an affine scaling interior trust region method for lc1 minimization subject to bounds on variables in this literature we extend the classical affine scaling interior trust region algorithm for smooth bounded constrained nonlinear programming to the nonsmooth case where the objective function is only locally lipschitzian we propose and analyze a new affine scaling trust region method in association with nonmonotonic interior backtracking line search technique for solving the linear equality constrained lc 1 optimization subject to bounds on variables at each iteration the trust region subproblem is defined by minimizing a quadratic function which is required both first and second order information of the objective function and a quadratic affine scaling matrix model subject only to an ellipsoidal constraint in a null space of the affine scaling linear equality constraints the second order derivative of the objective function is explicitly required to be lipschitzian each iterate switches to a very general setting of computing trial backtracking steps generated by the general trust region subproblem and then the line search technique projects the trial backtracking steps onto the strictly feasible set of the bounded constraints on variables the global convergence and fast local convergence rate of the proposed algorithm are established under some reasonable conditions where twice smoothness of the objective function is not required a nonmonotonic criterion should bring about speeding up the convergence progress in some ill conditioned cases applications of the algorithm to various nonsmooth optimization problems are discussed

estimates and bounds of dynamic permeability of granular media this paper presents a study of dynamic permeability of porous media combining homogenization of periodic media hpm and the self consistent method scm by taking advantage of the physical principles identified with hpm the application of scm leads to the determination of two physically admissible dynamic permeability assessments both different from that given by the cell model a comparison with numerical modeling demonstrates the fairly good reliability of the three estimates for granular media consisting of a periodic array of spherical grains furthermore the self consistent values enable exact bounds for the dynamic permeability of a wide class of porous media to be derived with a clear identification of their microstructure grain and fluid size distribution

on structure and computation of generalized nash equilibria we consider generalized nash equilibrium problems gnep from a structural and computational point of view in gnep the players feasible sets may depend on the other players strategies moreover the players may share common constraints in particular the latter leads to the stable appearance of nash equilibria which are fritz john fj points but not karush kuhn tucker kkt points basic in our approach is the representation of fj points as zeros of an appropriate underdetermined system of nonsmooth equations here additional nonsmooth variables are taken into account we prove that the set of fj points together with corresponding active lagrange multipliers generically constitutes a lipschitz manifold its dimension is n 1 j 0 where n is the number of players and j 0 is the number of active common constraints in a structural analysis of nash equilibria the number n 1 j 0 plays a crucial role in fact the latter number encodes both the possible degeneracies for the players parametric subproblems and the dimension of the set of nash equilibria in particular in the nondegenerate case the dimension of the set of nash equilibria equals locally n 1 j 0 for the computation of fj points we propose a nonsmooth projection method npm which aims at nding solutions of an underdetermined system of nonsmooth equations npm is shown to be well dened for gnep local convergence of npm is conjectured for gnep under generic assumptions and its proof is challenging however we indicate special cases known from the literature in which convergence holds

on h p adaptive bem for frictional contact problems in linear elasticity a mixed formulation for a tresca frictional contact problem in linear elasticity is considered in the context of boundary integral equations which is later extended to coulomb friction the discrete lagrange multiplier an approximation of the surface traction on the contact boundary part is a linear combination of biorthogonal basis functions in case of curved isoparametric elements these are the solutions of local problems in particular the biorthogonality allows to rewrite the variational inequality constraints as a simple set of complementarity problems thus enabling an efficient application of a semi smooth newton solver for the discrete mixed problem converging locally super linearly in the frictional case and quadratically in the frictionless case typically the solution of frictional contact problems is of reduced regularity at the interface between contact to non contact and from stick to slip to identify the a priori unknown locations of these interfaces two a posteriori error estimations are introduced in a first step the error is split into specific error contributions resulting from the contact and friction conditions and from the discretization error of a variational equation for the latter a residual and a bubble error estimation are considered explicitly the numerical experiments show the applicability of the derived error estimations and the superiority of h p adaptivity compared to low order uniform and adaptive approaches

a preconditioning proximal newton method for nondifferentiable convex optimization we propose a proximal newton method for solving nondifferentiable convex optimization this method combines the generalized newton method with rockafellar s proximal point algorithm at each step the proximal point is found approximately and the regularization matrix is preconditioned to overcome inexactness of this approximation we show that such a preconditioning is possible within some accuracy and the second order differentiability properties of the moreau yosida regularization are invariant with respect to this preconditioning based upon these superlinear convergence is established under a semismoothness condition

practical aspects of the moreau yosida regularization theoretical preliminaries when computing the infimal convolution of a convex function f with the squared norm the so called moreau yosida regularization of f is obtained among other things this function has a lipschitzian gradient we investigate some more of its properties relevant for optimization the most important part of our study concerns second order differentiability existence of a second order development of f implies that its regularization has a hessian for the converse we disclose the importance of the decomposition of bbb r n along cal u the subspace where f is smooth and cal v the subspace parallel to the subdifferential of f

identifying structure of nonsmooth convex functions by the bundle technique we consider the problem of minimizing nonsmooth convex functions defined piecewise by a finite number of functions each of which is either convex quadratic or twice continuously differentiable with positive definite hessian on the set of interest this is a particular case of functions with primal dual gradient structure a notion closely related to the so called cal vu space decomposition at a given point nonsmoothness is locally restricted to the directions of the subspace cal v while along the subspace cal u the behavior of the function is twice differentiable constructive identification of the two subspaces is important because it opens the way to devising fast algorithms for nonsmooth optimization by following iteratively the manifold of smoothness on which superlinear cal u newton steps can be computed in this work we show that for the class of functions in consideration the information needed for this identification can be obtained from the output of a standard bundle method for computing proximal points provided a minimizer satisfies the nondegeneracy and strong transversality conditions

inverse semidefinite quadratic programming problem with l1 norm measure abstract we consider an inverse problem arising from a semidefinite quadratic programming sdqp problem which is a minimization problem involving l 1 vector norm with positive semidefinite cone constraint by using convex optimization theory the first order optimality condition of the problem can be formulated as a semismooth equation under two assumptions we prove that any element of the generalized jacobian of the equation at its solution is nonsingular based on this a smoothing approximation operator is given and a smoothing newton method is proposed for solving the solution of the semismooth equation we need to compute the directional derivative of the smoothing operator at the corresponding point and to solve one linear system per iteration in the newton method and its global convergence is demonstrated finally we give the numerical results to show the effectiveness and stability of the smoothing newton method for this inverse problem

differentiable exact penalty functions for nonlinear second order cone programs we propose a method for solving nonlinear second order cone programs socps based on a continuously differentiable exact penalty function the construction of the penalty function is given by incorporating a multipliers estimate in the augmented lagrangian for socps under the nondegeneracy assumption and the strong second order sufficient condition we show that a generalized newton method has global and superlinear convergence we also present some preliminary numerical experiments

finite deformation frictional mortar contact using a semi smooth newton method with consistent linearization a two dimensional finite deformation frictional contact formulation with coulomb s law is presented the approach considers multibody contact and is based on a mortar formulation the enforcement of contact constraints is realized with dual lagrange multipliers these alternative multiplier spaces are constructed in a way that the multipliers can easily be eliminated from the global system of equations by static condensation such that the system size does not increase friction kinematic variables are formulated in an objective way and enter non smooth complementarity functions for expressing the contact constraints an active set strategy is derived by applying a semismooth newton method which treats contact nonlinearities material and geometrical nonlinearities in one single iterative scheme by further carrying out a consistent linearization for both normal and frictional contact forces and constraints a robust and highly efficient algorithm for linear and higher order quadratic interpolation is achieved efficiency of the proposed method and quality of results are demonstrated in serveral exemples

stochastic theory of dynamic permeability in poroelastic media a theory for the dynamic permeability in deformable porous media is developed the analysis is based on the momentum flux transfer from the slow compressional into the slow shear wave a proxy for the viscous wave in a newtonian fluid in the presence of random pore scale heterogeneities a first order statistical smoothing approximation is used to infer a dynamic permeability in the form of an integral over the covariance function modulated by the slow shear wave in a smooth pore throat limit the results reproduce the model proposed by johnson et al j fluid mech 176 379 1987

a algorithm for convex minimization for convex minimization we introduce an algorithm based on space decomposition the method uses a bundle subroutine to generate a sequence of approximate proximal points when a primal dual track leading to a solution and zero subgradient pair exists these points approximate the primal track points and give the algorithm s or corrector steps the subroutine also approximates dual track points that are gradients needed for the method s newton predictor steps with the inclusion of a simple line search the resulting algorithm is proved to be globally convergent the convergence is superlinear if the primal dual track points and the objective s hessian are approximated well enough

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

a generalized newton method for absolute value equations associated with second order cones in this paper we introduce the absolute value equations associated with second order cones socave in short which is a generalization of the absolute value equations discussed recently in the literature it is proved that the socave is equivalent to a class of second order cone linear complementarity problems soclcp in short in particular we propose a generalized newton method for solving the socave and show that the proposed method is globally linearly and locally quadratically convergent under suitable assumptions we also report some preliminary numerical results of the proposed method for solving the socave and the soclcp which show the efficiency of the proposed method

solution dependence on initial conditions in differential variational inequalities in the first part of this paper we establish several sensitivity results of the solution x t to the ordinary differential equation ode initial value problem ivp dx dt f x x 0 as a function of the initial value for a nondifferentiable f x specifically we show that for xi t equiv x t xi 0 0 leq t leq t a if f is b differentiable on xi t then so is the solution operator x t at 0 b if f is semismooth on xi t then so is x t at 0 c if f has a linear newton approximation on xi t then so does x t at 0 moreover the linear newton approximation of the solution operator can be obtained from the solution of a linear differential inclusion in the second part of the paper we apply these ode sensitivity results to a differential variational inequality dvi and discuss a the existence uniqueness and lipschitz dependence of solutions to subclasses of the dvi subject to boundary conditions via an implicit function theorem for semismooth equations and b the convergence of a nonsmooth shooting method for numerically computing such boundary value solutions

on the computation of an element of clarke generalized jacobian for a vector valued max function abstract in this paper we present an algorithm for calculating an element of clarke generalized jacobian for a vector valued max type function the algorithm reduces the computational cost of an existing algorithm

a semismooth newton cg based dual ppa for matrix spectral norm approximation problems we consider a class of matrix spectral norm approximation problems for finding an affine combination of given matrices having the minimal spectral norm subject to some prescribed linear equality and inequality constraints these problems arise often in numerical algebra engineering and other areas such as finding chebyshev polynomials of matrices and fastest mixing markov chain models based on classical analysis of proximal point algorithms ppas and recent developments on semismooth analysis of nonseparable spectral operators we propose a semismooth newton cg based dual ppa for solving the matrix norm approximation problems furthermore when the primal constraint nondegeneracy condition holds for the subproblems our semismooth newton cg method is proven to have at least a superlinear convergence rate we also design efficient implementations for our proposed algorithm to solve a variety of instances and compare its performance with the nowadays popular first order alternating direction method of multipliers admm the results show that our algorithm which is warm started with an initial point obtained from the admm substantially outperforms the pure admm especially for the constrained cases and it is able to solve the problems robustly and efficiently to a relatively high accuracy

semismoothness and decomposition of maximal normal operators abstract a function is semismooth if and only if its subdifferential is semicontinuous all the semicontinuous operators form a big linear subspace of the maximal normal operator space by big we mean that this subspace includes all the continuous operators and all the maximal monotone operators all the decomposable maximal normal operators form another big subspace

finite termination of a newton type algorithm for a class of affine variational inequality problems abstract many optimization problems can be reformulated as a system of equations one may use the generalized newton method or the smoothing newton method to solve the reformulated equations so that a solution of the original problem can be found such methods have been powerful tools to solve many optimization problems in the literature in this paper we propose a newton type algorithm for solving a class of monotone affine variational inequality problems avips for short in the proposed algorithm the techniques based on both the generalized newton method and the smoothing newton method are used in particular we show that the algorithm can find an exact solution of the avip in a finite number of iterations under an assumption that the solution set of the avip is nonempty preliminary numerical results are reported

second order necessary conditions in constrained semismooth optimization first and second order conditions are given which must necessarily be satisfied by local minimizers for certain finite dimensional nonsmooth nonlinear programming problems the problems considered are of standard form having a finite number of equality and inequality constraints the principal result does not require a constraint qualification but does require that the functions be semismooth at the minimizer the necessary conditions are stated in terms of the generalized gradients of nonsmooth analysis and certain second order directional derivatives

a globally and superlinearly convergent algorithm for nonsmooth convex minimization it is well known that a possibly nondifferentiable convex minimization problem can be transformed into a differentiable convex minimization problem by way of the moreau yosida regularization this paper presents a globally convergent algorithm that is designed to solve the latter problem under additional semismoothness and regularity assumptions the proposed algorithm is shown to have a q superlinear rate of convergence

quasi newton trust region algorithm for non smooth least squares problems1this work was supported by cnpq of brazil and the national natural science foundation of china 1 in this paper we present a quasi newton trust region algorithm for non smooth least squares problems which possess the smooth plus non smooth decomposition this method uses a smooth subproblem with second order information to approximate the locally lipschitzian function which is important to increase the convergence rate of the algorithm and deal with the large residual non smooth least squares problems the global and superlinear convergence of this algorithm are established

an augmented primal dual method for linear conic programs we propose a new iterative approach for solving linear programs over convex cones assuming that slater s condition is satisfied the conic problem is transformed to the minimization of a convex differentiable function in the primal dual space this function shows similarities with the augmented lagrangian function and is called augmented primal dual function or apd function the evaluation of the function and its derivative is cheap if the projection of a given point onto the cone can be computed cheaply and if the projection of a given point onto the affine subspace defining the primal problem can be computed cheaply for the special case of a semidefinite program a certain regularization of the apd function is analyzed numerical examples minimizing the apd function with a conjugate gradient method illustrate the potential of the approach

superlinearly convergent approximate newton methods for lc 1 optimization problems in the literature the proof of superlinear convergence of approximate newton or sqp methods for solving nonlinear programming problems requires twice smoothness of the objective and constraint functions sometimes the second order derivatives of those functions are required to be lipschitzian in this paper we present approximate newton or sqp methods for solving nonlinear programming problems whose objective and constraint functions have locally lipschitzian derivatives and establishq superlinear convergence of these methods under the assumption that these derivatives are semismooth this assumption is weaker than the second order differentiability the extended linear quadratic programming problem in the fully quadratic case is an example of nonlinear programming problems whose objective functions have semismooth but not smooth derivatives

a squared smoothing newton method for nonsmooth matrix equations and its applications in semidefinite optimization problems we study a smoothing newton method for solving a nonsmooth matrix equation that includes semidefinite programming and the semidefinite complementarity problem as special cases this method if specialized for solving semidefinite programs needs to solve only one linear system per iteration and achieves quadratic convergence under strict complementarity and nondegeneracy we also establish quadratic convergence of this method applied to the semidefinite complementarity problem under the assumption that the jacobian of the problem is positive definite on the affine hull of the critical cone at the solution these results are based on the strong semismoothness and complete characterization of the b subdifferential of a corresponding squared smoothing matrix function which are of general theoretical interest

convergence analysis of some algorithms for solving nonsmooth equations this paper presents convergence analysis of some algorithms for solving systems of nonlinear equations defined by locally lipschitzian functions for the directional derivative based and the generalized jacobian based newton methods both the iterates and the corresponding function values are locally superlinearly convergent globally a limiting point of the iterate sequence generated by the damped directional derivative based newton method is a zero of the system if and only if the iterate sequence converges to this point and the stepsize eventually becomes one provided that the system is strongly bd regular and semismooth at this point in this case the convergence is superlinear a general attraction theorem is presented which can be applied to two algorithms proposed by han pang and rangaraj a hybrid method which is both globally convergent in the sense of finding a stationary point of the norm function of the system and locally quadratically convergent is also presented

amenability poincare series and quasiconformal maps any coveringy x of a hyperbolic riemann surfacex of finite area determines an inclusion of teichmuller spaces teich x teich y we show this map is an isometry for the teichmuller metric if the covering isamenable and contracting otherwise in particular we establish 1 for classical poincare series kra s theta conjecture

algebraic univalence theorems for nonsmooth functions abstract a well known univalence result due to d gale and h nikaido 1965 math ann 159 81 93 asserts that if the jacobian matrix of a differentiable function from a closed rectangle k in r n into r n is a p matrix at each point of k then f is one to one on k in this paper by introducing the concepts of h differentiability and h differential of a function as a set of matrices we generalize the gale nikaido result to nonsmooth functions our results further extend those of other authors valid for compact rectangles we show that our results are applicable when the h differential is any one of the following the jacobian matrix of a differentiable function the generalized jacobian of a locally lipschitzian function the bouligand subdifferential of a semismooth function and the c differential of l qi 1993 math oper res 18 227 244

weak directional closedness and generalized subdifferentials abstract spingarn introduced the notion of a submonotone operator and showed that the clarke subdifferential is submonotone if and only if it is semismooth in the sense of mifflin and regular in the sense of clarke in this article a property of operators referred to as weak directional closedness wdc is introduced the wdc property is used to extend spingarn s result to a broad class of generalized subdifferentials for locally lipschitz functions two members of this class of subdifferentials are the clarke subdifferential which is always wdc and the michel penot subdifferential which may or may not be wdc a subdifferential that is wdc and is contained in the clarke subdifferential is constructed it is shown that this subdifferential coincides with the michel penot subdifferential whenever the michel penot subdifferential is wdc and submonotone

newton s method for quadratic stochastic programs with recourse abstract quadratic stochastic programs qsp with recourse can be formulated as nonlinear convex programming problems by attaching a lagrange multiplier vector to the nonlinear convex program a qsp is written as a system of nonsmooth equations a newton like method for solving the qsp is proposed and global convergence and local super linear convergence of the method are established the current method is more general than previous methods which were developed for box diagonal and fully quadratic qsp numerical experiments are given to demonstrate the efficiency of the algorithm and to compare the use of monte carlo rules and lattice rules for multiple integration in the algorithm

poincare series and holomorphic averaging we provide an alternate proof of mcmullen s theorem on contractive properties of the poincare series operator in the special case of the universal covering this case includes in particular kra s theta conjecture

affine scaling interior levenberg marquardt method for bound constrained semismooth equations under local error bound conditions we develop and analyze a new affine scaling levenberg marquardt method with nonmonotonic interior backtracking line search technique for solving bound constrained semismooth equations under local error bound conditions the affine scaling levenberg marquardt equation is based on a minimization of the squared euclidean norm of linear model adding a quadratic affine scaling matrix to find a solution that belongs to the bounded constraints on variable the global convergence results are developed in a very general setting of computing trial directions by a semismooth levenberg marquardt method where a backtracking line search technique projects trial steps onto the feasible interior set we establish that close to the solution set the affine scaling interior levenberg marquardt algorithm is shown to converge locally q superlinearly depending on the quality of the semismooth and levenberg marquardt parameter under an error bound assumption that is much weaker than the standard nonsingularity condition that is bd regular condition under nonsmooth case a nonmonotonic criterion should bring about speed up the convergence progress in the contours of objective function with large curvature

a quasi second order proximal bundle algorithm this paper introduces an algorithm for convex minimization which includes quasi newton updates within a proximal point algorithm that depends on a preconditioned bundle subalgorithm the method uses the hessian of a certain outer function which depends on the jacobian of a proximal point mapping which in turn depends on the preconditioner matrix and on a lagrangian hessian relative to a certain tangent space convergence is proved under boundedness assumptions on the preconditioner sequence

dynamics on teichmuller spaces and self covering of riemann surfaces a non injective holomorphic self cover of a riemann surface induces a non surjective holomorphic self embedding of its teichmuller space we investigate the dynamics of such self embeddings by applying our structure theorem of self covering of riemann surfaces and examine the distribution of its isometric vectors on the tangent bundle over the teichmuller space we also extend our observation to quasiregular self covers of riemann surfaces and give an answer to a certain problem on quasiconformal equivalence to a holomorphic self cover

on the convergence of modified newton methods for solving equations containing a non differentiable term we provide a semilocal convergence analysis for certain modified newton methods for solving equations containing a non differentiable term the sufficient convergence conditions of the corresponding newton methods are often taken as the sufficient conditions for the modified newton methods that is why the latter methods are not usually treated separately from the former however here we show that weaker conditions as well as a finer error analysis than before can be obtained for the convergence of modified newton methods numerical examples are also provided

a sequential homotopy method for mathematical programming problems we propose a sequential homotopy method for the solution of mathematical programming problems formulated in abstract hilbert spaces under the guignard constraint qualification the method is equivalent to performing projected backward euler timestepping on a projected gradient antigradient flow of the augmented lagrangian the projected backward euler equations can be interpreted as the necessary optimality conditions of a primal dual proximal regularization of the original problem the regularized problems are always feasible satisfy a strong constraint qualification guaranteeing uniqueness of lagrange multipliers yield unique primal solutions provided that the stepsize is sufficiently small and can be solved by a continuation in the stepsize we show that equilibria of the projected gradient antigradient flow and critical points of the optimization problem are identical provide sufficient conditions for the existence of global flow solutions and show that critical points with emanating descent curves cannot be asymptotically stable equilibria of the projected gradient antigradient flow practically eradicating convergence to saddle points and maxima the sequential homotopy method can be used to globalize any locally convergent optimization method that can be used in a homotopy framework we demonstrate its efficiency for a class of highly nonlinear and badly conditioned control constrained elliptic optimal control problems with a semismooth newton approach for the regularized subproblems

a bundle modification strategy for convex minimization we present a new bundle algorithm for minimizing convex not necessarily smooth functions the novelty of our approach is based on a bundle modification strategy that we apply whenever the stability center is updated and which is aimed at substituting the points of the bundle by new points characterized by possibly better values of the objective function convergence of the algorithm is proved and numerical results are presented

periodic homogenization and consistent estimates of transport parameters through sphere and polyhedron packings in the whole porosity range this paper presents a study of transport parameters diffusion dynamic permeability thermal permeability trapping constant of porous media by combining the homogenization of periodic media hpm and the self consistent scheme scm based on a bicomposite spherical pattern the link between the hpm and scm approaches is first established by using a systematic argument independent of the problem under consideration it is shown that the periodicity condition can be replaced by zero flux and energy through the whole surface of the representative elementary volume consequently the scm solution can be considered as a geometrical approximation of the local problem derived through hpm for materials such that the morphology of the period is close to the scm pattern these results are then applied to derive the estimates of the effective diffusion the dynamic permeability the thermal permeability and the trapping constant of porous media these scm estimates are compared with numerical hpm results obtained on periodic arrays of spheres and polyhedrons it is shown that scm estimates provide good analytical approximations of the effective parameters for periodic packings of spheres at porosities larger than 0 6 while the agreement is excellent for periodic packings of polyhedrons in the whole range of porosity

cell model calculations of dynamic drag parameters in packings of spheres an external flow approach is used to predict the viscous drag due to oscillating flow in an air filled stack of fixed identical rigid spheres analytical expressions for dynamic and direct current dc permeability high frequency limit of tortuosity and the characteristic viscous dimension are derived using a cell model with an adjustable cell radius which allows for hydrodynamic interactions between the spherical particles the resulting theory requires knowledge of two fixed parameters the volume porosity and the particle radius the theory also requires a value for the cell radius use of the cell radius corresponding to that of the sphere circumscribing a unit cell of a cubic lattice arrangement is proposed this is found to enable good agreement between predictions of the new theory and both published data and numerical results for simple cubic and random spherical packings

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

karush kuhn tucker systems regularity conditions error bounds and a class of newton type methods we consider optimality systems of karush kuhn tucker kkt type which arise for example as primal dual conditions characterizing solutions of optimization problems or variational inequalities in particular we discuss error bounds and newton type methods for such systems an exhaustive comparison of various regularity conditions which arise in this context is given we obtain a new error bound under an assumption which we show to be strictly weaker than assumptions previously used for kkt systems such as quasi regularity or semistability equivalently the r0 property error bounds are useful among oth er things for identifying active constraints and developing efficient local algorithms we propose a family of local newton type algorithms this family contains some known active set newton methods as well as some new methods regularity conditions required for local superlinear convergence compare favorably with convergence conditions of nonsmooth newton methods and sequential quadratic programming methods

hedging with a correlated asset solution of a nonlinear pricing pde hedging a contingent claim with an asset which is not perfectly correlated with the underlying asset results in unhedgeable residual risk even if the residual risk is considered diversifiable the option writer is faced with the problem of uncertainty in the estimation of the drift rates of the underlying and the hedging instrument if the residual risk is not considered diversifiable then this risk can be priced using an actuarial standard deviation principle in infinitesimal time in both cases these models result in the same nonlinear partial differential equation pde a fully implicit monotone discretization method is developed for solution of this pricing pde this method is shown to converge to the viscosity solution certain grid conditions are required to guarantee monotonicity an algorithm is derived which given an initial grid inserts a finite number of nodes in the grid to ensure that the monotonicity condition is satisfied at each timestep the nonlinear discretized algebraic equations are solved using an iterative algorithm which is shown to be globally convergent monte carlo hedging examples are given to illustrate the profit and loss distribution at the expiry of the option

a family of second order methods for convex ell _1 l1 regularized optimization this paper is concerned with the minimization of an objective that is the sum of a convex function f and an ell 1 l1 regularization term our interest is in active set methods that incorporate second order information about the function f to accelerate convergence we describe a semismooth newton framework that can be used to generate a variety of second order methods including block active set methods orthant based methods and a second order iterative soft thresholding method the paper proposes a new active set method that performs multiple changes in the active manifold estimate at every iteration and employs a mechanism for correcting these estimates when needed this corrective mechanism is also evaluated in an orthant based method numerical tests comparing the performance of three active set methods are presented

a new exact penalty function for constrained smooth or nonsmooth optimization problems new continuously differentiable penalty functions are derived they are proved exact in the sense that under some nondegeneracy assumption local optimizers of a nonlinear program are precisely the optimizers of the associated penalty function this is achieved by augmenting the dimension of the program by a variable that controls both the weight of the penalty terms and the regularization of the nonsmooth terms

a new non interior continuation method for solving the second order cone complementarity problem abstract in this paper based on a symmetrically perturbed smoothing fischer burmeister function a non interior continuation method is proposed for solving the second order cone complementarity problem soccp the proposed algorithm solves only one linear system of equations and performs only one line search at each iteration under monotonicity it is shown that our algorithm is globally and locally superlinearly convergent without requiring strict complementarity assumption at the soccp solution furthermore the proposed algorithm has local quadratic convergence under mild conditions some numerical results are reported which indicate the effectiveness of the proposed algorithm

a class of inexact variable metric proximal point algorithms for the problem of solving maximal monotone inclusions we present a rather general class of algorithms which contains hybrid inexact proximal point methods as a special case and allows for the use of a variable metric in subproblems the global convergence and local linear rate of convergence are established under standard assumptions we demonstrate the advantage of variable metric implementation in the case of solving systems of smooth monotone equations by the proximal newton method

a regularization semismooth newton method based on the generalized fischer burmeister function for p0 ncps we consider a regularization method for nonlinear complementarity problems with f being a p 0 function which replaces the original problem with a sequence of the regularized complementarity problems in this paper this sequence of regularized complementarity problems are solved approximately by applying the generalized newton method for an equivalent augmented system of equations constructed by the generalized fischer burmeister fb ncp functions f p with p 1 we test the performance of the regularization semismooth newton method based on the family of ncp functions through solving all test problems from mcplib numerical experiments indicate that the method associated with a smaller p for example email protected 1 1 2 usually has better numerical performance and the generalized fb functions f p with email protected 1 1 2 can be used as the substitutions for the fb function f 2

a nonsmooth version of newton s method newton s method for solving a nonlinear equation of several variables is extended to a nonsmooth case by using the generalized jacobian instead of the derivative this extension includes the b derivative version of newton s method as a special case convergence theorems are proved under the condition of semismoothness it is shown that the gradient function of the augmented lagrangian forc2 nonlinear programming is semismooth thus the extended newton s method can be used in the augmented lagrangian method for solving nonlinear programs

on the superlinear convergence of the variable metric proximal point algorithm using broyden and bfgs matrix secant updating in previous work the authors provided a foundation for the theory of variable metric proximal point algorithms in hilbert space in that work conditions are developed for global linear and super linear convergence this paper focuses attention on two matrix secant updating strategies for the finite dimensional case these are the broyden and bfgs updates the bfgs update is considered for application in the symmetric case e g convex programming applications while the broyden update can be applied to general monotone operators subject to the linear convergence of the iterates and a quadratic growth condition on the inverse of the operator at the solution super linear convergence of the iterates is established for both updates these results are applied to show that the chen fukushima variable metric proximal point algorithm is super linearly convergent when implemented with the bfgs update

riemann surfaces and the geometrization of 3 manifolds about a decade ago thurston proved that a vast collection of 3 manifolds carry metrics of constant negative curvature these manifolds are thus elements of hyperbolic geometry as natural as euclid s regular polyhedra for a closed manifold mostow rigidity assures that a hyperbolic structure is unique when it exists mos so topology and geometry mesh harmoniously in dimension 3 this remarkable theorem applies to all 3 manifolds which can be built up in an inductive way from 3 balls i e haken manifolds thurston s construction of a hyperbolic structure is also inductive at the inductive step one must find the right geometry on an open 3 manifold so that its ends may be glued together using quasiconformal deformations the gluing problem can be formulated as a fixed point problem for a map of teichmuller space to itself thurston proposes to find the fixed point by iterating this map here we outline thurston s construction and sketch a new proof that the iteration converges our argument rests on a result entirely in the theory of riemann surfaces an extremal quasiconformal mapping can be relaxed isotoped to a map of lesser dilatation when lifted to a sufficiently

second order necessary conditions in semismooth optimization first and second order conditions are given which are necessary for a functionf to have a local minimal value atx inrn it is assumed thatf is locally lipschitzian nearx and semismooth atx the necessary conditions are expressed in terms of the generalized gradients of nonsmooth analysis and certain second order directional derivatives the method of proof bears no resemblance to standard methods three special cases are discussed here but applications to constrained problems are made elsewhere

the relaxed nonlinear phss like iteration method for absolute value equations finding the solution of the absolute value equation ave a x x b has attracted much attention in recent years in this paper we propose a relaxed nonlinear phss like iterative method which is more efficient than the picard hss iterative method for the ave and is a generalization of the nonlinear hss like iteration method for the ave by using the theory of nonsmooth analysis we prove the convergence of the relaxed nonlinear phss like iterative method for the ave numerical experiments are given to demonstrate the feasibility robustness and effectiveness of the relaxed nonlinear hss like method

generalized hessian properties of regularized nonsmooth functions we take up the question of second order expansions for a class of functions of importance in optimization namely moreau envelope regularizations of nonsmooth functions f it is shown that when f is prox regular which includes convex functions and the extended real valued functions representing problems of nonlinear programming the many second order properties that can be formulated around the existence and stability of expansions of the envelopes of f or of their gradient mappings are linked by surprisingly extensive lists of equivalences with each other and with generalized differentiation properties of f itself this clarifies the circumstances conducive to developing computational methods based on envelope functions such as second order approximations in nonsmooth optimization and variants of the proximal point algorithm the results establish that generalized second order expansions of moreau envelopes at least can be counted on in most situations of interest in finite dimensional optimization

a primal dual active set method for non negativity constrained total variation deblurring problems this paper studies image deblurring problems using a total variation based model with a non negativity constraint the addition of the non negativity constraint improves the quality of the solutions but makes the solution process a difficult one the contribution of our work is a fast and robust numerical algorithm to solve the non negatively constrained problem to overcome the nondifferentiability of the total variation norm we formulate the constrained deblurring problem as a primal dual program which is a variant of the formulation proposed by chan golub and mulet for unconstrained problems here dual refers to a combination of the lagrangian and fenchel duals to solve the constrained primal dual program we use a semi smooth newton s method we exploit the relationship between the semi smooth newton s method and the primal dual active set method to achieve considerable simplification of the computations the main advantages of our proposed scheme are no parameters need significant adjustment a standard inverse preconditioner works very well quadratic rate of local convergence theoretical and numerical numerical evidence of global convergence and high accuracy of solving the optimality system the scheme shows robustness of performance over a wide range of parameters a comprehensive set of numerical comparisons are provided against other methods to solve the same problem which show the speed and accuracy advantages of our scheme

amenable coverings of complex manifolds and holomorphic probability measures let y x be a covering map between complex manifolds x and y for many holomorphic objects such as functions or forms one can define a pushforward operator carrying objects on y to objects on x by summing over the fiber the pushforward is efficient if there is little cancellation in the sum to achieve efficiency there must be coherence in phase on different sheets of the covering in this paper we will demonstrate a direct connection between efficiency of the push forward and amenability of the covering the latter is a purely combinatorial property of the pair of groups 1 y 1 x the covering is amenable if there exists a 1 x invariant finitely additive probability measure on the coset space 1 x 1 y amenability is discussed in more detail in 3 the connection will be established by studying complex valued measures on the fibers of the covering varying holomorphically with respect to the base

semismooth newton and augmented lagrangian methods for a simplified friction problem in this paper a simplified friction problem and iterative second order algorithms for its solution are analyzed in infinite dimensional function spaces motivated from the dual formulation a primal dual active set strategy and a semismooth newton method for a regularized problem as well as an augmented lagrangian method for the original problem are presented and their close relation is analyzed local as well as global convergence results are given by means of numerical tests we discuss among others convergence properties the dependence on the mesh and the role of the regularization and illustrate the efficiency of the proposed methodologies