approximate primal solutions and rate analysis for dual subgradient methods in this paper we study methods for generating approximate primal solutions as a byproduct of subgradient methods applied to the lagrangian dual of a primal convex possibly nondifferentiable constrained optimization problem our work is motivated by constrained primal problems with a favorable dual problem structure that leads to efficient implementation of dual subgradient methods such as the recent resource allocation problems in large scale networks for such problems we propose and analyze dual subgradient methods that use averaging schemes to generate approximate primal optimal solutions these algorithms use a constant stepsize in view of its simplicity and practical significance we provide estimates on the primal infeasibility and primal suboptimality of the generated approximate primal solutions these estimates are given per iteration thus providing a basis for analyzing the trade offs between the desired level of error and the selection of the stepsize value our analysis relies on the slater condition and the inherited boundedness properties of the dual problem under this condition it also relies on the boundedness of subgradients which is ensured by assuming the compactness of the constraint set

a morse theoretical approach for the boundary mean curvature problem on b4 abstract in this paper we investigate existence as well as multiplicity of scalar flat metric of prescribed boundary mean curvature on the standard 4 dimensional ball due to the existence of critical point at infinity the standard variational methods cannot be applied to overcome this difficulty we prove that in a neighborhood of critical points at infinity a morse lemmas at infinity reduction holds then develop a whole morse theory of this noncompact variational problem in particular we establish under generic boundary condition morse inequalities at infinity which give a lower bound on the number of solutions to the above problem in terms of the total contribution of the critical point at infinity to the difference of topology between the level sets of the associated euler lagrange functional as further application of this morse theoretical approach we prove more existence results and extend a topological invariant introduced by a bahri

ergodic primal convergence in dual subgradient schemes for convex programming lagrangean dualization and subgradient optimization techniques are frequently used within the field of computational optimization for finding approximate solutions to large structured optimization prob lems the dual subgradient scheme does not automatically produce primal feasible solutions there is an abundance of techniques for computing such solutions via penalty functions tangential approximation schemes or the solution of auxiliary primal programs all of which require a fair amount of computational effort we consider a subgradient optimization scheme applied to a lagrangean dual formulation of a convex program and construct at minor cost an ergodic sequence of subproblem solutions which converges to the primal solution set numerical experiments performed on a traffic equilibrium assignment problem under road pricing show that the computation of the ergodic sequence results in a considerable improvement in the quality of the primal solutions obtained compared to those generated in the basic subgradient scheme

the nirenberg problem and its generalizations a unified approach making use of integral representations we develop a unified approach to establish blow up profiles compactness and existence of positive solutions of the conformally invariant equations p sigma v kv frac n 2 sigma n 2 sigma on the standard unit sphere mathbb s n for all sigma in 0 n 2 where p sigma is the intertwining operator of order 2 sigma finding positive solutions of these equations is equivalent to seeking metrics in the conformal class of the standard metric on spheres with prescribed certain curvatures when sigma 1 it is the prescribing scalar curvature problem or the nirenberg problem and when sigma 2 it is the prescribing q curvature problem

texture segmentation using voronoi polygons textures are defined in terms of primitives called tokens a texture segmentation algorithm based on the voronoi tessellation is discussed the algorithm first builds the voronoi tessellation of the tokens that make up the textured image it then computes a feature vector for each voronoi polygon these feature vectors are used in a probabilistic relaxation labeling on the tokens to identify the interior and the border regions of the textures some experimental results are shown

a perturbation result for prescribing mean curvature in this article we consider a boundary value problem associated with the conformal deformation of metrics we are interested in the case in which a noncompact group of conformal transformation acts on the equation so that the kazdanwarner condition gives rise to obstruction as in the nirenberg problem the simplest situation is the following let bn 1 be the unit ball in rn 1 with euclidean metric g0 its boundary will be denoted by s n bn 1 let h be a smooth function on s n we study the problem of finding a conformal metric gu u4 n 1 g0 whose scalar curvature vanishes in bn 1 and on the boundary bn 1 the mean curvature is given by h this problem is equivalent to solving the following boundary value problem

submodular functions from discrete to continous domains submodular set functions have many applications in combinatorial optimization as they can be minimized and approximately maximized in polynomial time a key element in many of the algorithms and analyses is the possibility of extending the submodular set function to a convex function which opens up tools from convex optimization submodularity goes beyond set functions and has naturally been considered for problems with multiple labels or for functions defined on continuous domains where it corresponds essentially to cross second derivatives being nonpositive in this paper we show that most results relating submodularity and convexity for set functions can be extended to all submodular functions in particular a we naturally define a continuous extension in a set of probability measures b show that the extension is convex if and only if the original function is submodular c prove that the problem of minimizing a submodular function is equivalent to a typically non smooth convex optimization problem and d propose another convex optimization problem with better computational properties e g a smooth dual problem most of these extensions from the set function situation are obtained by drawing links with the theory of multi marginal optimal transport which provides also a new interpretation of existing results for set functions we then provide practical algorithms to minimize generic submodular functions on discrete domains with associated convergence rates

distributed sub optimal resource allocation over weight balanced graph via singular perturbation in this paper we consider distributed optimization design for resource allocation problems over weight balanced graphs with the help of singular perturbation analysis we propose a simple sub optimal continuous time optimization algorithm moreover we prove the existence and uniqueness of the algorithm equilibrium and then show the convergence with an exponential rate finally we verify the sub optimality of the algorithm which can approach the optimal solution as an adjustable parameter tends to zero

the prescribed boundary mean curvature problem on b4 abstract in this paper we perform a fine blow up analysis for a boundary value elliptic equation involving the critical trace sobolev exponent related to the conformal deformation of the metrics on the standard ball namely the problem of prescribing the boundary mean curvature from this analysis some a priori estimates in low dimension are obtained with these estimates we prove the existence of at least one solution when an index counting formula associated to the prescribed mean curvature is different from zero

distributed robust adaptive equilibrium computation for generalized convex games this paper considers a class of generalized convex games where each player is associated with a convex objective function a convex inequality constraint and a convex constraint set the players aim to compute a nash equilibrium through communicating with neighboring players the particular challenge we consider is that the component functions are unknown a priori to associated players we study two distributed computation algorithms and analyze their convergence properties in the presence of data transmission delays and dynamic changes of network topologies the algorithm performance is verified through demand response on the ieee 30 bus test system our technical tools integrate convex analysis variational inequalities and simultaneous perturbation stochastic approximation

primal convergence from dual subgradient methods for convex optimization when solving a convex optimization problem through a lagrangian dual reformulation subgradient optimization methods are favorably utilized since they often find near optimal dual solutions quickly however an optimal primal solution is generally not obtained directly through such a subgradient approach unless the lagrangian dual function is differentiable at an optimal solution we construct a sequence of convex combinations of primal subproblem solutions a so called ergodic sequence which is shown to converge to an optimal primal solution when the convexity weights are appropriately chosen we generalize previous convergence results from linear to convex optimization and present a new set of rules for constructing the convexity weights that define the ergodic sequence of primal solutions in contrast to previously proposed rules they exploit more information from later subproblem solutions than from earlier ones we evaluate the proposed rules on a set of nonlinear multicommodity flow problems and demonstrate that they clearly outperform the ones previously proposed

approximate projected consensus for convex intersection computation convergence analysis and critical error angle in this paper we study an approximate projected consensus algorithm for a network to cooperatively compute the intersection of convex sets where each set corresponds to one network node instead of assuming exact convex projection that each node can compute we allow each node to compute an approximate projection with respect to its own set after receiving the approximate projection information nodes update their states by weighted averaging with the neighbors over a directed and time varying communication graph the approximate projections are related to projection angle errors which introduces state dependent disturbance in the iterative algorithm projection accuracy conditions are presented for the considered algorithm to converge the results indicate how much projection accuracy is required to ensure global consensus to a point in the intersection set when the communication graph is uniformly jointly strongly connected in addition we show that pi 4 is a critical angle for the error of the projection approximation to ensure the boundedness finally the results are illustrated by simulations

convergence of limited communication gradient methods distributed optimization increasingly plays a central role in economical and sustainable operation of cyber physical systems nevertheless the complete potential of the technology has not yet been fully exploited in practice due to communication limitations posed by the real world infrastructures this work investigates fundamental properties of distributed optimization based on gradient methods where gradient information is communicated using a limited number of bits in particular a general class of quantized gradient methods are studied where the gradient direction is approximated by a finite quantization set sufficient and necessary conditions are provided on such a quantization set to guarantee that the methods minimize any convex objective function with lipschitz continuous gradient and a nonempty and bounded set of optimizers a lower bound on the cardinality of the quantization set is provided along with specific examples of minimal quantizations convergence rate results are established that connect the fineness of the quantization and the number of iterations needed to reach a predefined solution accuracy generalizations of the results to a relevant class of constrained problems using projections are considered finally the results are illustrated by simulations of practical systems

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 posteriori error bounds for the linearly constrained varitional inequality problem in this paper we derive a posteriori error bound of an approximate solution to a linearly constrained variational inequality problem three measures of error are used with each depending on a particular characterization of an exact solution and related to a certain class of iterative methods

computational complexity certification for dual gradient method application to embedded mpc abstract in this paper we analyze the computational complexity of the dual gradient method for solving linearly constrained convex problems when it is difficult to project on the primal feasible set described by linear constraints we use the lagrangian relaxation to handle the complicated constraints and then we apply the dual gradient algorithm for solving the corresponding dual we give a unified convergence rate analysis for the dual gradient algorithm we provide sublinear or linear estimates on the primal suboptimality and feasibility violation of the generated approximate primal solutions our analysis relies on the lipschitz property of the dual function or an error bound property furthermore the iteration complexity analysis is based on two types of approximate primal solutions an average primal sequence or the last primal iterate sequence we also discuss complexity certifications and implementation aspects of the dual gradient algorithm on constrained mpc problems for embedded linear systems

non euclidean restricted memory level method for large scale convex optimization we propose a new subgradient type method for minimizing extremely large scale nonsmooth convex functions over simple domains the characteristic features of the method are a the possibility to adjust the scheme to the geometry of the feasible set thus allowing to get nearly dimension independent and nearly optimal in the large scale case rate of convergence results for minimization of a convex lipschitz continuous function over a euclidean ball a standard simplex and a spectahedron the set of positive semidefinite symmetric matrices of given size with unit trace b flexible handling of accumulated information allowing for tradeoff between the level of utilizing this information and iteration s complexity we present extensions of the scheme for the cases of minimizing non lipschitzian convex objectives finding saddle points of convex concave functions and solving variational inequalities with monotone operators finally we report on encouraging numerical results of experiments with test problems of dimensions up to 66 000

on distributed convex optimization under inequality and equality constraints we consider a general multi agent convex optimization problem where the agents are to collectively minimize a global objective function subject to a global inequality constraint a global equality constraint and a global constraint set the objective function is defined by a sum of local objective functions while the global constraint set is produced by the intersection of local constraint sets in particular we study two cases one where the equality constraint is absent and the other where the local constraint sets are identical we devise two distributed primal dual subgradient algorithms based on the characterization of the primal dual optimal solutions as the saddle points of the lagrangian and penalty functions these algorithms can be implemented over networks with dynamically changing topologies but satisfying a standard connectivity property and allow the agents to asymptotically agree on optimal solutions and optimal values of the optimization problem under the slater s condition

blow up phenomena for scalar flat metrics on manifolds with boundary abstract let m n g be a compact riemannian manifold with boundary m this article is concerned with the set of scalar flat metrics on m which are in the conformal class of g and have m as a constant mean curvature hypersurface we construct examples of metrics on the unit ball b n in dimensions n 25 for which this set is non compact these manifolds have umbilic boundary but they are not conformally equivalent to b n

incremental subgradient methods for nondifferentiable optimization we consider a class of subgradient methods for minimizing a convex function that consists of the sum of a large number of component functions this type of minimization arises in a dual context from lagrangian relaxation of the coupling constraints of large scale separable problems the idea is to perform the subgradient iteration incrementally by sequentially taking steps along the subgradients of the component functions with intermediate adjustment of the variables after processing each component function this incremental approach has been very successful in solving large differentiable least squares problems such as those arising in the training of neural networks and it has resulted in a much better practical rate of convergence than the steepest descent method in this paper we establish the convergence properties of a number of variants of incremental subgradient methods including some that are stochastic based on the analysis and computational experiments the methods appear very promising and effective for important classes of large problems a particularly interesting discovery is that by randomizing the order of selection of component functions for iteration the convergence rate is substantially improved

cooperative distributed model predictive control abstract in this paper we propose a cooperative distributed linear model predictive control strategy applicable to any finite number of subsystems satisfying a stabilizability condition the control strategy has the following features hard input constraints are satisfied terminating the iteration of the distributed controllers prior to convergence retains closed loop stability in the limit of iterating to convergence the control feedback is plantwide pareto optimal and equivalent to the centralized control solution no coordination layer is employed we provide guidance in how to partition the subsystems within the plant we first establish exponential stability of suboptimal model predictive control and show that the proposed cooperative control strategy is in this class we also establish that under perturbation from a stable state estimator the origin remains exponentially stable for plants with sparsely coupled input constraints we provide an extension in which the decision variable space of each suboptimization is augmented to achieve pareto optimality we conclude with a simple example showing the performance advantage of cooperative control compared to noncooperative and decentralized control strategies

duality between subgradient and conditional gradient methods given a convex optimization problem and its dual there are many possible first order algorithms in this paper we show the equivalence between mirror descent algorithms and algorithms generalizing the conditional gradient method this is done through convex duality and implies notably that for certain problems such as for supervised machine learning problems with nonsmooth losses or problems regularized by nonsmooth regularizers the primal subgradient method and the dual conditional gradient method are formally equivalent the dual interpretation leads to a form of line search for mirror descent as well as guarantees of convergence for primal dual certificates

distributed constrained optimization by consensus based primal dual perturbation method various distributed optimization methods have been developed for solving problems which have simple local constraint sets and whose objective function is the sum of local cost functions of distributed agents in a network motivated by emerging applications in smart grid and distributed sparse regression this paper studies distributed optimization methods for solving general problems which have a coupled global cost function and have inequality constraints we consider a network scenario where each agent has no global knowledge and can access only its local mapping and constraint functions to solve this problem in a distributed manner we propose a consensus based distributed primal dual perturbation pdp algorithm in the algorithm agents employ the average consensus technique to estimate the global cost and constraint functions via exchanging messages with neighbors and meanwhile use a local primal dual perturbed subgradient method to approach a global optimum the proposed pdp method not only can handle smooth inequality constraints but also non smooth constraints such as some sparsity promoting constraints arising in sparse optimization we prove that the proposed pdp algorithm converges to an optimal primal dual solution of the original problem under standard problem and network assumptions numerical results illustrating the performance of the proposed algorithm for a distributed demand response control problem in smart grid are also presented

the prescribed boundary mean curvature problem on the standard n dimensional ball abstract in this paper we consider a boundary value problem associated with the conformal deformation of metrics on the unit ball b n n 3 using dynamical and topological methods involving the study of critical points at infinity of the associated variational problem we prove the existence of a metric conformally equivalent to the euclidean metric with zero scalar curvature and prescribed mean curvature on the boundary

a decomposition method for large scale milps with performance guarantees and a power system application lagrangian duality in mixed integer optimization is a useful framework for problem decomposition and for producing tight lower bounds to the optimal objective however in contrast to the convex case it is generally unable to produce optimal solutions directly in fact solutions recovered from the dual may not only be suboptimal but even infeasible in this paper we concentrate on large scale mixed integer programs with a specific structure that appears in a variety of application domains such as power systems and supply chain management we propose a solution method for these structures in which the primal problem is modified in a certain way guaranteeing that the solutions produced by the corresponding dual are feasible for the original unmodified primal problem the modification is simple to implement and the method is amenable to distributed computation we also demonstrate that the quality of the solutions recovered using our procedure improves as the problem size increases making it particularly useful for large scale problem instances for which commercial solvers are inadequate we illustrate the efficacy of our method with extensive experimentations on a problem stemming from power systems

conformal metrics on the unit ball with prescribed mean curvature this paper focuses on the study of the prescribed mean curvature problem on the unit ball if the difference between the mean curvature candidate f and mean curvature of the standard metric in the supremum norm is sufficiently small then the existence of positive solutions of conformal mean curvature equation has been known the purpose of the paper is to investigate quantitatively how large that difference can be by using a flow method

power watershed a unifying graph based optimization framework in this work we extend a common framework for graph based image segmentation that includes the graph cuts random walker and shortest path optimization algorithms viewing an image as a weighted graph these algorithms can be expressed by means of a common energy function with differing choices of a parameter q acting as an exponent on the differences between neighboring nodes introducing a new parameter p that fixes a power for the edge weights allows us to also include the optimal spanning forest algorithm for watershed in this same framework we then propose a new family of segmentation algorithms that fixes p to produce an optimal spanning forest but varies the power q beyond the usual watershed algorithm which we term the power watershed in particular when q 2 the power watershed leads to a multilabel scale and contrast invariant unique global optimum obtained in practice in quasi linear time placing the watershed algorithm in this energy minimization framework also opens new possibilities for using unary terms in traditional watershed segmentation and using watershed to optimize more general models of use in applications beyond image segmentation

a class of convergent primal dual subgradient algorithms for decomposable convex programs in this paper we develop a primal dual subgradient algorithm for preferably decomposable generally nondifferentiable convex programming problems under usual regularity conditions the algorithm employs a lagrangian dual function along with a suitable penalty function which satisfies a specified set of properties in order to generate a sequence of primal and dual iterates for which some subsequence converges to a pair of primal dual optimal solutions several classical types of penalty functions are shown to satisfy these specified properties a geometric convergence rate is established for the algorithm under some additional assumptions this approach has three principal advantages firstly both primal and dual solutions are available which prove to be useful in several contexts secondly the choice of step sizes which plays an important role in subgradient optimization is guided more determinably in this method via primal and dual information thirdly typical subgradient algorithms suffer from the lack of an appropriate stopping criterion and so the quality of the solution obtained after a finite number of steps is usually unknown in contrast by using the primal dual gap the proposed algorithm possesses a natural stopping criterion

metric selection in fast dual forward backward splitting the performance of fast forward backward splitting or equivalently fast proximal gradient methods depends on the conditioning of the optimization problem data this conditioning is related to a metric that is defined by the space on which the optimization problem is stated selecting a space on which the optimization data is better conditioned improves the performance of the algorithm in this paper we propose several methods with different computational complexity to find a space on which the algorithm performs well we evaluate the proposed metric selection procedures by comparing the performance to the case when the euclidean space is used for the most ill conditioned problem we consider the computational complexity is improved by two to three orders of magnitude we also report comparable to superior performance compared to state of the art optimization software

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

dual decomposition for multi agent distributed optimization with coupling constraints abstract we study distributed optimization in a cooperative multi agent setting where agents have to agree on the usage of shared resources and can communicate via a time varying network to this purpose each agent has its own decision variables that should be set so as to minimize its individual objective function subject to local constraints resource sharing is modeled via coupling constraints that involve the non positivity of the sum of agents individual functions each one depending on the decision variables of one single agent we propose a novel distributed algorithm to minimize the sum of the agents objective functions subject to both local and coupling constraints where dual decomposition and proximal minimization are combined in an iterative scheme notably privacy of information is guaranteed since only the dual optimization variables associated with the coupling constraints are exchanged by the agents under convexity assumptions jointly with suitable connectivity properties of the communication network we are able to prove that agents reach consensus to some optimal solution of the centralized dual problem counterpart while primal variables converge to the set of optimizers of the centralized primal problem the efficacy of the proposed approach is demonstrated on a plug in electric vehicles charging problem

a compactness theorem for scalar flat metrics on 3 manifolds with boundary abstract let m g be a compact riemannian three dimensional manifold with boundary we prove the compactness of the set of scalar flat metrics which are in the conformal class of g and have the boundary as a constant mean curvature hypersurface this involves a blow up analysis of a yamabe type equation with critical sobolev exponent on the boundary

accelerated dual descent for network flow optimization we present a fast distributed solution to the convex network flow optimization problem our approach uses a family of dual descent algorithms that approximate the newton direction to achieve faster convergence rates than existing distributed methods the approximate newton directions are obtained through matrix splitting techniques and sparse taylor approximations of the inverse hessian we couple this descent direction with a distributed line search algorithm which requires the same information as our descent direction to compute we show that similarly to conventional newton methods the proposed algorithm exhibits super linear convergence within a neighborhood of the optimal value numerical experiments corroborate that convergence times are between one to two orders of magnitude faster than existing distributed optimization methods a connection with recent developments that use consensus to compute approximate newton directions is also presented

distributed model predictive control of linear discrete time systems with local and global constraints abstract this paper proposes a distributed model predictive control dmpc approach for a family of discrete time linear systems with local uncoupled and global coupled constraints the proposed approach is based on the dual problem of a mpc optimization problem involving all systems this dual problem is then distributedly solved based on the alternating direction multiplier method admm with several known simplifications when the network of systems is large or sparsely connected the computation of the optimal control using admm can be expensive the proposed approach mitigates this problem by allowing early termination of the admm process this is made possible via a finite time consensus algorithm that determines the satisfaction of the termination condition and by appropriate tightening of the coupled constraints under reasonable assumptions the approach is guaranteed to converge to a small neighborhood of the optimal so long as the network is connected recursive feasibility and exponential stability of the closed loop system are shown the performance of the proposed approach is demonstrated by a numerical example

linear convergence of randomized feasible descent methods under the weak strong convexity assumption in this paper we generalize the framework of the feasible descent method fdm to a randomized r fdm and a randomized coordinate wise feasible descent method rcfdm framework we show that many machine learning algorithms including the famous sdca algorithm for optimizing the svm dual problem or the stochastic coordinate descent method for the lasso problem fits into the framework of rc fdm we prove linear convergence for both r fdm and rc fdm under the weak strong convexity assumption moreover we show that the duality gap converges linearly for rc fdm which implies that the duality gap also converges linearly for sdca applied to the svm dual problem

a metric approach to vector valued image segmentation we address the issue of low level segmentation of vector valued images focusing on the case of color natural images the proposed approach relies on the formulation of the problem in the metric framework as a voronoi tessellation of the image domain in this context a segmentation is determined by a distance transform and a set of sites our method consists in dividing the segmentation task in two successive sub tasks pre segmentation and hierarchical representation we design specific distances for both sub problems by considering low level image attributes and particularly color and lightness information then the interpretation of the metric formalism in terms of boundaries allows the definition of a soft contour map that has the property of producing a set of closed curves for any threshold finally we evaluate the quality of our results with respect to ground truth segmentation data

problemes de neumann non lineaires sur les varietes riemanniennes nonlinear neumann problems on riemannian manifolds let m g be a c compact riemannian manifold of dimension n 2 whose boundary b is an n 1 dimensional submanifold and let m m b be the interior of m study of neumann problems of the form x 0 in m d dn g y 0 on b where for every t x y r m b t x and g t y are bounded by c 1 t a or c exp t a application to the determination of a conformal metric for which the scalar curvature of m and the mean curvature of b take prescribed values

on the convergence time of dual subgradient methods for strongly convex programs this paper studies the convergence time of dual gradient methods for general possibly nondifferentiable strongly convex programs for general convex programs the convergence time of dual subgradient gradient methods with simple running averages running averages started from iteration 0 is known to be o frac 1 epsilon 2 this paper shows that the convergence time for general strongly convex programs is o frac 1 epsilon this paper also considers a variation of the average scheme called the sliding running averages and shows that if the dual function of the strongly convex program is locally quadratic then the convergence time of the dual gradient method with sliding running averages is o log frac 1 epsilon the convergence time analysis is further verified by numerical experiments

the volume algorithm producing primal solutions with a subgradient method we present an extension to the subgradient algorithm to produce primal as well as dual solutions it can be seen as a fast way to carry out an approximation of dantzig wolfe decomposition this gives a fast method for producing approximations for large scale linear programs it is based on a new theorem in linear programming duality we present successful experience with linear programs coming from set partitioning set covering max cut and plant location

graph based tools for microscopic cellular image segmentation we propose a framework of graph based tools for the segmentation of microscopic cellular images this framework is based on an object oriented analysis of imaging problems in pathology our graph tools rely on a general formulation of discrete functional regularization on weighted graphs of arbitrary topology it leads to a set of useful tools which can be combined together to address various image segmentation problems in pathology to provide fast image segmentation algorithms we also propose an image simplification based on graphs as a pre processing step the abilities of this set of image processing discrete tools are illustrated through automatic and interactive segmentation schemes for color cytological and histological images segmentation problems

rate analysis of inexact dual first order methods application to dual decomposition we propose and analyze two dual methods based on inexact gradient information and averaging that generate approximate primal solutions for smooth convex problems the complicating constraints are moved into the cost using the lagrange multipliers the dual problem is solved by inexact first order methods based on approximate gradients for which we prove sublinear rate of convergence in particular we provide a complete rate analysis and estimates on the primal feasibility violation and primal and dual suboptimality of the generated approximate primal and dual solutions moreover we solve approximately the inner problems with a linearly convergent parallel coordinate descent algorithm our analysis relies on the lipschitz property of the dual function and inexact dual gradients further we combine these methods with dual decomposition and constraint tightening and apply this framework to linear model predictive control obtaining a suboptimal and feasible control scheme

lagrangian relaxation via ballstep subgradient methods we exhibit useful properties of ballstep subgradient methods for convex optimization using level controls for estimating the optimal value augmented with simple averaging schemes they asymptotically find objective and constraint subgradients involved in optimality conditions when applied to lagrangian relaxation of convex programs they find both primal and dual solutions and have practicable stopping criteria up until now similar results have only been known for proximal bundle methods and for subgradient methods with divergent series stepsizes whose convergence can be slow encouraging numerical results are presented for large scale nonlinear multicommodity network flow problems

conformal metrics on the ball with zero scalar curvature and prescribed mean curvature on the boundary abstract this paper considers the prescribed zero scalar curvature and mean curvature problem on the n dimensional euclidean ball for n 3 we consider the limits of solutions of the regularization obtained by decreasing the critical exponent we characterize those subcritical solutions which blow up at the least possible energy level determining the points at which they can concentrate and their morse indices we show that when n 3 this is the only blow up which can occur for solutions we use this in combination with the morse inequalities for the subcritical problem to obtain a general existence theorem for the prescribed zero scalar curvature and mean curvature on the three dimensional euclidean ball in the higher dimensional case n 4 we give conditions on the function h to guarantee there is only one simple blow up point

on linear convergence of a distributed dual gradient algorithm for linearly constrained separable convex problems in this paper we propose a fully distributed dual gradient algorithm for minimizing linearly constrained separable convex problems and analyze its rate of convergence in particular we prove that under the assumption of strong convexity and lipschitz continuity of the gradient of the primal objective function we have a global error bound type property for the dual problem using this error bound property we devise a fully distributed dual gradient scheme i e a gradient scheme based on a weighted step size for which we derive global linear rate of convergence for both dual and primal suboptimality and for primal feasibility violation numerical simulations are also provided to confirm our theory

a quaternion framework for color image smoothing and segmentation in this paper we present feature detail preserving models for color image smoothing and segmentation using the hamiltonian quaternion framework first we introduce a novel quaternionic gabor filter qgf which can combine the color channels and the orientations in the image plane we show that these filters are optimally localized both in the spatial and frequency domains and provide a good approximation to quaternionic quadrature filters using the qgfs we extract the local orientation information in the color images second in order to model this derived orientation information we propose continuous mixtures of appropriate exponential basis functions and derive analytic expressions for these models these analytic expressions take the form of spatially varying kernels which when convolved with a color image or the signed distance function of an evolving contour placed in the color image yield a detail preserving smoothing and segmentation respectively several examples on widely used image databases are shown to depict the performance of our algorithms

a saddle point algorithm for networked online convex optimization an algorithm to learn optimal actions in convex distributed online problems is developed learning is online because cost functions are revealed sequentially and distributed because they are revealed to agents of a network that can exchange information with neighboring nodes only learning is measured in terms of the global network regret which is defined here as the accumulated loss of causal prediction with respect to a centralized clairvoyant agent to which the information of all times and agents is revealed at the initial time a variant of the arrow hurwicz saddle point algorithm is proposed to control the growth of global network regret this algorithm uses lagrange multipliers to penalize the discrepancies between agents and leads to an implementation that relies on local operations and exchange of variables between neighbors we show that decisions made with this saddle point algorithm lead to regret whose order is not larger than o sqrt t where t is the total operating time numerical behavior is illustrated for the particular case of distributed recursive least squares an application to computer network security in which service providers cooperate to detect the signature of malicious users is developed to illustrate the practical value of the proposed algorithm

suboptimal model predictive control feasibility implies stability practical difficulties involved in implementing stabilizing model predictive control laws for nonlinear systems are well known stabilizing formulations of the method normally rely on the assumption that global and exact solutions of nonconvex nonlinear optimization problems are possible in limited computational time in the paper we first establish conditions under which suboptimal model predictive control mpc controllers are stabilizing the conditions are mild holding out the hope that many existing controllers remain stabilizing even if optimality is lost second we present and analyze two suboptimal mpc schemes that are guaranteed to be stabilizing provided an initial feasible solution is available and for which the computational requirements are more reasonable

exponential convergence of distributed primal dual convex optimization algorithm without strong convexity abstract this paper establishes exponential convergence rates for a class of primal dual gradient algorithms in distributed optimization without strong convexity the convergence analysis is based on a carefully constructed lyapunov function by evaluating metric subregularity of the primal dual gradient map we present a general criterion under which the algorithm achieves exponential convergence to facilitate practical applications of this criterion several simplified sufficient conditions are derived we also prove that although these results are developed for the continuous time algorithms they carry over in a parallel manner to the discrete time algorithms constructed by using euler s approximation method

geodesic saliency of watershed contours and hierarchical segmentation the watershed is one of the latest segmentation tools developed in mathematical morphology in order to prevent its oversegmentation the notion of dynamics of a minimum based on geodesic reconstruction has been proposed in this paper we extend the notion of dynamics to the contour arcs this notion acts as a measure of the saliency of the contour contrary to the dynamics of minima our concept reflects the extension and shape of the corresponding object in the image this representation is also much more natural because it is expressed in terms of partitions of the plane i e segmentations a hierarchical segmentation process is then derived which gives a compact description of the image containing all the segmentations one can obtain by the notion of dynamics by means of a simple thresholding finally efficient algorithms for computing the geodesic reconstruction as well as the dynamics of contours are presented

compactness for a class of yamabe type problems on manifolds with boundary abstract in this paper we are interested in studying the set of positive solutions to the following yamabe type problem g u a x u 0 in m u g b x u n 2 u n n 2 on m where m n g is a compact riemannian n manifold n 3 with boundary g is the laplace beltrami operator for g g is the pointing unit outward normal to m a c m and b c m assuming suitable hypotheses on the functions a n 2 4 n 1 r g and b n 2 2 h g where r g denotes the scalar curvature of the metric g and h g is the mean curvature of m we prove some results of compactness for the set of positive solutions to the problem