newton type methods for optimization problems without constraint qualifications we consider equality constrained optimization problems where a given solution may not satisfy any constraint qualification but satisfies the standard second order sufficient condition for optimality based on local identification of the rank of the constraints degeneracy via the singular value decomposition we derive a modified primal dual optimality system whose solution is locally unique nondegenerate and thus can be found by standard newton type techniques using identification of active constraints we further extend our approach to mixed equality and inequality constrained problems and to mathematical programs with complementarity constraints mpcc in particular for mpcc we obtain a local algorithm with quadratic convergence under the second order sufficient condition only without any constraint qualifications not even the special mpcc constraint qualifications

generalized nash equilibrium problems and newton methods the generalized nash equilibrium problem where the feasible sets of the players may depend on the other players strategies is emerging as an important modeling tool however its use is limited by its great analytical complexity we consider several newton methods analyze their features and compare their range of applicability we illustrate in detail the results obtained by applying them to a model for internet switching

levenberg marquardt methods for constrained nonlinear equations with strong local convergence properties we consider the problem of finding a solution of a constrained and not necessarily square system of equations i e we consider systems of nonlinear equations and want to find a solution that belongs to a certain feasible set to this end we present two levenberg marquardt type algorithms that differ in the way they compute their search directions the first method solves a strictly convex minimization problem at each iteration whereas the second one solves only one system of linear equations in each step both methods are shown to converge locally quadratically under an error bound assumption that is much weaker than the standard nonsingularity condition both methods can be globalized in an easy way some numerical results for the second method indicate that the algorithm works quite well in practice

stabilized sequential quadratic programming for optimization and a stabilized newton type method for variational problems the stabilized version of the sequential quadratic programming algorithm ssqp had been developed in order to achieve fast convergence despite possible degeneracy of constraints of optimization problems when the lagrange multipliers associated to a solution are not unique superlinear convergence of ssqp had been previously established under the strong second order sufficient condition for optimality without any constraint qualification assumptions we prove a stronger superlinear convergence result than the above assuming the usual second order sufficient condition only in addition our analysis is carried out in the more general setting of variational problems for which we introduce a natural extension of ssqp techniques in the process we also obtain a new error bound for karush kuhn tucker systems for variational problems that holds under an appropriate second order condition

a trigonometric interpolation approach to mixed type boundary problems associated with permeameter shape factors 1 hydraulic conductivity is a fundamental hydrogeological parameter whose in situ measurement at a local scale is principally performed through injection tests from screened probes or using impermeable packers in screened wells the shape factor f l is a proportionality constant required to estimate conductivity from observed flow rate to injection head ratios and it depends on the geometric properties of the flow field existing approaches for determination of f are either based on geometric or mathematical simplifications and are limited to particular assumptions about the flow domain s external boundaries the present work presents a general semianalytical solution to steady state axisymmetric flow problems where external boundaries may be nearby and of arbitrary combinations of impermeable and constant head type the inner boundary along the probe or well may consist of an arbitrary number of impermeable and constant head intervals resulting in a mixed type boundary value problem for which a novel and direct solution method based on trigonometric interpolation is presented the approach is applied to generate practical nondimensional charts of f for different field and laboratory situations results show that f is affected by less than 5 if a minimum distance of 10 probe or well diameters is kept between the injection screen and a nearby boundary similarly minimum packer lengths of two well diameters are required to avoid increasing f by more than 10 furthermore f is determined for laboratory barrel experiments giving guidelines for achieving equal shape factors as in field situations without nearby boundaries f for the theoretical case of infinitely short packers is shown to be infinitely large

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

a family of inexact sqa methods for non smooth convex minimization with provable convergence guarantees based on the luo tseng error bound property we propose a new family of inexact sequential quadratic approximation sqa methods which we call the inexact regularized proximal newton irpn method for minimizing the sum of two closed proper convex functions one of which is smooth and the other is possibly non smooth our proposed method features strong convergence guarantees even when applied to problems with degenerate solutions while allowing the inner minimization to be solved inexactly specifically we prove that when the problem possesses the so called luo tseng error bound eb property irpn converges globally to an optimal solution and the local convergence rate of the sequence of iterates generated by irpn is linear superlinear or even quadratic depending on the choice of parameters of the algorithm prior to this work such eb property has been extensively used to establish the linear convergence of various first order methods however to the best of our knowledge this work is the first to use the luo tseng eb property to establish the superlinear convergence of sqa type methods for non smooth convex minimization as a consequence of our result irpn is capable of solving regularized regression or classification problems under the high dimensional setting with provable convergence guarantees we compare our proposed irpn with several empirically efficient algorithms by applying them to the ell 1 regularized logistic regression problem experiment results show the competitiveness of our proposed method

computing b stationary points of nonsmooth dc programs motivated by a class of applied problems arising from physical layer based security in a digital communication system in particular by a secrecy sum rate maximization problem this paper studies a nonsmooth difference of convex dc minimization problem the contributions of this paper are i clarify several kinds of stationary solutions and their relations ii develop and establish the convergence of a novel algorithm for computing a d stationary solution of a problem with a convex feasible set that is arguably the sharpest kind among the various stationary solutions iii extend the algorithm in several directions including a randomized choice of the subproblems that could help the practical convergence of the algorithm a distributed penalty approach for problems whose objective functions are sums of dc functions and problems with a specially structured nonconvex dc constraint for the latter class of problems a pointwise slater constraint qualification is introduced that facilitates the ver

a stabilized sqp method superlinear convergence stabilized sequential quadratic programming ssqp methods for nonlinear optimization generate a sequence of iterates with fast local convergence regardless of whether or not the active constraint gradients are linearly dependent this paper concerns the local convergence analysis of an ssqp method that uses a line search with a primal dual augmented lagrangian merit function to enforce global convergence the method is provably well defined and is based on solving a strictly convex quadratic programming subproblem at each iteration it is shown that the method has superlinear local convergence under assumptions that are no stronger than those required by conventional stabilized sqp methods the fast local convergence is obtained by allowing a small relaxation of the optimality conditions for the quadratic programming subproblem in the neighborhood of a solution in the limit the line search selects the unit step length which implies that the method does not suffer from the maratos effect the analysis indicates that the method has the same strong first and second order global convergence properties that have been established for augmented lagrangian methods yet is able to transition seamlessly to ssqp with fast local convergence in the neighborhood of a solution numerical results on some degenerate problems are reported

an lp newton method nonsmooth equations kkt systems and nonisolated solutions we define a new newton type method for the solution of constrained systems of equations and analyze in detail its properties under suitable conditions that do not include differentiability or local uniqueness of solutions the method converges locally quadratically to a solution of the system thus filling an important gap in the existing theory the new algorithm improves on known methods and when particularized to kkt systems derived from optimality conditions for constrained optimization or variational inequalities it has theoretical advantages even over methods specifically designed to solve such systems

on the identification of zero variables in an interior point framework we consider column sufficient linear complementarity problems and study the problem of identifying those variables that are zero at a solution to this end we propose a new computationally inexpensive technique that is based on growth functions we analyze in detail the theoretical properties of the identification technique and test it numerically the identification technique is particularly suited to interior point methods but can be applied to a wider class of methods

local behavior of an iterative framework for generalized equations with nonisolated solutions an iterative framework for solving generalized equations with nonisolated solutions is presented for generalized equations with the structure where is a multifunction and f is single valued the framework covers methods that at each step solve subproblems of the type the multifunction approximates f around s besides a condition on the quality of this approximation two other basic assumptions are employed to show q superlinear or q quadratic convergence of the iterates to a solution a key assumption is the upper lipschitz continuity of the solution set map of the perturbed generalized equation moreover the solvability of the subproblems is required conditions that ensure these assumptions are discussed in general and by means of several applications they include monotone mixed complementarity problems karush kuhn tucker systems arising from nonlinear programs and nonlinear equations particular results deal with error bounds and upper lipschitz continuity properties for these problems

an algorithm for degenerate nonlinear programming with rapid local convergence this paper describes and analyzes an algorithmic framework for solving nonlinear programming problems in which strict complementarity conditions and constraint qualifications are not necessarily satisfied at a solution the framework is constructed from three main algorithmic ingredients the first is any conventional method for nonlinear programming that produces estimates of the lagrange multipliers at each iteration the second is a technique for estimating the set of active constraint indices the third is a stabilized lagrange newton algorithm with rapid local convergence properties results concerning rapid local convergence and global convergence of the proposed framework are proved the approach improves on existing approaches in that less restrictive assumptions are needed for convergence and or the computational workload at each iteration is lower

solving karush kuhn tucker systems via the trust region and the conjugate gradient methods a popular approach to solving the karush kuhn tucker kkt system mainly arising from the variational inequality problem is to reformulate it as a constrained minimization problem with simple bounds in this paper we propose a trust region method for solving the reformulation problem with the trust region subproblems being solved by the truncated conjugate gradient cg method which is cost effective other advantages of the proposed method over existing ones include the fact that a good approximated solution to the trust region subproblem can be found by the truncated cg method and is judged in a simple way also the working matrix in each iteration is h instead of the condensed hth where h is a matrix element of the generalized jacobian of the function used in the reformulation as a matter of fact the matrix used is of reduced dimension we pay extra attention to ensure the success of the truncated cg method as well as the feasibility of the iterates with respect to the simple constraints another feature of the proposed method is that we allow the merit function value to be increased at some iterations to speed up the convergence global and superlinear quadratic convergence is shown under standard assumptions numerical results are reported on a subset of problems from the mcplib collection s p dirkse and m c ferris optim methods softw 5 1995 pp 319 345

an integral transform approach for a mixed boundary problem involving a flowing partially penetrating well with infinitesimal well skin 1 a flowing partially penetrating well with infinitesimal well skin is a mixed boundary because a cauchy condition is prescribed along the screen length and a neumann condition of no flux is stipulated over the remaining unscreened part an analytical approach based on the integral transform technique is developed to determine the laplace domain solution for such a mixed boundary problem in a confined aquifer of finite thickness first the mixed boundary is changed into a homogeneous neumann boundary by substituting the cauchy condition with a neumann condition in terms of well bore flux that varies along the screen length and is time dependent despite the well bore flux being unknown a priori the modified model containing this homogeneous neumann boundary can be solved with the laplace and the finite fourier cosine transforms to determine well bore flux screen length is discretized into a finite number of segments to which the cauchy condition is reinstated this reinstatement also restores the relation between the original model and the solutions obtained for a given time the numerical inversion of the laplace domain solution yields the drawdown distributions well bore flux and the well discharge this analytical approach provides an alternative for dealing with the mixed boundary problems especially when aquifer thickness is assumed to be finite

modified wilson s method for nonlinear programswith nonunique multipliers this paper first investigates newton type methods for generalized equations with compact solution sets the analysis of their local convergence behavior is based besides other conditions on the upper lipschitz continuity of the local solution set mapping of a simply perturbed generalized equation this approach is then applied to the kkt conditions of a nonlinear program with inequality constraints and leads to a modified version of the classical wilson method it is shown that the distances of the iterates to the set of kkt points converge q quadratically to zero under conditions that do not imply a unique multiplier vector additionally to the mangasarian fromovitz constraint qualification and to a second order sufficiency condition the local minimizer is required to fulfill a constant rank condition weaker than the constant rank constraint qualification one of janin and a so called weak complementarity condition

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

the analysis of constant drawdown wells using instantaneous source functions wells are commonly held at constant drawdown to maximize their discharge or to recover nonaqueous phase liquids but only the simplest cases of constant drawdown wells have been treated analytically in this paper the method of instantaneous source functions which was originally limited to cases where the discharges of wells are specified is extended to predict transient discharge and drawdown in the vicinity of wells held at constant drawdown the analysis requires applying a laplace transform to the duhamel integral rearranging the resulting algebraic expression and then numerically inverting to obtain the discharge function that will result in a constant drawdown at a particular location this approach allows flow problems with a wide range of geometries to be analyzed and it also allows superposition to be used for transient problems without violating the condition of constant drawdown at the well

critical solutions of nonlinear equations local attraction for newton type methods we show that if the equation mapping is 2 regular at a solution in some nonzero direction in the null space of its jacobian in which case this solution is critical in particular the local lipschitzian error bound does not hold then this direction defines a star like domain with nonempty interior from which the iterates generated by a certain class of newton type methods necessarily converge to the solution in question this is despite the solution being degenerate and possibly non isolated so that there are other solutions nearby in this sense newtonian iterates are attracted to the specific critical solution those results are related to the ones due to a griewank for the basic newton method but are also applicable for example to some methods developed specially for tackling the case of potentially non isolated solutions including the levenberg marquardt and the lp newton methods for equations and the stabilized sequential quadratic programming for optimization

a regularized smoothing newton type algorithm for quasi variational inequalities abstract this paper is concerned with developing an efficient regularized smoothing newton type algorithm for quasi variational inequalities the proposed algorithm takes the advantage of newly introduced smoothing functions and a non monotone line search strategy it is proved to be globally and locally superlinearly quadratically convergent under suitable assumptions numerical results demonstrate that the algorithm generally outperforms the existing interior point method and semismooth method facchinei et al 2014

erratum to levenberg marquardt methods with strong local convergence properties for solving nonlinear equations with convex constraints we consider the problem of finding a solution of a constrained and not necessarily square system of equations i e we consider systems of nonlinear equations and want to find a solution that belongs to a certain feasible set to this end we present two levenberg marquardt type algorithms that differ in the way they compute their search directions the first method solves a strictly convex minimization problem at each iteration whereas the second one solves only one system of linear equations in each step both methods are shown to converge locally quadratically under an error bound assumption that is much weaker than the standard nonsingularity condition both methods can be globalized in an easy way some numerical results for the second method indicate that the algorithm works quite well in practice

generalized derivatives for hybrid systems established sensitivity results for hybrid discrete continuous dynamic systems are generalized by relaxing smoothness assumptions on the functions governing the systems continuous evolution and discrete event handling the new results only require l smoothness of these functions in the sense of nesterov instead of continuous differentiability parametric lexicographic derivatives for such a hybrid system provide useful local first order sensitivity information and are described as the unique solutions of auxiliary hybrid systems this sensitivity analysis framework permits generalized derivative evaluation even for certain hybrid systems in which small changes in parameters can change the sequence of discrete modes visited to handle parametric sensitivities of event times that are not known explicitly conditions are provided under which a local inverse function or implicit function is l smooth with lexicographic derivatives that are described as the unique solutions of certain equation systems these equation systems are readily solved when the functions involved are piecewise differentiable

convergence acceleration of fourier series by analytical and numerical application of poisson s formula a form of the poisson summation formula appropriate for the transformation of infinite fourier series is derived and shown to be applicable to slowly converging series arising in the solution of laplace s equation in a rectangular electrochemical cell the result is a rapidly convergent fourier series which is particularly useful in giving the potential close to the electrode surface results obtained from the analytical expression are found to be in close agreement with those resulting from numerical evaluation of the fourier integrals using simpson s rule and the euler transformation and direct summation by using goertzel s 1958 algorithm

stabilized sqp revisited the stabilized version of the sequential quadratic programming algorithm ssqp had been developed in order to achieve superlinear convergence in situations when the lagrange multipliers associated to a solution are not unique within the framework of fischer math program 94 91 124 2002 the key to local superlinear convergence of ssqp are the following two properties upper lipschitzian behavior of solutions of the karush kuhn tucker kkt system under canonical perturbations and local solvability of ssqp subproblems with the associated primal dual step being of the order of the distance from the current iterate to the solution set of the unperturbed kkt system according to fernandez and solodov math program 125 47 73 2010 both of these properties are ensured by the second order sufficient optimality condition sosc without any constraint qualification assumptions in this paper we state precise relationships between the upper lipschitzian property of solutions of kkt systems error bounds for kkt systems the notion of critical lagrange multipliers a subclass of multipliers that violate sosc in a very special way the second order necessary condition for optimality and solvability of ssqp subproblems moreover for the problem with equality constraints only we prove superlinear convergence of ssqp under the assumption that the dual starting point is close to a noncritical multiplier since noncritical multipliers include all those satisfying sosc but are not limited to them we believe this gives the first superlinear convergence result for any newtonian method for constrained optimization under assumptions that do not include any constraint qualifications and are weaker than sosc in the general case when inequality constraints are present we show that such a relaxation of assumptions is not possible we also consider applying ssqp to the problem where inequality constraints are reformulated into equalities using slack variables and discuss the assumptions needed for convergence in this approach we conclude with consequences for local regularization methods proposed in izmailov and solodov siam j optim 16 210 228 2004 wright siam j optim 15 673 676 2005 in particular we show that these methods are still locally superlinearly convergent under the noncritical multiplier assumption weaker than sosc employed originally

on the accurate identification of active constraints we consider nonlinear programs with inequality constraints and we focus on the problem of identifying those constraints which will be active at an isolated local solution the correct identification of active constraints is important from both a theoretical and a practical point of view such an identification removes the combinatorial aspect of the problem and locally reduces the inequality constrained minimization problem to an equality constrained problem which can be more easily dealt with we present a new technique which identifies active constraints in a neighborhood of a solution and which requires neither complementary slackness nor uniqueness of the multipliers we also present extensions to variational inequalities and numerical examples illustrating the identification technique

generalized sensitivity analysis of nonlinear programs this paper extends classical sensitivity results for nonlinear programs to cases in which parametric perturbations cause changes in the active set this is accomplished using lexicographic directional derivatives a recently developed tool in nonsmooth analysis based on nesterov s lexicographic differentiation a nonsmooth implicit function theorem is augmented with generalized derivative information and applied to a standard nonsmooth reformulation of the parametric kkt system it is shown that the sufficient conditions for this implicit function theorem variant are implied by a kkt point satisfying the linear independence constraint qualification and strong second order sufficiency mirroring the classical theory the resulting sensitivity system is a nonsmooth equation system which admits primal and dual sensitivities as its unique solution practically implementable algorithms are provided for calculating the nonsmooth sensitivity system s unique solution which is then used to furnish b subdifferenti

symplectic integration of magnetic systems abstract simulation of the long time behavior of systems requires more than just numerical stability to return dependable results it must preserve the underlying geometric structure of the continuous equations symplectic integrators are the most common form of geometric integrator and are therefore of interest in simulating plasmas for many plasma periods for example we present here results on generating symplectic integrators for magnetic systems and in particular show that the algorithms due to boris and vay are symplectic

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

adjusting dual iterates in the presence of critical lagrange multipliers it is a well known phenomenon that the presence of critical lagrange multipliers in constrained optimization problems may cause a deterioration of the convergence speed of primal dual newton type m

shape factors for constant head double packer permeameters 1 local estimates of hydraulic conductivity are often obtained from constant head double packer tests the tests yield ratios of flow against head difference these are converted to values of hydraulic conductivity by dividing by a shape factor previously used shape factors have generally assumed that the packers are infinitely large such shape factors lead to overestimates of hydraulic conductivity as they ignore the effect of water flow around the packers a semianalytical solution is derived for a constant head double packer permeameter test the solution is compared with existing approximate techniques including the spheroidal equipotential solution the narrow packer approximation of rehbinder 1996 and the long packer approximation of rehbinder 2005 generally it is shown that ignoring the length of the packer and assuming a fully cased well lead to an overestimate of hydraulic conductivity the significance of this reduces with increasing well screen aspect ratio evaluation of the solution is nontrivial for this reason a polynomial relationship between shape factor well screen aspect ratio and packer length obtained by curve fitting with the semianalytical solution is provided