computations actually require matrix vector products and only one cholesky factorization resp with an additional solution of a convex program in the l 1 approach resp the l 1 l infty approach and allow the exploitation of sparsity in the large scale setting two techniques respectively using shortest paths between all pairs of atoms to generate the complete dissimilarity matrix and the spanning trees procedure are investigated in order to compute a good starting point for the dca finally many numerical simulations of the molecular optimization problems with up to 12567 variables are reported which prove the practical usefulness of the nonstandard nonsmooth reformulations the globality of found solutions and the robustness and efficiency of our algorithms

convex proximal bundle methods in depth a unified analysis for inexact oracles the last few years have seen the advent of a new generation of bundle methods capable to handle inexact oracles polluted by noise proving convergence of a bundle method is never simple and coping with inexact oracles substantially increases the technicalities besides several variants exist to deal with noise each one needing an ad hoc proof to show convergence we state a synthetic convergence theory in which we highlight the main arguments and specify which assumption is used to establish each intermediate result the framework is comprehensive and generalizes in various ways a number of algorithms proposed in the literature based on the ingredients of our synthetic theory we consider various bundle methods adapted to oracles for which high accuracy is possible yet it is preferable not to make exact calculations often because they are too time consuming

a practical relative error criterion for augmented lagrangians this paper develops a new error criterion for the approximate minimization of augmented lagrangian subproblems this criterion is practical since it is readily testable given only a gradient or subgradient of the augmented lagrangian it is also relative in the sense of relative error criteria for proximal point algorithms in particular it uses a single relative tolerance parameter rather than a summable parameter sequence our analysis first describes an abstract version of the criterion within rockafellar s general parametric convex duality framework and proves a global convergence result for the resulting algorithm specializing this algorithm to a standard formulation of convex programming produces a version of the classical augmented lagrangian method with a novel inexact solution condition for the subproblems finally we present computational results drawn from the cute test set including many nonconvex problems indicating that the approach works well in practice

data driven wavefield focusing and imaging with multidimensional deconvolution numerical examples for reflection data with internal multiples standard imaging techniques rely on the single scattering assumption this requires that the recorded data do not include internal multiples i e waves that have bounced multiple times between reflectors before reaching the receivers at the acquisition surface when multiple reflections are present in the data standard imaging algorithms incorrectly image them as ghost reflectors these artifacts can mislead interpreters in locating potential hydrocarbon reservoirs recently we introduced a new approach for retrieving the green s function recorded at the acquisition surface due to a virtual source located at depth we refer to this approach as data driven wavefield focusing additionally after applying source receiver reciprocity this approach allowed us to decompose the green s function at a virtual receiver at depth in its downgoing and upgoing components these wavefields were then used to create a ghost free image of the medium with either crosscorrelation or multidimensional deconvolution presenting an advantage over standard prestack migration we tested the robustness of our approach when an erroneous background velocity model is used to estimate the first arriving waves which are a required input for the datadriven wavefield focusing process we tested the new method with a numerical example based on a modification of the amoco model

iteration complexity of block decomposition algorithms and the alternating direction method of multipliers in this paper we consider the monotone inclusion problem consisting of the sum of a continuous monotone map and a point to set maximal monotone operator with a separable two block structure and introduce a framework of block decomposition prox type algorithms for solving it which allows for each one of the single block proximal subproblems to be solved in an approximate sense moreover by showing that any method in this framework is also a special instance of the hybrid proximal extragradient hpe method introduced by solodov and svaiter we derive corresponding convergence rate results we also describe some instances of the framework based on specific and inexpensive schemes for solving the single block proximal subproblems finally we consider some applications of our methodology to establish for the first time i the iteration complexity of an algorithm for finding a zero of the sum of two arbitrary maximal monotone operators and as a consequence the ergodic iteration complexity of the douglas

a fast approximation algorithm for solving the complete set packing problem we study the complete set packing problem cspp where the family of feasible subsets may include all possible combinations of objects this setting arises in applications such as combinatorial auctions for selecting optimal bids and cooperative game theory for finding optimal coalition structures although the set packing problem has been well studied in the literature where exact and approximation algorithms can solve very large instances with up to hundreds of objects and thousands of feasible subsets these methods are not extendable to the cspp since the number of feasible subsets is exponentially large formulating the cspp as an milp and solving it directly using cplex for example is impossible for problems with more than 20 objects we propose a new mathematical formulation for the cspp that directly leads to an efficient algorithm for finding feasible set packings upper bounds we also propose a new formulation for finding tighter lower bounds compared to lp relaxation and develop an efficient method for solving the corresponding large scale milp we test the algorithm with the winner determination problem in spectrum auctions the coalition structure generation problem in coalitional skill games and a number of other simulated problems that appear in the literature

a parallelizable augmented lagrangian method applied to large scale non convex constrained optimization problems we contribute improvements to a lagrangian dual solution approach applied to large scale optimization problems whose objective functions are convex continuously differentiable and possibly nonlinear while the non relaxed constraint set is compact but not necessarily convex such problems arise for example in the split variable deterministic reformulation of stochastic mixed integer optimization problems we adapt the augmented lagrangian method framework to address the presence of nonconvexity in the non relaxed constraint set and to enable efficient parallelization the development of our approach is most naturally compared with the development of proximal bundle methods and especially with their use of serious step conditions however deviations from these developments allow for an improvement in efficiency with which parallelization can be utilized pivotal in our modification to the augmented lagrangian method is an integration of the simplicial decomposition method and the nonlinear block gauss seidel method an adaptation of a serious step condition associated with proximal bundle methods allows for the approximation tolerance to be automatically adjusted under mild conditions optimal dual convergence is proven and we report computational results on test instances from the stochastic optimization literature we demonstrate improvement in parallel speedup over a baseline parallel approach

elastodynamic green s function retrieval through single sided marchenko inverse scattering the solution of the inverse scattering problem for the one dimensional schr odinger equation is given by the marchenko equation recently a marchenko type equation has been derived for three dimensional 3d acoustic wave fields whose solution has been shown to recover the green s functions from points within the medium to its exterior using only single sided scattered data here we extend this approach to 3d vectorial wave fields that satisfy the elastodynamic wave equation and recover green s functions from points interior to an elastic solid state medium from purely external and one sided measurements the method is demonstrated in a solid earth like model to construct green s functions using only subsurface sources from earth surface force and deformation sources and particle velocity and stress measurements

neural network for nonsmooth nonconvex constrained minimization via smooth approximation a neural network based on smoothing approximation is presented for a class of nonsmooth nonconvex constrained optimization problems where the objective function is nonsmooth and nonconvex the equality constraint functions are linear and the inequality constraint functions are nonsmooth convex this approach can find a clarke stationary point of the optimization problem by following a continuous path defined by a solution of an ordinary differential equation the global convergence is guaranteed if either the feasible set is bounded or the objective function is level bounded specially the proposed network does not require 1 the initial point to be feasible 2 a prior penalty parameter to be chosen exactly 3 a differential inclusion to be solved numerical experiments and comparisons with some existing algorithms are presented to illustrate the theoretical results and show the efficiency of the proposed network

an augmented lagrangian method for distributed optimization we propose a novel distributed method for convex optimization problems with a certain separability structure the method is based on the augmented lagrangian framework we analyze its convergence and provide an application to two network models as well as to a two stage stochastic optimization problem the proposed method compares favorably to two augmented lagrangian decomposition methods known in the literature as well as to decomposition methods based on the ordinary lagrangian function

proximal quasi newton methods for nondifferentiable convex optimization the method is based on rockafellar s proximal point algorithm and a cutting plane technique at each step we use an approximate proximal point pa xk of xk to define a vk ekf pa xk with ek vk where is a constant the method monitors the reduction in the value of vk to identify when a line search on f should be used the quasi newton step is used to reduce the value of vk without the differentiability of f the method converges globally and the rate of convergence is q linear superlinear convergence is also discussed to extend the characterization result of dennis and more numerical results show the good performance of the method

a cholesky dual method for proximal piecewise linear programming a quadratic programming method is given for minimizing a sum of piecewise linear functions and a proximal quadratic term subject to simple bounds on variables it may be used for search direction finding in nondifferentiable optimization algorithms an efficient implementation is described that updates a cholesky factorization of active constraints and provides good accuracy via iterative refinement numerical experience is reported for some large problems

a proximal cutting plane method using chebychev center for nonsmooth convex optimization an algorithm is developed for minimizing nonsmooth convex functions this algorithm extends elzinga moore cutting plane algorithm by enforcing the search of the next test point not too far from the previous ones thus removing compactness assumption our method is to elzinga moore s algorithm what a proximal bundle method is to kelley s algorithm instead of lower approximations used in proximal bundle methods the present approach is based on some objects regularizing translated functions of the objective function we propose some variants and using some academic test problems we conduct a numerical comparative study with elzinga moore algorithm and two other well known nonsmooth methods

on convergence of an augmented lagrangian decomposition method for sparse convex optimization a decomposition method for large scale convex optimization problems with block angular structure and many linking constraints is analysed the method is based on a separable approximation of the augmented lagrangian function weak global convergence of the method is proved and speed of convergence analysed it is shown that convergence properties of the method are heavily dependent on sparsity of the linking constraints application to large scale linear programming and stochastic programming is discussed

application of a smoothing technique to decomposition in convex optimization dual decomposition is a powerful technique for deriving decomposition schemes for convex optimization problems with separable structure although the augmented lagrangian is computationally more stable than the ordinary lagrangian the ldquoprox termrdquo destroys the separability of the given problem in this technical note we use another approach to obtain a smooth lagrangian based on a smoothing technique developed by nesterov which preserves separability of the problem with this approach we derive a new decomposition method called ldquoproximal center algorithm rdquo which from the viewpoint of efficiency estimates improves the bounds on the number of iterations of the classical dual gradient scheme by an order of magnitude

a practical general approximation criterion for methods of multipliers based on bregman distances this paper demonstrates that for generalized methods of multipliers for convex programming based on bregman distance kernels including the classical quadratic method of multipliers the minimization of the augmented lagrangian can be truncated using a simple generally implementable stopping criterion based only on the norms of the primal iterate and the gradient or a subgradient of the augmented lagrangian at that iterate previous results in this and related areas have required conditions that are much harder to verify such as e optimality with respect to the augmented lagrangian or strong conditions on the convex program to be solved here only existence of a kkt pair is required and the convergence properties of the exact form of the method are preserved the key new element in the analysis is the use of a full conjugate duality framework as opposed to mainly examining the action of the method on the standard dual function of the convex program an existence result for the iterates stronger than those possible for the exact form of the algorithm is also included

epsilon proximal decomposition method we propose a modification of the proximal decomposition method investigated by spingarn 30 and mahey et al 19 for minimizing a convex function on a subspace for the method to be favorable from a computational point of view particular importance is the introduction of approximations in the proximal step first we couple decomposition on the graph of the epsilon subdifferential mapping and cutting plane approximations to get an algorithmic pattern that falls in the general framework of rockafellar inexact proximal point algorithms 26 recently solodov and svaiter 27 proposed a new proximal point like algorithm that uses improved error criteria and an enlargement of the maximal monotone operator defining the problem we combine their idea with bundle mecanism to devise an inexact proximal decomposition method with error condition which is not hard to satisfy in practice then we present some applications favorable to our development first we give a new regularized version of benders decomposition method in convex programming called the proximal convex benders decomposition algorithm second we derive a new algorithm for nonlinear multicommodity flow problems among which the message routing problem in telecommunications data networks

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

optimal algorithms for l_ 1 subspace signal processing we describe ways to define and calculate l 1 norm signal subspaces that are less sensitive to outlying data than l 2 calculated subspaces we start with the computation of the l 1 maximum projection principal component of a data matrix containing n signal samples of dimension d we show that while the general problem is formally np hard in asymptotically large n d the case of engineering interest of fixed dimension d and asymptotically large sample size n is not in particular for the case where the sample size is less than the fixed dimension we present in explicit form an optimal algorithm of computational cost 2 n for the case n d we present an optimal algorithm of complexity o n d we generalize to multiple l 1 max projection components and present an explicit optimal l 1 subspace calculation algorithm of complexity o n dk k 1 where k is the desired number of l 1 principal components subspace rank we conclude with illustrations of l 1 subspace signal processing in the fields of data dimensionality reduction direction of arrival estimation and image conditioning restoration

spectral bounds for unconstrained 1 1 quadratic optimization problems abstract given an unconstrained quadratic optimization problem in the following form qp min x t qx x 1 1 n with q r n n we present different methods for computing bounds on its optimal objective value some of the lower bounds introduced are shown to generally improve over the one given by a classical semidefinite relaxation we report on theoretical results on these new bounds and provide preliminary computational experiments on small instances of the maximum cut problem illustrating their performance

variable metric bundle methods from conceptual to implementable forms to minimize a convex function we combine moreau yosida regularizations quasi newton matrices and bundling mechanisms first we develop conceptual forms using reversal quasi newton formulae and we state their global and local convergence then to produce implementable versions we incorporate a bundle strategy together with a curve search no convergence results are given for the implementable versions however some numerical illustrations show their good behaviour even for large scale problems

applications of splitting algorithm to decomposition in convex programming and variational inequalities recently han and lou proposed a highly parallelizable decomposition algorithm for minimizing a strongly convex cost over the intersection of closed convex sets it is shown that their algorithm is in fact a special case of a splitting algorithm analyzed by gabay for finding a zero of the sum of two maximal monotone operators gabay s convergence analysis for the splitting algorithm is sharpened and new applications of this algorithm to variational inequalities convex programming and the solution of linear complementarily problems are proposed for convex programs with a certain separable structure a multiplier method that is closely related to the alternating direction method of multipliers of gabay mercier and of glowinski marrocco but which uses both ordinary and augmented lagrangians is obtained

on the gap between the quadratic integer programming problem and its semidefinite relaxation consider the semidefinite relaxation sdr of the quadratic integer program qip where q is a given symmetric matrix and d is diagonal we consider the sdr gap we establish the uniqueness of the sdr solution and prove that if and only if r n 1max xtvvtx x 1 1 n 1 where v is an orthogonal matrix whose columns span the r dimensional null space of d q and where d is the unique sdr solution we also give a test for establishing whether that involves 2r 1 function evaluations in the case that r 1 we derive an upper bound on which is tighter than thus we show that breaching the sdr gap for the qip problem is as difficult as the solution of a qip with the rank of the cost function matrix equal to the dimension of the null space of d q this reduced rank qip problem has been recently shown to be solvable in polynomial time for fixed r

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

projection free parallel quadratic programming for linear model predictive control a key component in enabling the application of model predictive control mpc in fields such as automotive aerospace and factory automation is the availability of low complexity fast optimisation algorithms to solve the mpc finite horizon optimal control problem in architectures with reduced computational capabilities in this paper we introduce a projection free iterative optimisation algorithm and discuss its application to linear mpc the algorithm originally developed by brand for non negative quadratic programs is based on a multiplicative update rule and it is shown to converge to a fixed point which is the optimum an acceleration technique based on a projection free line search is also introduced to speed up the convergence to the optimum the algorithm is applied to mpc through the dual of the quadratic program qp formulated from the mpc finite time optimal control problem we discuss how termination conditions with guaranteed degree of suboptimality can be enforced and how the algorithm

application of computational geometry to multiuser detection in cdma the maximum likelihood multiuser detection problem in code division multiple access is known to be an optimization problem with an objective function that is required to be optimized over a combinatorial decision region conventional suboptimal detectors relax the combinatorial decision region by a convex region without altering the objective function to be optimized we take an approach wherein the objective function is reduced to a form appropriate for the application of a polynomial complexity algorithm in computational geometry while keeping the decision region combinatorial the resulting detector allows a tradeoff between performance and computational complexity the bit error rate performance of the detector has been found to be better than the decorrelator and the linear minimum mean square error detectors for the same level of complexity

on the convergence of a distributed augmented lagrangian method for nonconvex optimization in this paper we propose a distributed algorithm for optimization problems that involve a separable possibly nonconvex objective function subject to convex local constraints and linear coupling constraints the method is based on the accelerated distributed augmented lagrangians adal algorithm that was recently developed by the authors to address convex problems here we extend this line of work in two ways first we establish convergence of the method to a local minimum of the problem using assumptions that are common in the analysis of nonconvex optimization methods to the best of our knowledge this is the first work that shows convergence to local minima specifically for a distributed augmented lagrangian al method applied to nonconvex optimization problems distributed al methods are known to perform very well when used to solve convex problems second we propose a more general and decentralized rule to select the stepsizes of the method this improves on the authors original adal method where the stepsize selection used global information at initialization numerical results are included to verify the correctness and efficiency of the proposed distributed method

on the link between l1 pca and ica principal component analysis pca based on l1 norm maximization is an emerging technique that has drawn growing interest in the signal processing and machine learning research communities especially due to its robustness to outliers the present work proves that l1 norm pca can perform independent component analysis ica under the whitening assumption however when the source probability distributions fulfil certain conditions the l1 norm criterion needs to be minimized rather than maximized which can be accomplished by simple modifications on existing optimal algorithms for l1 pca if the sources have symmetric distributions we show in addition that l1 pca is linked to kurtosis optimization a number of numerical experiments illustrate the theoretical results and analyze the comparative performance of different algorithms for ica via l1 pca although our analysis is asymptotic in the sample size this equivalence opens interesting new perspectives for performing ica using optimal algorithms for l1 pca with guaranteed global convergence while inheriting the increased robustness to outliers of the l1 norm criterion

internal multiple prediction and removal using marchenko autofocusing and seismic interferometry standard seismic processing steps such as velocity analysis and reverse time migration imaging usually assume that all reflections are primaries multiples represent a source of coherent noise and must be suppressed to avoid imaging artifacts many suppression methods are relatively ineffective for internal multiples we show how to predict and remove internal multiples using marchenko autofocusing and seismic interferometry we first show how internal multiples can theoretically be reconstructed in convolutional interferometry by combining purely reflected up and downgoing green s functions from virtual sources in the subsurface we then generate the relevant up and downgoing wavefields at virtual sources along discrete subsurface boundaries using autofocusing then we convolve purely scattered components of up and downgoing green s functions to reconstruct only the internal multiple field which is adaptively subtracted from the measured data crucially this is all possible without detailed modeled information about the earth s subsurface the method only requires surface reflection data and estimates of direct nonreflected arrivals between subsurface virtual sources and the acquisition surface the method is demostrated on a stratified synclinal model and shown to be particularly robust against errors in the reference velocity model used

a polynomial case of unconstrained zero one quadratic optimization unconstrained zero one quadratic maximization problems can be solved in polynomial time when the symmetric matrix describing the objective function is positive semidefinite of fixed rank with known spectral decomposition

two level primal dual proximal decomposition technique to solve large scale optimization problems we describe a primal dual application of the proximal point algorithm to nonconvex minimization problems motivated by the work of spingarn and more recently by the work of hamdi et al about the primal resource directive decomposition scheme to solve nonlinear separable problems this paper discusses some local results of a primal dual regularization approach that leads to a decomposition algorithm

alternating linearization for structured regularization problems we adapt the alternating linearization method for proximal decomposition to structured regularization problems in particular to the generalized lasso problems the method is related to two well known operator splitting methods the douglas rachford and the peaceman rachford method but it has descent properties with respect to the objective function this is achieved by employing a special update test which decides whether it is beneficial to make a peaceman rachford step any of the two possible douglas rachford steps or none the convergence mechanism of the method is related to that of bundle methods of nonsmooth optimization we also discuss implementation for very large problems with the use of specialized algorithms and sparse data structures finally we present numerical results for several synthetic and real world examples including a three dimensional fused lasso problem which illustrate the scalability efficacy and accuracy of the method

fixed rank rayleigh quotient maximization by an mpsk sequence certain optimization problems in communication systems such as limited feedback constant envelope beamforming or noncoherent m ary phase shift keying mpsk sequence detection result in the maximization of a fixed rank positive semidefinite quadratic form over the mpsk alphabet this form is a special case of the rayleigh quotient of a matrix and in general its maximization by an mpsk sequence is mathcal np hard however if the rank of the matrix is not a function of its size then the optimal solution can be computed with polynomial complexity in the matrix size in this work we develop a new technique to efficiently solve this problem by utilizing auxiliary continuous valued angles and partitioning the resulting continuous space of solutions into a polynomial size set of regions each of which corresponds to a distinct mpsk sequence the sequence that maximizes the rayleigh quotient is shown to belong to this polynomial size set of sequences thus efficiently reducing the size of the feasible set from exponential to polynomial based on this analysis we also develop an algorithm that constructs this set in polynomial time and show that it is fully parallelizable memory efficient and rank scalable the proposed algorithm compares favorably with other solvers for this problem that have appeared recently in the literature

accounting for free surface multiples in marchenko imaging imagine placing a receiver at any location in the earth and recording the response at that location to sources on the surface in such a world we could place receivers around our reservoir to better image the reservoir and understand its properties realistically this is not a feasible approach for understanding the subsurface we have developed an alternative and realizable approach to obtain the response of a buried virtual receiver for sources at the surface our method is capable of retrieving the green s function for a virtual point in the subsurface to the acquisition surface in our case a physical receiver is not required at the subsurface point instead we require the reflection measurements for sources and receivers at the surface of the earth and a macromodel of the velocity no small scale details of the model are necessary we can interpret the retrieved green s function as the response to sources at the surface for a virtual receiver in the subsurface we obtain this green s function by solving the marchenko equation an integral equation pertinent to inverse scattering problems our derivation of the marchenko equation for the green s function retrieval takes into account the free surface reflections present in the reflection response previous work considered a response without free surface multiples we decompose the marchenko equation into up and downgoing fields and solve for these fields iteratively the retrieved green s function not only includes primaries and internal multiples as do previous methods but it also includes freesurface multiples we use these up and downgoing fields to obtain a 2d image of our area of interest in this case below a synclinal structure

an illustration of adaptive marchenko imaging in marchenko imaging wavefields are retrieved at specified focal points in the subsurface through an iterative scheme derived from the multidimensional marchenko equation the method requires seismic reflection data at the earth s surface after free surface multiple elimination and an estimate of the direct wavefield from the surface to each focal point which can be computed for instance in a macrovelocity model in the first iteration the direct wavefield is crosscorrelated with the reflection data this operation is identical to inverse wavefield extrapolation as is applied commonly in various imaging schemes for instance in reverse time migration rtm at each succeeding iteration the result of the previous iteration is truncated in time and crosscorrelated with the reflection data again to obtain a seismic image a multidimensional deconvolution based imaging condition can be applied to the retrieved wavefields by this approach both primary reflections and internal multiples contribute to the construction of the image alternatively a crosscorrelation based imaging condition can be used in which only the primary reflections are imaged and the contributions of internal multiples are subtracted the latter strategy offers more flexibility because the subtraction of redatumed internal multiples can be implemented adaptively through this approach the artifacts from internal multiples can be removed effectively from a conventional rtm image

efficient computation of the binary vector that maximizes a rank deficient quadratic form the maximization of a full rank quadratic form over the binary alphabet can be performed through exponential complexity exhaustive search however if the rank of the form is not a function of the problem size then it can be maximized in polynomial time by introducing auxiliary spherical coordinates we show that the rank deficient quadratic form maximization problem is converted into a double maximization of a linear form over a multidimensional continuous set the multidimensional set is partitioned into a polynomial size set of regions which are associated with distinct candidate binary vectors and the optimal binary vector belongs to the polynomial size set of candidate vectors thus the size of the candidate set is reduced from exponential to polynomial we also develop an algorithm that constructs the polynomial size candidate set in polynomial time and show that it is fully parallelizable and rank scalable finally we demonstrate the efficiency of the proposed algorithm in the context of adaptive spreading code design

generalized internal multiple imaging various examples are provided for generalized internal multiple imaging gimi in one example among others a method includes generating a higher order internal multiple image using a background green s function and rendering the higher order internal multiple image for presentation in another example a system includes a computing device and a generalized internal multiple imaging gimi application executable in the computing device the gimi application includes logic that generates a higher order internal multiple image using a background green s function and logic that renders the higher order internal multiple image for display on a display device in another example a non transitory computer readable medium has a program executable by processing circuitry that generates a higher order internal multiple image using a background green s function and renders the higher order internal multiple image for display on a display device

a parallel bundle framework for asynchronous subspace optimization of nonsmooth convex functions an algorithmic framework is presented for optimizing general convex functions by nonsynchronized parallel processes each process greedily picks a suitable adaptive subset of coordinates and runs a bundle method on a corresponding restricted problem stopping whenever a descent step is encountered or predicted decrease is reduced sufficiently no prior knowledge on the dependencies between variables is assumed instead dependency information is collected automatically by analyzing aggregate subgradient properties required for ensuring convergence within this framework three strategies are discussed for supporting varying scenarios of structural independence a single convex function the sum of partially separable convex functions and a scenario tuned to problem decomposition by lagrangian relaxation of packing type constraints the theoretical framework presented here generalizes a practical method proposed by the authors for lagrangian relaxation of large scale packing problems and simplifies the anal

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

solving the fixed rank convex quadratic maximization in binary variables by a parallel zonotope construction algorithm abstract we address the weighted max cut problem or equivalently the problem of maximizing a quadratic form in n binary variables if the underlying symmetric matrix is positive semidefinite of fixed rank d then the problem can be reduced to searching the extreme points of a zonotope thus becoming of polynomial complexity in o n d 1 reverse search is an efficient and practical means for enumerating the cells of a regular hyperplane arrangement or equivalently the extreme points of a zonotope we present an enhanced version of reverse search of significantly reduced computational complexity that uses ray shooting and is well suited for parallel computation furthermore a neighborhood zonotope edge following descent heuristic can be devised we report preliminary computational experiments of a parallel implementation of our algorithms

a descent algorithm for nonsmooth convex optimization this paper presents a new descent algorithm for minimizing a convex function which is not necessarily differentiable the algorithm can be implemented and may be considered a modification of the e subgradient algorithm and lemarechal s descent algorithm also our algorithm is seen to be closely related to the proximal point algorithm applied to convex minimization problems a convergence theorem for the algorithm is established under the assumption that the objective function is bounded from below limited computational experience with the algorithm is also reported

obtaining lower bounds from the progressive hedging algorithm for stochastic mixed integer programs we present a method for computing lower bounds in the progressive hedging algorithm pha for two stage and multi stage stochastic mixed integer programs computing lower bounds in the pha allows one to assess the quality of the solutions generated by the algorithm contemporaneously the lower bounds can be computed in any iteration of the algorithm by using dual prices that are calculated during execution of the standard pha we report computational results on stochastic unit commitment and stochastic server location problem instances and explore the relationship between key pha parameters and the quality of the resulting lower bounds

accelerating block decomposition first order methods for solving composite saddle point and two player nash equilibrium problems this article considers the two player composite nash equilibrium cne problem with a separable nonsmooth part which is known to include the composite saddle point csp problem as a special case due to its two block structure this problem can be solved by any algorithm belonging to the block decomposition hybrid proximal extragradient bd hpe framework proposed in r d c monteiro and b f svaiter siam j optim 23 2013 pp 475 507 the framework consists of a family of inexact proximal point methods for solving a more general two block structured monotone inclusion problem which at every iteration solves two prox subinclusions according to a certain relative error criterion by exploiting the fact that the two prox subinclusions in the context of the cne problem are equivalent to two composite convex programs this article proposes a new instance of the bd hpe framework that approximately solves them using an accelerated gradient method it is shown that the new instance is able to tak

proximity control in bundle methods for convex proximal bundle methods for minimizing a convex functionf generate a sequence x k by takingxk 1 to be the minimizer of hat f k x u k x x k 2 2 where hat f k is a sufficiently accurate polyhedral approximation tof andu k 0 the usual choice ofu k 1 may yield very slow convergence a technique is given for choosing u k adaptively that eliminates sensitivity to objective scaling some encouraging numerical experience is reported

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

autofocus imaging image reconstruction based on inverse scattering theory conventional imaging algorithms assume single scattering and therefore cannot image multiply scattered waves correctly the multiply scattered events in the data are imaged at incorrect locations resulting in spurious subsurface structures and erroneous interpretation this drawback of current migration imaging algorithms is especially problematic for regions where illumination is poor e g subsalt in which the spurious events can mask true structure here we discuss an imaging technique that not only images primaries but also internal multiples accurately using only surfacereflection data and direct arrivals we generate the up and down going wavefields at every image point in the subsurface an imaging condition is applied to these up and downgoing wavefields directly to generate the image because the above algorithm is based on inverse scattering theory the reconstructed wavefields are accurate and contain multiply scattered energy in addition to the primary event as corroborated by our synthetic examples imaging of these multiply scattered energy helps eliminate spurious reflectors in the image other advantages of this imaging algorithm over existing imaging algorithms include more accurate amplitudes target oriented imaging and a highly parallelizable algorithm

monotone operator theory in convex optimization several aspects of the interplay between monotone operator theory and convex optimization are presented the crucial role played by monotone operators in the analysis and the numerical solution of convex minimization problems is emphasized we review the properties of subdifferentials as maximally monotone operators and in tandem investigate those of proximity operators as resolvents in particular we study new transformations which map proximity operators to proximity operators and establish connections with self dual classes of firmly nonexpansive operators in addition new insights and developments are proposed on the algorithmic front

adaptive overburden elimination with the multidimensional marchenko equation abstractiterative substitution of the multidimensional marchenko equation has been introduced recently to integrate internal multiple reflections in the seismic imaging process in so called marchenko imaging a macro velocity model of the subsurface is required to meet this objective the model is used to back propagate the data during the first iteration and to truncate integrals in time during all successive iterations in case of an erroneous model the image will be blurred akin to conventional imaging and artifacts may arise from inaccurate integral truncations however the scheme is still successful in removing artifacts from internal multiple reflections inspired by these observations we rewrote the marchenko equation such that it can be applied early in a processing flow without the need of a macro velocity model instead we have required an estimate of the two way traveltime surface of a selected horizon in the subsurface we have introduced an approximation such that adaptive subtracti

implementation of the marchenko method the marchenko method makes it possible to compute subsurface to surface green s functions from reflection measurements at the surface applications of the marchenko method have already been discussed in many papers but its implementation aspects have not yet been discussed in detail solving the marchenko equation is an inverse problem the marchenko method computes a solution of the marchenko equation by an adaptive iterative scheme or by a direct inversion we have evaluated the iterative implementation based on a neumann series which is considered to be the conventional scheme at each iteration of this scheme a convolution in time and an integration in space are performed between a so called focusing update function and the reflection response in addition by applying a time window one obtains an update which becomes the input for the next iteration in each iteration upgoing and downgoing focusing functions are updated with these terms after convergence of the scheme the resulting upgoing and downgoing focusing functions are used to compute the upgoing and downgoing green s functions with a virtual source position in the subsurface and receivers at the surface we have evaluated this algorithm in detail and developed an implementation that reproduces our examples the software fits into the seismic unix software suite of the colorado school of mines

fast distributed gradient methods we study distributed optimization problems when n nodes minimize the sum of their individual costs subject to a common vector variable the costs are convex have lipschitz continuous gradient with constant l and bounded gradient we propose two fast distributed gradient algorithms based on the centralized nesterov gradient algorithm and establish their convergence rates in terms of the per node communications k and the per node gradient evaluations k our first method distributed nesterov gradient achieves rates o logk k and o logk k our second method distributed nesterov gradient with consensus iterations assumes at all nodes knowledge of l and w the second largest singular value of the n n doubly stochastic weight matrix w it achieves rates o 1 k2 and o 1 k2 0 arbitrarily small further we give for both methods explicit dependence of the convergence constants on n and w simulation examples illustrate our findings

wave focusing on the line focusing of waves in one dimension is analyzed for the plasma wave equation and the wave equation with variable speed the existence of focusing causal solutions to these equations is established and such wave solutions are constructed explicitly by deriving an orthogonality relation for the time independent schrodinger equation the connection between wave focusing and inverse scattering is studied the potential at any point is recovered from the incident wave that leads to focusing to that point it is shown that focusing waves satisfy certain temporal antisymmetry and support properties discontinuities in the spatial and temporal derivatives of the focusing waves are examined and related to the discontinuities in the potential of the schrodinger equation the theory is illustrated with some explicit examples