numerical solution of a nonlinear wave equation in polar coordinates we study a finite element scheme and a difference scheme for the radial solutions of a nonlinear klein gordon equation

near equality near convexity sums of maximally monotone operators and averages of firmly nonexpansive mappings we study nearly equal and nearly convex sets ranges of maximally monotone operators and ranges and fixed points of convex combinations of firmly nonexpansive mappings the main result states that the range of an average of firmly nonexpansive mappings is nearly equal to the average of the ranges of the mappings a striking application of this result yields that the average of asymptotically regular firmly nonexpansive mappings is also asymptotically regular throughout examples are provided to illustrate the theory we also obtain detailed information on the domain and range of the resolvent average

solvability theory for a class of hemivariational inequalities involving copositive plus matrices applications in robotics the study of the equilibrium of an object robotic hand system including nonmonotone adhesive effects and nonclassical friction effects leads to new inequality methods in robotics the aim of this paper is to describe these inequality methods and provide a corresponding suitable mathematical theory

minimum cost spanning tree games we consider the problem of cost allocation among users of a minimum cost spanning tree network it is formulated as a cooperative game in characteristic function form referred to as a minimum cost spanning tree m c s t game we show that the core of a m c s t game is never empty in fact a point in the core can be read directly from any minimum cost spanning tree graph associated with the problem for m c s t games with efficient coalition structures we define and construct m c s t games on the components of the structure we show that the core and the nucleolus of the original game are the cartesian products of the cores and the nucleoli respectively of the induced games on the components of the efficient coalition structure

duality for e variational inequalities via the subdifferential calculus abstract based on the properties of the convex e subdifferential calculus we introduce to a general e variational inequality formulated with the help of a set valued operator and a perturbation function a dual one expressed by making use of the fenchel conjugate of the perturbation function under convexity hypotheses we show that the fulfillment of a regularity condition guarantees that the primal e variational inequality is solvable if and only if its dual one is solvable by particularizing the perturbation function we obtain several dual statements and we succeed to generalize and improve a duality scheme recently given by kum kim and lee an example justifying this generalization is also provided among the special instances of the general result we rediscover also the duality scheme concerning variational inequalities due to mosco

a method for the integration in time of certain partial differential equations abstract a method for the numerical solution of ordinary differential equations is analyzed that is explicit and yet can conserve the quadratic quantities conserved by the equations the method can be a useful alternative to the usual leapfrog technique in that it does not suffer from the occurrence of blowup phenomena numerical examples concerning the korteweg de vries equation and the nonlinear schrodinger equation are given

on reduced convex qp formulations of monotone lcps techniques for transforming convex quadratic programs qps into monotone linear complementarity problems lcps and vice versa are well known we describe a class of lcps for which a reduced qp formulation one that has fewer constraints than the standard qp formulation is available we mention several instances of this class including the known case in which the coefficient matrix in the lcp is symmetric

general over relaxed proximal point algorithm involving a maximal relaxed monotone mappings with applications abstract a general framework for the over relaxed proximal point algorithm based on the notion of a maximal relaxed monotonicity is developed and then the convergence analysis in the context of solving a general class of nonlinear implicit inclusion problems is explored furthermore some auxiliary results of interest involving a maximal relaxed monotone mappings in a hilbert space setting are included

matching games the least core and the nucleolus a matching game is a cooperative game defined by a graph g n e the player set is n and the value of a coalition s n is defined as the size of a maximum matching in the subgraph induced by s we show that the nucleolus of such games can be computed efficiently the result is based on an alternative characterization of the least core which may be of independent interest the general case of weighted matching games remains unsolved

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

dualization of generalized equations of maximal monotone type this paper develops a simple duality framework for generalized equations defined by set valued mappings from a linear space to another the original problem is related to two auxiliary problems of the similar form corresponding to lagrangian and dual problems in the theory of convex programming as in convex programming the alternative formulations can be used to obtain information about a given problem and then used to solve it numerically in particular dualization can be used in deriving existence criteria for a given problem indirectly by considering one of the alternative formulations also a given problem can often be solved more easily by way of a dual method the strongest results of this paper concern monotone mappings in this context the duality framework yields several new criteria for maximal monotonicity of composite mappings these results are useful theoretically as well as in numerical solution of generalized equations the duality framework can also be used in problem decomposition since dualization can lead to reformulations to which operator splitting methods and other special methods can be applied

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

approximation solvability of a class of nonlinear set valued variational inclusions involving a η monotone mappings abstract a new class of nonlinear set valued variational inclusions involving a monotone mappings in a hilbert space setting is introduced and then based on the generalized resolvent operator technique associated with a monotonicity the existence and approximation solvability of solutions using an iterative algorithm is investigated

variational convergence of a new proximal algorithm for nonlinear general a monotone operator equation systems in banach spaces abstract the purpose of this paper is to introduce the notion of general a monotone operators in banach spaces and the new proximal mapping associated with the general a monotone operators by using alber s inequalities and the new proximal mapping technique the variational convergence of a new proximal algorithm for nonlinear general a monotone operator equation systems with relaxed cocoercive operators in banach spaces is introduced and studied our results improve and generalize the corresponding results of recent works

some optimal inapproximability results we prove optimal up to an arbitrary e 0 inapproximability results for max e k sat for k 3 maximizing the number of satisfied linear equations in an over determined system of linear equations modulo a prime p and set splitting as a consequence of these results we get improved lower bounds for the efficient approximability of many optimization problems studied previously in particular for max e2 sat max cut max di cut and vertex cover

variational composition of a monotone operator and a linear mapping with applications to elliptic pdes with singular coefficients abstract this paper proposes a regularized notion of a composition of a monotone operator with a linear mapping this new concept called variational composition can be shown to be maximal monotone in many cases where the usual composition is not the two notions coincide however whenever the latter is maximal monotone the utility of the variational composition is demonstrated by applications to subdifferential calculus theory of measurable multifunctions and elliptic pdes with singular coefficients

convergence rates in forward backward splitting forward backward splitting methods provide a range of approaches to solving large scale optimization problems and variational inequalities in which structures conducive to decomposition can be utilized apart from special cases where the forward step is absent and a version of the proximal point algorithm comes out efforts at evaluating the convergence potential of such methods have so far relied on lipschitz properties and strong monotonicity or inverse strong monotonicity of the mapping involved in the forward step the perspective mainly being that of projection algorithms here convergence is analyzed by a technique that allows properties of the mapping in the backward step to be brought in as well for the first time in such a general setting global and local contraction rates are derived moreover they are derived in a form which makes it possible to determine the optimal step size relative to certain constants associated with the given problem insights are thereby gained into the effects of shifting strong monotonicity between the forward and backward mappings when a splitting is selected

the resolvent average of monotone operators dominant and recessive properties within convex analysis a rich theory with various applications has been evolving since the proximal average of convex functions was first introduced over a decade ago when one considers the subdifferential of the proximal average a natural averaging operation of the subdifferentials of the averaged functions emerges in the present paper we extend the reach of this averaging operation to the framework of monotone operator theory in hilbert spaces transforming it into the resolvent average the theory of resolvent averages contains many desirable properties in particular we study a detailed list of properties of monotone operators and classify them as dominant or recessive with respect to the resolvent average as a consequence we not only recover a significant part of the theory of proximal averages but also obtain results and properties which have not been previously available

on a new system of nonlinear a monotone multivalued variational inclusions in this paper we introduce and study a new system of nonlinear a monotone multivalued variational inclusions in hilbert spaces by using the concept and properties of a monotone mappings and the resolvent operator technique associated with a monotone mappings due to verma we construct a new iterative algorithm for solving this system of nonlinear multivalued variational inclusions associated with a monotone mappings in hilbert spaces we also prove the existence of solutions for the nonlinear multivalued variational inclusions and the convergence of iterative sequences generated by the algorithm our results improve and generalize many known corresponding results

asynchronous block iterative primal dual decomposition methods for monotone inclusions we propose new primal dual decomposition algorithms for solving systems of inclusions involving sums of linearly composed maximally monotone operators the principal innovation in these algorithms is that they are block iterative in the sense that at each iteration only a subset of the monotone operators needs to be processed as opposed to all operators as in established methods flexible strategies are used to select the blocks of operators activated at each iteration in addition we allow lags in operator processing permitting asynchronous implementation the decomposition phase of each iteration of our methods is to generate points in the graphs of the selected monotone operators in order to construct a half space containing the kuhn tucker set associated with the system the coordination phase of each iteration involves a projection onto this half space we present two related methods the first method provides weakly convergent primal and dual sequences under general conditions while the second is a variant in which strong convergence is guaranteed without additional assumptions neither algorithm requires prior knowledge of bounds on the linear operators involved or the inversion of linear operators our algorithmic framework unifies and significantly extends the approaches taken in earlier work on primal dual projective splitting methods

new properties of forward backward splitting and a practical proximal descent algorithm abstract in this paper we discuss a proximal descent algorithm for finding a zero of the sum of two maximal monotone operators in a real hilbert space some new properties of forward backward splitting are given which extend the well known properties of the usual projection then they are used to analyze the weak convergence of the proximal descent algorithm without assuming lipschitz continuity of the forward operator we also give a new technique of choosing trial values of the step length involved in an armijo like condition which returns the not necessarily decreasing step length self adaptively rudimentary numerical experiments show that it is effective in practical implementations

a new relaxed proximal point procedure and applications to nonlinear variational inclusions a new relaxed algorithmic procedure based on the notion of a maximal relaxed monotonicity is developed and then the convergence analysis in the context of solving a general class of nonlinear inclusion problems is examined furthermore some results involving a maximal relaxed monotone mappings in a hilbert space setting are included

new approach to the η proximal point algorithm and nonlinear variational inclusion problems abstract a new approach using the over relaxed proximal point algorithm in the context of solving a class of inclusion problems based on the notion of maximal monotonicity is developed and examined convergence analysis seems to be reasonable and finally some specializations are included furthermore the model developed in this communication seems to be appropriate to the yosida approximation in the sense that it can be applied to first order evolution equations inclusions as well

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

general implicit variational inclusion problems based on a maximal m relaxed monotonicity amrm frameworks a general framework for an algorithmic procedure based on the variational convergence of operator sequences involving a maximal m relaxed monotone amrm mappings in a hilbert space setting is developed and then it is applied to approximating the solution of a general class of nonlinear implicit inclusion problems involving a maximal m relaxed monotone mappings furthermore some specializations of interest on existence theorems and corresponding approximation solvability theorems on h maximal monotone mappings are included that may include several other results for general variational inclusion problems on general maximal monotonicity in the literature

mixed equilibrium problems sensitivity analysis and algorithmic aspect abstract the aim of this paper is twofold first it is to extend the sensitivity analysis framework developed recently for variational inequalities to mixed equilibrium problems the second is to propose iterative methods for solving this kind of problems in the process we establish an equivalence between an extended version of wiener hopf equations and the given problems relying on a generalization of the yosida approximation notion our results generalize results obtained for optimization variational inequalities complementarity problems and problems of nash equilibria

sensitivity analysis for generalized nonlinear parametric a η m maximal monotone operator inclusion systems with relaxed cocoercive type operators abstract by using lim s inequalities nadler s results the new parametric resolvent operator technique associated with a m maximal monotone operators in this paper the existence theorem for a new class of generalized nonlinear parametric a m maximal monotone operator inclusion systems with relaxed cocoercive type operators in hilbert spaces is analyzed and established our results generalize sensitivity analysis results of other recent works on strongly monotone quasi variational inclusions nonlinear implicit quasi variational inclusions and nonlinear mixed quasi variational inclusion systems in hilbert spaces

petrov galerkin methods for nonlinear dispersive waves abstract petrov galerkin methods based on piecewise linear interpolants for the korteweg de vries and related equations are studied both accuracy and stability are analysed for the linearised case it is shown that in the nonlinear case the order of accuracy of the standard galerkin procedure is reduced and an alternative technique is therefore proposed which retains the fourth order accuracy in space this is found to perform well when compared with finite difference or other finite element schemes

convergence of the lambert mcleod trajectory solver and of the celf method a trajectory problem is an initial value problemd y dt f y y 0 where the interest lies in obtaining the curve traced by the solution the trajectory rather than in finding the actual correspondanc between values of the parametert and points on that curve we prove the convergence of the lambert mcleod scheme for the numerical integration of trajectory problems we also study the celf method an explicit procedure for the integration in time of semidiscretizations of pdes which has some useful conservation properties the proofs rely on the concept of restricted stability introduced by stetter in order to show the convergence of the methods an idea of strang is also employed whereby the numerical solution is compared with a suitable perturbation of the theoretical solution rather than with the theoretical solution itself

the primal douglas rachford splitting algorithm for a class of monotone mappings with application to the traffic equilibrium problem we apply the douglas rachford splitting algorithm to a class of multi valued equations consisting of the sum of two monotone mappings compared with the dual application of the same algorithm which is known as the alternating direction method of multipliers the primal application yields algorithms that seem somewhat involved however the resulting algorithms may be applied effectively to problems with certain special structure in particular we show that they can be used to derive decomposition algorithms for solving the variational inequality formulation of the traffic equilibrium problem

a new system of variational inclusions with h η monotone operators in hilbert spaces in this paper we introduce and study a new system of variational inclusions involving h h monotone operators in hilbert space using the resolvent operator associated with h h monotone operators we prove the existence and uniqueness of solutions for this new system of variational inclusions we also construct a new algorithm for approximating the solution of this system and discuss the convergence of the sequence of iterates generated by the algorithm

a new condition for maximal monotonicity via representative functions abstract in this paper we give a weaker sufficient condition for the maximal monotonicity of the operator s a t a where s x x t y y are two maximal monotone operators a x y is a linear continuous mapping and x y are reflexive banach spaces we prove that our condition is weaker than the generalized interior point conditions given so far in the literature this condition is formulated using the representative functions of the operators involved in particular we rediscover some sufficient conditions given in the past using the so called fitzpatrick function for the maximal monotonicity of the sum of two maximal monotone operators and for the precomposition of a maximal monotone operator with a linear operator respectively

a new method for solving a system of generalized nonlinear variational inequalities in banach spaces abstract the purpose of this paper is by using the generalized projection approach to introduce an iterative scheme for finding a solution to a system of generalized nonlinear variational inequality problem under suitable conditions some existence and strong convergence theorems are established in uniformly smooth and strictly convex banach spaces the results presented in the paper improve and extend some recent results

maximal monotonicity for the precomposition with a linear operator we give the weakest constraint qualification known to us that ensures the maximal monotonicity of the operator a circ t circ a when a is a linear continuous mapping between two reflexive banach spaces and t is a maximal monotone operator as a special case we get the weakest constraint qualification that guarantees the maximal monotonicity of the sum of two maximal monotone operators on a reflexive banach space then we give a weak constraint qualification assuring the bre zis haraux type approximation of the range of the subdifferential of the precomposition to a of a proper convex lower semicontinuous function in nonreflexive banach spaces extending and correcting in a special case an older result due to riahi

generalized linear complementarity problems it has been shown by lemke that if a matrix is copositive plus on n then feasibility of the corresponding linear complementarity problem implies solvability in this article we show under suitable conditions that feasibility of ageneralized linear complementarity problem i e defined over a more general closed convex cone in a real hilbert space implies solvability whenever the operator is copositive plus on that cone we show that among all closed convex cones in a finite dimensional real hilbert space polyhedral cones are theonly ones with the property that every copositive plus feasible glcp is solvable we also prove a perturbation result for generalized linear complementarity problems

on the maximality of the sum of two maximal monotone operators abstract a wide variety of problems involving nonlinear partial differential equations subject to boundary conditions may be shown to have solutions by establishing that the associated differential operators satisfy a certain technical condition this condition called maximal monotonicity allows the use of a well developed abstract theory which includes results about existence regularity etc of solutions of operator equations it is quite useful therefore to have easily verified conditions which imply that an operator is maximal monotone frequently an operator may be regarded as the sum of simpler components this paper gives a new sufficient condition which guarantees that the sum of two maximal monotone operators is again maximal monotone author

smooth methods of multipliers for complementarity problems this paper describes several methods for solving nonlinear complementarity problems a general duality framework for pairs of monotone operators is developed and then applied to the monotone complementarity problem obtaining primal dual and primal dual formulations we derive bregman function based generalized proximal algorithms for each of these formulations generating three classes of complementarity algorithms the primal class is well known the dual class is new and constitutes a general collection of methods of multipliers or augmented lagrangian methods for complementarity problems in a special case it corresponds to a class of variational inequality algorithms proposed by gabay by appropriate choice of bregman function the augmented lagrangian subproblem in these methods can be made continuously differentiable the primal dual class of methods is entirely new and combines the best theoretical features of the primal and dual methods some preliminary computation shows that this class of algorithms is effective at solving many of the standard complementarity test problems

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

robust linear and support vector regression the robust huber m estimator a differentiable cost function that is quadratic for small errors and linear otherwise is modeled exactly in the original primal space of the problem by an easily solvable simple convex quadratic program for both linear and nonlinear support vector estimators previous models were significantly more complex or formulated in the dual space and most involved specialized numerical algorithms for solving the robust huber linear estimator numerical test comparisons with these algorithms indicate the computational effectiveness of the new quadratic programming model for both linear and nonlinear support vector problems results are shown on problems with as many as 20000 data points with considerably faster running times on larger problems

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

structure of a simple scheduling polyhedron in a one machine nonpreemptive scheduling problem the feasible schedules may be defined by the vector of the corresponding job completion times for given positive processing times the associated simple scheduling polyhedronp is the convex hull of these feasible completion time vectors the main result of this paper is a complete description of the minimal linear system definingp we also give a complete combinatorial description of the face lattice ofp and a simple o n logn separation algorithm this algorithm has potential usefulness in cutting plane type algorithms for more difficult scheduling problems

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

polymatroid optimization submodularity and joint replenishment games in this paper we consider the problem of maximizing a separable concave function over a polymatroid more specifically we study the submodularity of its optimal objective value in the parameters of the objective function this question is interesting in its own right and is encountered in many applications but our research has been motivated mainly by a cooperative game associated with the well known joint replenishment model by applying our general results on polymatroid optimization we prove that this cooperative game is submodular i e its characteristic cost function is submodular if the joint setup cost is a normalized and nondecreasing submodular function furthermore the same result holds true for a more general one warehouse multiple retailer game which affirmatively answers an open question posed by anily and haviv anily s m haviv 2007 the cost allocation problem for the first order interaction joint replenishment model oper res 55 2 292 302

a characterization of boundary conditions yielding maximal monotone operators abstract we provide a characterization for maximal monotone realizations for a certain class of nonlinear operators in terms of their corresponding boundary data spaces the operators under consideration naturally arise in the study of evolutionary problems in mathematical physics we apply our abstract characterization result to port hamiltonian systems and a class of frictional boundary conditions in the theory of contact problems in visco elasticity

the linear l 1 estimator and the huber m estimator relationships between a linear l1 estimation problem and the huber m estimator problem can be easily established by their dual formulations the least norm solution of a linear programming problem studied by mangasarian and meyer siam j control optim 17 1979 pp 745 752 provides a key link between the dual problems based on the dual formulations we establish a local linearity property of the huber m estimators with respect to the tuning parameter gamma and prove that the solution set of the huber m estimator problem is lipschitz continuous with respect to perturbations of the tuning parameter gamma as a consequence the set of the linear l1 estimators is the limit of the set of the huber m estimators as gamma to 0 thus the huber m estimator problem has many solutions for small tuning parameter gamma if the linear l1 estimation problem has multiple solutions a recursive version of madsen and nielsen s algorithm siam j optim 3 1993 pp 223 235 based on computation of the huber m estimator is proposed for finding a linear l1 estimator

finding the nucleoli of large cooperative games the nucleolus is one of the most important solution concepts in cooperative game theory as a result of its attractive properties it always exists if the imputation is non empty is unique and is always in the core if the core is non empty however computing the nucleolus is very challenging because it involves the lexicographical minimization of an exponentially large number of excess values we present a method for computing the nucleoli of large games including some structured games with more than 50 players using nested linear programs lp although different variations of the nested lp formulation have been documented in the literature they have not been used for large games because of the large size and number of lps involved in addition subtle issues such as how to deal with multiple optimal solutions and with tight constraint sets need to be resolved in each lp in order to formulate and solve the subsequent ones unfortunately this technical issue has been largely overlooked in the literature we treat these issues rigorously and provide a new nested lp formulation that is smaller in terms of the number of large lps and their sizes we provide numerical tests for several games including the general flow games the coalitional skill games and the weighted voting games with up to 100 players

a monotone skew splitting model for composite monotone inclusions in duality the principle underlying this paper is the basic observation that the problem of simultaneously solving a large class of composite monotone inclusions and their duals can be reduced to that of finding a zero of the sum of a maximally monotone operator and a linear skew adjoint operator an algorithmic framework is developed for solving this generic problem in a hilbert space setting new primal dual splitting algorithms are derived from this framework for inclusions involving composite monotone operators and convergence results are established these algorithms draw their simplicity and efficacy from the fact that they operate in a fully decomposed fashion in the sense that the monotone operators and the linear transformations involved are activated separately at each iteration comparisons with existing methods are made and applications to composite variational problems are demonstrated

nearly convex sets fine properties and domains or ranges of subdifferentials of convex functions nearly convex sets play important roles in convex analysis optimization and theory of monotone operators we give a systematic study of nearly convex sets and construct examples of subdifferentials of lower semicontinuous convex functions whose domains or ranges are nonconvex

totally balanced games and games of flow a class of characteristic function games arising from maximum flow problems is introduced and is shown to coincide with the class of totally balanced games the proof relies on the max flow min cut theorem of ford and fulkerson and on the observation that the class of totally balanced games is the span of the additive games with the minimum operation

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

a reduction method for variational inequalities this paper explains a method by which the number of variables in a variational inequality having a certain form can be substantially reduced by changing the set over which the variational inequality is posed the method applies in particular to certain economic equilibrium problems occurring in applications we explain and justify the method and give examples of its application including a numerical example in which the solution time for the reduced problem was approximately 2 of that for the problem in its original form 1998 the mathematical programming society inc published by elsevier science b v

the relevance of convex analysis for the study of monotonicity abstract we introduce new representations for maximal monotone operators we relate them to previous representations given by krauss fitzpatrick martinez legaz and thera we show their usefulness for the study of compositions and sums of maximal monotone operators

sharing supermodular costs we study cooperative games with supermodular costs we show that supermodular costs arise in a variety of situations in particular we show that the problem of minimizing a linear function over a supermodular polyhedron a problem that often arises in combinatorial optimization has supermodular optimal costs in addition we examine the computational complexity of the least core and least core value of supermodular cost cooperative games we show that the problem of computing the least core value of these games is strongly np hard and in fact is inapproximable within a factor strictly less than 17 16 unless p np for a particular class of supermodular cost cooperative games that arises from a scheduling problem we show that the shapley value which in this case is computable in polynomial time is in the least core while computing the least core value is np hard

traveling salesman games in this paper we discuss the problem of how to divide the total cost of a round trip along several institutes among the institutes visited we introduce two types of cooperative games fixed route traveling salesman games and traveling salesman games as a tool to attack this problem under very mild conditions we prove that fixed route traveling salesman games have non empty cores if the fixed route is a solution of the classical traveling salesman problem core elements provide us with fair cost allocations a traveling salesman game may have an empty core even if the cost matrix satisfies the triangle inequality in this paper we introduce a class of matrices defining ts games with non empty cores

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

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

generalized eckstein bertsekas proximal point algorithm based on a maximal monotonicity design a general design for the eckstein bertsekas proximal point algorithm using the notion of the a maximal monotonicity is developed convergence analysis for the generalized eckstein bertsekas proximal point algorithm in the context of solving a class of nonlinear inclusion problems is explored some auxiliary results of interest involving a maximal monotone mappings are also included the obtained results generalize investigations on general maximal monotonicity and beyond

the kernel average for two convex functions and its application to the extension and representation of monotone operators we provide and analyze an average for two convex functions based on a kernel function it covers several known averages such as the arithmetic average epigraphical average and the proximal average when applied to the fitzpatrick function and the conjugate of fitzpatrick function associated with a monotone operator our average produces an autoconjugate also known as selfdual lagrangian which can be used for finding an explicit maximal monotone extension of the given monotone operator this completely settles one of the open problems posed by fitzpatrick in the setting of reflexive banach spaces

mann and ishikawa type perturbed iterative algorithms for generalized nonlinear implicit quasi variational inclusions abstract in this paper we introduce a new class of generalized nonlinear implicit quasi variational inclusions and prove its equivalence with a class of fixed point problems by making use of the properties of maximal monotone we also prove the existence of solutions for this generalized nonlinear implicit quasi variational inclusions and the convergence of iterative sequences generated by the perturbed algorithms

alternating projection proximal methods for convex programming and variational inequalities we consider a mixed problem composed in part of finding a zero of a maximal monotone operator and in part of solving a monotone variational inequality problem we propose a solution method for this problem that alternates between a proximal step for the maximal monotone operator part and a projection type step for the monotone variational inequality part and analyze its convergence and rate of convergence this method extends a decomposition method of chen and teboulle math programming 64 1994 pp 81 101 for convex programming and yields as a by product new decomposition methods

the over relaxed proximal point algorithm based on h maximal monotonicity design and applications first the general framework for a generalized over relaxed proximal point algorithm using the notion of h maximal monotonicity also referred to as h monotonicity is developed and then the convergence analysis for this algorithm in the context of solving a general class of nonlinear inclusion problems is examined along with some auxiliary results on the resolvent operators corresponding to h maximal monotonicity

composition duality and maximal monotonicity this paper shows how the attouch thera duality principle for operator inclusions can be extended to compositions of multifunctions so that the primal and dual inclusions may involve operators between different pairs of spaces we first present the extension and give an example from computational economics to demonstrate that it has practical utility then we consider a particular case often found in applications involving real hilbert spaces and maximal monotone operators we show that in this case our duality framework includes those previously developed by rockafellar mosco and gabay finally we demonstrate that in this case under very simple hypotheses the duality transformation preserves the maximal monotonicity of the operators involved this proof uses an apparently new criterion for maximal monotonicity of operators of the form l tl where t is maximal monotone and l is linear and continuous with adjoint l

the numerical study of a nonlinear one dimensional dirac equation we study the convergence of a difference scheme for a nonlinear one dimensional dirac equation also we show some numerical experiments by using this scheme

convergence analysis of new over relaxed proximal point algorithm frameworks with errors and applications to general a monotone nonlinear inclusion forms abstract the purpose of this paper is to introduce and study a new class of over relaxed proximal point algorithm frameworks with errors based on general a monotonicity further by using alber s inequalities the definition of normalized duality mapping on the dual spaces of banach spaces and the new proximal mapping technique associated with the general a monotone operators we discuss the approximation solvability of general a monotone nonlinear inclusion forms in banach spaces and prove the convergence analysis of iterative sequences generated by the algorithm frameworks via applying the lipschitz continuity of m 1 that is the inverse of multi valued operator m and the lipschitz continuity of proximal mapping associated with the general a monotone operators respectively finally some applications are given to show that the results presented in this paper improve generalize and unify the corresponding results of recent works