metric subregularity and the proximal point method we examine the linear convergence rates of variants of the proximal point method for finding zeros of maximal monotone operators we begin by showing how metric subregularity is sufficient for local linear convergence to a zero of a maximal monotone operator this result is then generalized to obtain convergence rates for the problem of finding a common zero of multiple monotone operators by considering randomized and averaged proximal methods

compositions and averages of two resolvents relative geometry of fixed points sets and a partial answer to a question by c byrne abstract we show that the set of fixed points of the average of two resolvents can be found from the set of fixed points for compositions of two resolvents associated with scaled monotone operators recently the proximal average has attracted considerable attention in convex analysis our results imply that the minimizers of proximal average functions can be found from the set of fixed points for compositions of two proximal mappings associated with scaled convex functions when both convex functions in the proximal average are indicator functions of convex sets least squares solutions can be completely recovered from the limiting cycles given by compositions of two projection mappings this provides a partial answer to a question posed by c byrne a novelty of our approach is to use the notion of resolvent average and proximal average

the asymptotic behavior of the composition of two resolvents abstract the asymptotic behavior of the composition of two resolvents in a hilbert space is investigated connections are made between the solutions of associated monotone inclusion problems and their dual versions the applications provided include a study of an alternating minimization procedure and a new proof of von neumann s classical result on the method of alternating projections

block iterative surrogate projection methods for convex feasibility problems abstract a unified framework is presented for studying the convergence of projection methods for finding a common point of finitely many closed convex sets in r n every iteration approximates each set by a half space given either by an approximate projection of the current iterate or by an aggregate inequality derived from the convex inequalities describing this set the next iterate is found by projecting the current one on a surrogate half space formed by taking a convex combination of the half space inequalities convergence to a solution is established under weak conditions that allow various acceleration techniques and choices of aggregating weights the resulting methods are block iterative and hence lend themselves to parallel implementation we show that the idea of forming cut maps via surrogate inequalities encompasses many classical as well as recently proposed methods for set intersection problems and convex feasibility problems with nondifferentiable inequalities and linear equations and inequalities

linear and strong convergence of algorithms involving averaged nonexpansive operators abstract we introduce regularity notions for averaged nonexpansive operators combined with regularity notions of their fixed point sets we obtain linear and strong convergence results for quasicyclic cyclic and random iterations new convergence results on the borwein tam method btm and on the cyclically anchored douglas rachford algorithm cadra are also presented finally we provide a numerical comparison of btm cadra and the classical method of cyclic projections for solving convex feasibility problems

axisymmetric shape of gas bubbles in a uniform flow this paper investigates axisymmetric shapes of an incompressible inviscid fluid bubble in a uniform flow approximate solutions are constructed for the governing nonlinear boundary value problem using the domain perturbation method bifurcation and turning points are located by applying a special type of hermite pade approximation

state local optimality of a bang bang trajectory a hamiltonian approach we give a sufficient condition for a bang bang extremal to be a strong local optimizer for the minimum time problem with fixed endpoints we underline that the conditions imply that the optimum is local with respect to the state and not necessarily to the final time moreover it is given through a finite dimensional minimization problem hence is suited for numerical verification a geometric interpretation through the projection of the hamiltonian flow on the state space is also given

prox penalization and splitting methods for constrained variational problems this paper is concerned with the study of a class of prox penalization methods for solving variational inequalities of the form ax n c x ni0 where mathcal h is a real hilbert space a mathcal h rightrightarrows mathcal h is a maximal monotone operator and n c is the outward normal cone to a closed convex set c subset mathcal h given psi mathcal h to mathbb r cup infty which acts as a penalization function with respect to the constraint x in c and a penalization parameter beta n we consider a diagonal proximal algorithm of the form x n i lambda n a beta n partial psi 1 x n 1 and an algorithm which alternates proximal steps with respect to a and penalization steps with respect to c and reads as x n i lambda n beta n partial psi 1 i lambda na 1 x n 1 we obtain weak ergodic convergence for a general maximal monotone operator a and weak convergence of the whole sequence x n when a is the subdifferential of a proper lower semicontinuous convex function mixing with passty s idea we can extend the ergodic convergence theorem thus obtaining the convergence of a prox penalization splitting algorithm for constrained variational inequalities governed by the sum of several maximal monotone operators our results are applied to an optimal control problem where the state variable and the control are coupled by an elliptic equation we also establish robustness and stability results that account for numerical approximation errors

on projection algorithms for solving convex feasibility problems due to their extraordinary utility and broad applicability in many areas of classical mathematics and modern physical sciences most notably computerized tomography algorithms for solving convex feasibility problems continue to receive great attention to unify generalize and review some of these algorithms a very broad and flexible framework is investigated several crucial new concepts which allow a systematic discussion of questions on behaviour in general hilbert spaces and on the quality of convergence are brought out numerous examples are given

on the r superlinear convergence of the kkt residuals generated by the augmented lagrangian method for convex composite conic programming due to the possible lack of primal dual type error bounds it was not clear whether the karush kuhn tucker kkt residuals of the sequence generated by the augmented lagrangian method alm for solving convex composite conic programming cccp problems converge superlinearly in this paper we resolve this issue by establishing the r superlinear convergence of the kkt residuals generated by the alm under only a mild quadratic growth condition on the dual of cccp with easy to implement stopping criteria for the augmented lagrangian subproblems this discovery may help to explain the good numerical performance of several recently developed semismooth newton cg based alm solvers for linear and convex quadratic semidefinite programming

iterative selection methods for common fixed point problems many problems encountered in applied mathematics can be recast as the problem of selecting a particular common fixed point of a countable family of quasi nonexpansive operators in a hilbert space we propose two iterative methods to solve such problems our convergence analysis is shown to cover a variety of existing results in this area applications to solving monotone inclusion and equilibrium problems are considered

surrogate projection methods for finding fixed points of firmly nonexpansive mappings we present methods for finding common fixed points of finitely many firmly nonexpansive mappings on a hilbert space at every iteration an approximation to each mapping generates a halfspace containing its set of fixed points the next iterate is found by projecting the current iterate on a surrogate halfspace formed by taking a convex combination of the halfspace inequalities this acceleration technique extends one for convex feasibility problems cfps since projection operators onto closed convex sets are firmly nonexpansive the resulting methods are block iterative and hence lend themselves to parallel implementation we extend to accelerated methods some recent results of bauschke and borwein siam rev 38 1996 pp 367 426 on the convergence of projection methods

convergence rate analysis for averaged fixed point iterations in common fixed point problems in this paper we establish sublinear and linear convergence of fixed point iterations generated by averaged operators in a hilbert space our results are achieved under a bounded holder regularity

faster convergence rates of relaxed peaceman rachford and admm under regularity assumptions in this paper we provide a comprehensive convergence rate analysis of the douglas rachford splitting drs peaceman rachford splitting prs and alternating direction method of multipliers admm algorithms under various regularity assumptions including strong convexity lipschitz differentiability and bounded linear regularity the main consequence of this work is that relaxed prs and admm automatically adapt to the regularity of the problem and achieve convergence rates that improve upon the tight worst case rates that hold in the absence of such regularity all of the results are obtained using simple techniques

asymptotic convergence analysis of the proximal point algorithm the asymptotic convergence of the proximal point algorithm ppa for the solution of equations of type 0 in tz where t is a multivalued maximal monotone operator in a real hilbert space is analyzed when 0 in tz has a nonempty solution set bar z convergence rates are shown to depend on how rapidly t 1 grows away from bar z in a neighbourhood of 0 when this growth is bounded by a power function with exponent s then for a sequence z k generated by the ppa z k bar z converges to zero like o k s 2 linearly superlinearly or in a finite number of steps according to whether s in 0 1 s 1 s in 1 infty or s infty

strong conical hull intersection property bounded linear regularity jameson s property g and error bounds in convex optimization the strong conical hull intersection property and bounded linear regularity are properties of a collection of finitely many closed convex intersecting sets in euclidean space these fundamental notions occur in various branches of convex optimization constrained approximation convex feasibility problems linear inequalities for instance it is shown that the standard constraint qualification from convex analysis implies bounded linear regularity which in turn yields the strong conical hull intersection property jameson s duality for two cones which relates bounded linear regularity to property g is re derived and refined for polyhedral cones a statement dual to hoffman s error bound result is obtained a sharpening of a result on error bounds for convex inequalities by auslender and crouzeix is presented finally for two subspaces property g is quantified by the angle between the subspaces

new douglas rachford algorithmic structures and their convergence analyses in this paper we study new algorithmic structures with douglas rachford dr operators to solve convex feasibility problems we propose to embed the basic two set dr algorithmic operator into the string averaging projections and into the block iterative projection algorithmic structures thereby creating new dr algorithmic schemes that include the recently proposed cyclic dr algorithm and the averaged dr algorithm as special cases we further propose and investigate a new multiple set dr algorithmic operator convergence of all these algorithmic schemes is studied by using properties of strongly quasi nonexpansive operators and firmly nonexpansive operators

convergence rates with inexact non expansive operators in this paper we present a convergence rate analysis for the inexact krasnosel ski mann iteration built from non expansive operators the presented results include two main parts we first establish the global pointwise and ergodic iteration complexity bounds then under a metric sub regularity assumption we establish a local linear convergence for the distance of the iterates to the set of fixed points the obtained results can be applied to analyze the convergence rate of various monotone operator splitting methods in the literature including the forward backward splitting the generalized forward backward the douglas rachford splitting alternating direction method of multipliers and primal dual splitting methods for these methods we also develop easily verifiable termination criteria for finding an approximate solution which can be seen as a generalization of the termination criterion for the classical gradient descent method we finally develop a parallel analysis for the non stationary krasnosel ski mann iteration

a norm convergence result on random products of relaxed projections in hilbert space suppose x is a hilbert space and c 1 ldots c n are closed convex intersecting subsets with projections p 1 ldots p n suppose further r is a mapping from bbb n onto 1 ldots n that assumes every value infinitely often we prove a more general version of the following result begin quote if the n tuple c 1 ldots c n is innately boundedly regular then the sequence x n defined by x 0 in x mbox arbitrary x n 1 p r n x n mbox for all n geq 0 converges in norm to some point in bigcap i 1 n c i end quote examples without the usual assumptions on compactness are given methods of this type have been used in areas like computerized tomography and signal processing

a new use of douglas rachford splitting for identifying infeasible unbounded and pathological conic programs in this paper we present a method for identifying infeasible unbounded and pathological conic programs based on douglas rachford splitting when an optimization program is infeasible unbounded or pathological the iterates of douglas rachford splitting diverge somewhat surprisingly such divergent iterates still provide useful information which our method uses for identification in addition for strongly infeasible problems the method produces a separating hyperplane and informs the user on how to minimally modify the given problem to achieve strong feasibility as a first order method the proposed algorithm relies on simple subroutines and therefore is simple to implement and has low per iteration cost

on the behavior of subgradient projections methods for convex feasibility problems in euclidean spaces we study some methods of subgradient projections for solving a convex feasibility problem with general not necessarily hyperplanes or half spaces convex sets in the inconsistent case and propose a strategy that controls the relaxation parameters in a specific self adapting manner this strategy leaves enough user flexibility but gives a mathematical guarantee for the algorithm s behavior in the inconsistent case we present the numerical results of computational experiments that illustrate the computational advantage of the new method

warped proximal iterations for monotone inclusions abstract resolvents of set valued operators play a central role in various branches of mathematics and in particular in the design and the analysis of splitting algorithms for solving monotone inclusions we propose a generalization of this notion called warped resolvent which is constructed with the help of an auxiliary operator the properties of warped resolvents are investigated and connections are made with existing notions abstract weak and strong convergence principles based on warped resolvents are proposed and shown to not only provide a synthetic view of splitting algorithms but to also constitute an effective device to produce new solution methods for challenging inclusion problems

iterating bregman retractions the notion of a bregman retraction of a closed convex set in euclidean space is introduced bregman retractions include backward bregman projections and forward bregman projections as well as their convex combinations and are thus quite flexible the main result on iterating bregman retractions unifies several convergence results on projection methods for solving convex feasibility problems it is also used to construct new sequential and parallel algorithms

an invariant second variation in optimal control for an optimal control problem we define a second variation which is invariant under change of coordinates and realize it as a linear quadratic problem when the strong legendre condition is satisfied we give a complete hamiltonian characterization of the index and of the nullity of the second variation

approximants de pade hermite 2eme partie programmation in this second paper we propose an algorithm for the construction of the tables and of the pade hermite tables 3 then we present some numerical applications

strong optimality for a bang bang trajectory in this paper we give sufficient conditions for a bang bang regular extremal to be a strong local optimum for a control problem in the mayer form strong means that we consider the c0 topology in the state space the controls appear linearly and take values in a polyhedron and the state space and the end point constraints are finite dimensional smooth manifolds in the case of bang bang extremals the kernel of the first variation of the problem is trivial and hence the usual second variation which is defined on the kernel of the first one does not give any information we consider the finite dimensional subproblem generated by perturbing the switching times and we prove that the sufficient second order optimality conditions for this finite dimensional subproblem yield local strong optimality we give an explicit algorithm to check the positivity of the second variation which is based on the properties of the hamiltonian fields

on the range of the douglas rachford operator the problem of finding a minimizer of the sum of two convex functions or more generally that of finding a zero of the sum of two maximally monotone operators is of central importance in variational analysis perhaps the most popular method of solving this problem is the douglas rachford splitting method surprisingly little is known about the range of the douglas rachford operator in this paper we set out to study this range systematically we prove that for 3 monotone operators a very pleasing formula can be found that reveals the range to be nearly equal to a simple set involving the domains and ranges of the underlying operators a similar formula holds for the range of the corresponding displacement mapping we discuss applications to subdifferential operators to the infimal displacement vector and to firmly nonexpansive mappings various examples and counterexamples are presented including some concerning the celebrated brezis haraux theorem

applications of hermite pade approximants to water waves and the harmonic oscillator on a lattice three types of hermite pade approximants are considered known respectively as quadratic integral and differential pade approximants the singularity structure of each type of approximant is described it is more complicated than that of the standard pade approximant and this property may often be used to estimate the types of singularity of a function from its power series expansion as well as evaluate it on its branch cuts two applications in different fields are described which illustrate these properties of the above types of hermite pade approximants the first concerns the characteristic values of mathieu s equation which are related to the energy eigenvalues of the harmonic oscillator on a lattice the second concerns the investigation of the singularity structure and values of various physical quantities associated with periodic and solitary water waves

distributed algorithms for computing a common fixed point of a group of nonexpansive operators this paper addresses the problem of seeking a common fixed point for a collection of nonexpansive operators over time varying multi agent networks in real hilbert spaces each operator is only privately and approximately known to each individual agent and all agents need to cooperate to solve this problem by local communications over time varying networks to handle this problem inspired by the centralized inexact krasnosel skii mann ikm iteration we propose a distributed algorithm called distributed inexact averaged operator algorithm d iao it is shown that the d iao algorithm can converge weakly to a common fixed point of the family of nonexpansive operators moreover under the linearly regular assumption it is proved that the d iao algorithm converges with a rate o gamma log 16 4k for some constant gamma in 0 1 where k is the iteration number the same converge speed can be ensured by a novel power regularity condition which is strictly weaker than linear regularity to reduce computational complexity another algorithm called d biao is developed and fully analyzed simulation examples are given to support the theoretical res

bregman monotone optimization algorithms a broad class of optimization algorithms based on bregman distances in banach spaces is unified around the notion of bregman monotonicity a systematic investigation of this notion leads to a simplified analysis of numerous algorithms and to the development of a new class of parallel block iterative surrogate bregman projection schemes another key contribution is the introduction of a class of operators that is shown to be intrinsically tied to the notion of bregman monotonicity and to include the operators commonly found in bregman optimization methods special emphasis is placed on the viability of the algorithms and the importance of legendre functions in this regard various applications are discussed

generalized solutions for the sum of two maximally monotone operators a common theme in mathematics is to define generalized solutions to deal with problems that potentially do not have solutions a classical example is the introduction of least squares solutions via the normal equations associated with a possibly infeasible system of linear equations in this paper we introduce a normal problem associated with finding a zero of the sum of two maximally monotone operators if the original problem admits solutions then the normal problem returns this same set of solutions the normal problem may yield solutions when the original problem does not admit any furthermore it has attractive variational and duality properties several examples illustrate our theory

on the douglas rachford algorithm the douglas rachford algorithm is a very popular splitting technique for finding a zero of the sum of two maximally monotone operators the behaviour of the algorithm remains mysterious in the general inconsistent case i e when the sum problem has no zeros however more than a decade ago it was shown that in the possibly inconsistent convex feasibility setting the shadow sequence remains bounded and its weak cluster points solve a best approximation problem in this paper we advance the understanding of the inconsistent case significantly by providing a complete proof of the full weak convergence in the convex feasibility setting in fact a more general sufficient condition for the weak convergence in the general case is presented our proof relies on a new convergence principle for fejer monotone sequences numerous examples illustrate our results

coincidence points by generalized mann iterates with applications in hilbert spaces this paper deals with the construction of the coincidence points by the generalized mann iterates and their applications in hilbert spaces our results generalize and improve a number of recent results

decomposition through formalization in a product space when an optimization problem is posed in a product space it is classical to decompose this problem the goal of this paper is to show how such an approach can be used when the problem to be solved is not naturally posed in a product space by associating systematically to this problem an equivalent one posed in ann fold cartesian product space we obtain by decomposition of the latter both a splitting of operators and a desintegration of constraints for the former applications to three rather classical mathematical programming problems are given

a weak to strong convergence principle for feje monotone methods in hilbert spaces we consider a wide class of iterative methods arising in numerical mathematics and optimization that are known to converge only weakly exploiting an idea originally proposed by haugazeau we present a simple modification of these methods that makes them strongly convergent without additional assumptions several applications are discussed

generalized mann iterates for constructing fixed points in hilbert spaces the mean iteration scheme originally proposed by mann is extended to a broad class of relaxed inexact fixed point algorithms in hilbert spaces weak and strong convergence results are established under general conditions on the underlying averaging process and the type of operators involved this analysis significantly widens the range of applications of mean iteration methods several examples are given