stability in the presence of degeneracy and error estimation given an approximation to a local minimizer to a nonlinear optimization problem and to associated multipliers we obtain a tight error estimate in terms of the violation of the first order conditions our results apply to degenerate optimization problems where independence of the active constraint gradients and strict complementarity can be violated

fusion of hard and soft information in nonparametric density estimation abstract this paper discusses univariate density estimation in situations when the sample hard information is supplemented by soft information about the random phenomenon these situations arise broadly in operations research and management science where practical and computational reasons severely limit the sample size but problem structure and past experiences could be brought in in particular density estimation is needed for generation of input densities to simulation and stochastic optimization models in analysis of simulation output and when instantiating probability models we adopt a constrained maximum likelihood estimator that incorporates any possibly random soft information through an arbitrary collection of constraints we illustrate the breadth of possibilities by discussing soft information about shape support continuity smoothness slope location of modes symmetry density values neighborhood of known density moments and distribution functions the maximization takes place over spaces of extended real valued semicontinuous functions and therefore allows us to consider essentially any conceivable density as well as convenient exponential transformations the infinite dimensionality of the optimization problem is overcome by approximating splines tailored to these spaces to facilitate the treatment of small samples the construction of these splines is decoupled from the sample we discuss existence and uniqueness of the estimator examine consistency under increasing hard and soft information and give rates of convergence numerical examples illustrate the value of soft information the ability to generate a family of diverse densities and the effect of misspecification of soft information

modifying sqp for degenerate problems most local convergence analyses of the sequential quadratic programming sqp algorithm for nonlinear programming make strong assumptions about the solution namely that the active constraint gradients are linearly independent and that there are no weakly active constraints in this paper we establish a framework for variants of sqp that retain the characteristic superlinear convergence rate even when these assumptions are relaxed proving general convergence results and placing some recently proposed sqp variants in this framework we discuss the reasons for which implementations of sqp often continue to exhibit good local convergence behavior even when the assumptions commonly made in the analysis are violated finally we describe a new algorithm that formalizes and extends standard sqp implementation techniques and we prove convergence results for this method also

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

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

on the constant positive linear dependence condition and its application to sqp methods in this paper we introduce a constant positive linear dependence condition cpld which is weaker than the mangasarian fromovitz constraint qualification mfcq and the constant rank constraint qualification crcq we show that a limit point of a sequence of approximating karush kuhn tucker kkt points is a kkt point if the cpld holds there we show that a kkt point satisfying the cpld and the strong second order sufficiency conditions ssosc is an isolated kkt point we then establish convergence of a general sequential quadratical programming sqp method under the cpld and the ssosc finally we apply these results to analyze the feasible sqp method proposed by panier and tits in 1993 for inequality constrained optimization problems we establish its global convergence under the ssosc and a condition slightly weaker than the mangasarian fromovitz constraint qualification and we prove superlinear convergence of a modified version of this algorithm under the ssosc and a condition slightly weaker than the linear independence constraint qualification

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

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

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

contiguous relations basic hypergeometric functions and orthogonal polynomials iii associated continuous dual q hahn polynomials explicit solutions for the three term recurrence satisfied by associated continuous dual q hahn polynomials are obtained a minimal solution is identified and an explicit expression for the related continued fraction is derived the absolutely continuous component of the spectral measure is obtained eleven limit cases are discussed in some detail these include associated big q laguerre associated wall associated al salam chihara associated al salam carlitz i and associated continuous q hermite polynomials

on a general q fourier transformation with nonsymmetric kernels wiener used the poisson kernel for the hermite polynomials to deal with the classical fourier transform askey atakishiyev and suslov used this approach to obtain a q fourier transform by using the continuous q hermite polynomials rahman and suslov extended this result by taking the askey wilson polynomials considered to be the most general continuous classical orthogonal polynomials the theory of q fourier transformation is further extended here by considering a nonsymmetric version of the poisson kernel with askey wilson polynomials this approach enables us to obtain some new results for example the complex and real orthogonalities of these kernels

a set of sums for continuous dual q 2 hahn polynomials an infinite set l y r z r l n0 of linear sums of continuous dual q 2 hahn polynomials with prefactors depending on a complex parameter z is studied the sums l y r z have an interpretation in context with tensor product representations of the quantum affine algebra uq sl 2 involving both a positive and a negative discrete series representation for each l 0 the sum l y r z can be expressed in terms of the sum 0 y r z continuous dual q2 hahn polynomials and their associated polynomials the sum 0 y r z is obtained as a combination of eight basic hypergeometric series moreover an integral representation is provided for the sums l y r z with the complex parameter restricted by zq 1 this gives rise to a bilinear summation formula for the continuous dual q 2 hahn polynomials

local convergence of exact and inexact augmented lagrangian methods under the second order sufficient optimality condition we establish local convergence and rate of convergence of the classical augmented lagrangian algorithm under the sole assumption that the dual starting point is close to a multiplier satisfying the second order sufficient optimality condition in particular no constraint qualifications of any kind are needed previous literature on the subject required in addition the linear independence constraint qualification and either the strict complementarity assumption or a stronger version of the second order sufficient condition that said the classical results allow the initial multiplier estimate to be far from the optimal one at the expense of proportionally increasing the threshold value for the penalty parameters although our primary goal is to avoid constraint qualifications if the stronger assumptions are introduced then starting points far from the optimal multiplier are allowed within our analysis as well using only the second order sufficient optimality condition for penalty parameters large

a globally convergent stabilized sqp method sequential quadratic programming sqp methods are a popular class of methods for nonlinearly constrained optimization they are particularly effective for solving a sequence of related problems such as those arising in mixed integer nonlinear programming and the optimization of functions subject to differential equation constraints recently there has been considerable interest in the formulation of stabilized sqp methods which are specifically designed to handle degenerate optimization problems existing stabilized sqp methods are essentially local in the sense that both the formulation and analysis focus on the properties of the methods in a neighborhood of a solution a new sqp method is proposed that has favorable global convergence properties yet under suitable assumptions is equivalent to a variant of the conventional stabilized sqp method in the neighborhood of a solution the method combines a primal dual generalized augmented lagrangian function with a flexible line search to obtain a seque

attraction of newton method to critical lagrange multipliers fully quadratic case in this paper we continue the studies of the persistent effect of attraction of newton type iterations for optimality systems to critical lagrange multipliers it appears very important to understand the nature of this striking phenomenon in particular because it is precisely the reason of slow convergence of such methods when applied to problems with degenerate constraints all previously known results concerned with this effect were a posteriori by nature they were showing that in case of convergence the dual limit is in a sense unlikely to be noncritical this paper suggests the first a priori result in this direction showing that critical multipliers actually serve as attractors for a fully quadratic optimization problem with equality constraints under certain reasonable assumptions we establish actual local convergence of the newton lagrange method to a critical multiplier starting from a dense set around a given critical multiplier this is an important step forward in understanding the attraction phenomenon

convergence of splitting and newton methods for complementarity problems an application of some sensitivity results this paper is concerned with two well known families of iterative methods for solving the linear and nonlinear complementarity problems for the linear complementarity problem we consider the class of matrix splitting methods and establish under a finiteness assumption on the number of solutions a necessary and sufficient condition for the convergence of the sequence of iterates produced a rate of convergence result for this class of methods is also derived under a stability assumption on the limit solution for the nonlinear complementarity problem we establish the convergence of the newton method under the assumption of a pseudo regular solution which generalizes robinson s concept of a strongly regular solution in both instances the convergence proofs rely on a common sensitivity result of the linear complementarity problem under perturbation

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

on the cost of solving augmented lagrangian subproblems at each iteration of the augmented lagrangian algorithm a nonlinear subproblem is being solved the number of inner iterations of some any method needed to obtain a solution of the subproblem or even a suitable approximate stationary point is in principle unknown in this paper we show that to compute an approximate stationary point sufficient to guarantee local superlinear convergence of the augmented lagrangian iterations it is enough to solve two quadratic programming problems or two linear systems in the equality constrained case in other words two inner newtonian iterations are sufficient to the best of our knowledge such results are not available even under the strongest assumptions of second order sufficiency strict complementarity and the linear independence constraint qualification our analysis is performed under second order sufficiency only which is the weakest assumption for obtaining local convergence and rate of convergence of outer iterations of the augmented lagrangian algorithm the structure of the quadratic problems in question is related to the stabilized sequential quadratic programming and to second order corrections

on attraction of linearly constrained lagrangian methods and of stabilized and quasi newton sqp methods to critical multipliers it has been previously demonstrated that in the case when a lagrange multiplier associated to a given solution is not unique newton iterations e g those of sequential quadratic programming sqp have a tendency to converge to special multipliers called critical multipliers when such critical multipliers exist this fact is of importance because critical multipliers violate the second order sufficient optimality conditions and this was shown to be the reason for slow convergence typically observed for problems with degenerate constraints convergence to noncritical multipliers results in superlinear rate despite degeneracy some theoretical and numerical validation of this phenomenon can be found in izmailov and solodov comput optim appl 42 231 264 2009 math program 117 271 304 2009 however previous studies concerned the basic forms of newton iterations the question remained whether the attraction phenomenon still persists for relevant modifications as well as in professional implementations in this paper we answer this question in the affirmative by presenting numerical results for the well known minos and snopt software packages applied to a collection of degenerate problems we also extend previous theoretical considerations to the linearly constrained lagrangian methods and to the quasi newton sqp on which minos and snopt are based experiments also show that in the stabilized version of sqp the attraction phenomenon still exists but appears less persistent

a note on upper lipschitz stability error bounds and critical multipliers for lipschitz continuous kkt systems we prove a new local upper lipschitz stability result and the associated local error bound for solutions of parametric karush kuhn tucker systems corresponding to variational problems with lipschitzian base mappings and constraints possessing lipschitzian derivatives and without any constraint qualifications this property is equivalent to the appropriately extended to this nonsmooth setting notion of noncriticality of the lagrange multiplier associated to the primal solution which is weaker than second order sufficiency all this extends several results previously known only for optimization problems with twice differentiable data or assuming some constraint qualifications in addition our results are obtained in the more general variational setting

q2 kampe de feriet series and sums of continuous dual q 2 hahn polynomials two types of bilinear sums involving continuous dual q2 and q 2 hahn polynomials with a particular choice of parameters are identified with bilinear sums of balanced basic hypergeometric 2 2 series a symmetry relation with respect to the eigenvalues of the polynomials is established for the sum q2r z depending on two further complex parameters r and z an explicit expression as a linear combination of four q2 kampe de feriet series is obtained for a bilateral bilinear sum of continuous dual q 2 hahn polynomials for generic parameters application of the symmetry relation allows to describe the sum q2r z in terms of eight q2 kampe de feriet series special cases include partial derivatives of the latter five term recurrence relations for q2r z can be derived with respect to each of its parameters considerable simplifications are observed at z 1 or z q 1 where the sum q2r z is essentially given by very well poised balanced 8w7 series with base q4 or q

tensor products of uq sl 2 modules and the big q2 jacobi function transform four tensor products of evaluation modules of the quantum affine algebra uq sl 2 obtained from the negative and positive series the complementary and the strange series representations are investigated linear operators r z satisfying the intertwining property on finite linear combinations of the canonical basis elements of the tensor products are described in terms of two sets of infinite sums r t r t z 0 and r t r t z 0 involving big q2 jacobi functions or related nonterminating basic hypergeometric series inhomogeneous recurrence relations can be derived for both sets evaluations of the simplest sums provide the corresponding initial conditions for the first set of sums the relations entail a big q2 jacobi function transform pair an integral decomposition is obtained for the sum r t a partial description of the relation between the decompositions of the tensor products with respect to uqsl 2 or with respect to its complement in uq sl 2 can be formulated in terms of askey wilson

on combining feasibility descent and superlinear convergence in inequality constrained optimization extension of quasi newton techniques from unconstrained to constrained optimization via sequential quadratic programming sqp presents several difficulties among these are the possible inconsistency away from the solution of first order approximations to the constraints resulting in infeasibility of the quadratic programs and the task of selecting a suitable merit function to induce global convergence in ths case of inequality constrained optimization both of these difficulties disappear if the algorithm is forced to generate iterates that all satisfy the constraints and that yield monotonically decreasing objective function values feasibility of the successive iterates is in fact required in many contexts such as in real time applications or when the objective function is not well defined outside the feasible set it has been recently shown that this can be achieved while preserving local two step superlinear convergence in this note the essential ingredients for an sqp based method exhibiting the desired properties are highlighted correspondingly a class of such algorithms is described and analyzed tests performed with an efficient implementation are discussed

on attraction of newton type iterates to multipliers violating second order sufficiency conditions assuming that the primal part of the sequence generated by a newton type e g sqp method applied to an equality constrained problem converges to a solution where the constraints are degenerate we investigate whether the dual part of the sequence is attracted by those lagrange multipliers which satisfy second order sufficient condition sosc for optimality or by those multipliers which violate it this question is relevant at least for two reasons one is speed of convergence of standard methods the other is applicability of some recently proposed approaches for handling degenerate constraints we show that for the class of damped newton methods convergence of the dual sequence to multipliers satisfying sosc is unlikely to occur we support our findings by numerical experiments we also suggest a simple auxiliary procedure for computing multiplier estimates which does not have this undesirable property finally some consequences for the case of mixed equality and inequality constraints are discussed

asymptotic behavior of statistical estimators and of optimal solutions of stochastic optimization problems we study the asymptotic behavior of the statistical estimators that maximize a not necessarily dieren tiable criterion function possibly subject to side constraints equalities and inequalities the consistency results generalize those of wald and huber conditions are also given under which one is still able to obtain asymptotic normality the analysis brings to the fore the relationship between the problem of nding statistical estimators and that of nding the optimal solutions of stochastic optimization problems with partial information the last section is devoted to the properties of the saddle points of the associated lagrangians

asymptotics of least squares estimators for constrained nonlinear regression this paper is devoted to studying the asymptotic behavior of ls estimators in constrained nonlinear regression problems here the constraints are given by nonlinear equalities and inequalities thus this is a very general setting essentially this kind of estimation problem is a stochastic optimization problem so we make use of methods in optimization to overcome the difficulty caused by nonlinearity in the regression model and given constraints

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

superlinear convergence of an interior point method despite dependent constraints we show that an interior point method for monotone variational inequalities exhibits superlinear convergence provided that all the standard assumptions hold except for the well known assumption that the jacobian of the active constraints has full rank at the solution we show that superlinear convergence occurs even when the constant rank condition on the jacobian assumed in an earlier work does not hold

sharp primal superlinear convergence results for some newtonian methods for constrained optimization as is well known q superlinear or q quadratic convergence of the primal dual sequence generated by an optimization algorithm does not in general imply q superlinear convergence of the primal part primal convergence however is often of particular interest for the sequential quadratic programming sqp algorithm local primal dual quadratic convergence can be established under the assumptions of uniqueness of the lagrange multiplier associated to the solution and the second order sufficient condition at the same time previous primal q superlinear convergence results for sqp required strengthening of the first assumption to the linear independence constraint qualification in this paper we show that this strengthening of assumptions is actually not necessary specifically we show that once primal dual convergence is assumed or already established for primal superlinear rate one needs only a certain error bound estimate this error bound holds for example under the second order sufficient condition which is needed for primal dual local analysis in any case moreover in some situations even second order sufficiency can be relaxed to the weaker assumption that the multiplier in question is noncritical our study is performed for a rather general perturbed sqp framework which covers in addition to sqp and quasi newton sqp some other algorithms as well for example as a byproduct we obtain primal q superlinear convergence results for the linearly constrained augmented lagrangian methods for which no primal q superlinear rate of convergence results were previously available another application of the general framework is sequential quadratically constrained quadratic programming methods finally we discuss some difficulties with proving primal superlinear convergence for the stabilized version of sqp

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

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

critical solutions of nonlinear equations stability issues it is known that when the set of lagrange multipliers associated with a stationary point of a constrained optimization problem is not a singleton this set may contain so called critical multipliers this special subset of lagrange multipliers defines to a great extent stability pattern of the solution in question subject to parametric perturbations criticality of a lagrange multiplier can be equivalently characterized by the absence of the local lipschitzian error bound in terms of the natural residual of the optimality system in this work taking the view of criticality as that associated to the error bound we extend the concept to general nonlinear equations not necessarily with primal dual optimality structure among other things we show that while singular noncritical solutions of nonlinear equations can be expected to be stable only subject to some poor asymptotically thin classes of perturbations critical solutions can be stable under rich classes of perturbations this fact is quite remarkable considering that in the case of nonisolated solutions critical solutions usually form a thin subset within all the solutions we also note that the results for general equations lead to some new insights into the properties of critical lagrange multipliers i e solutions of equations with primal dual structure

a quasi newton strategy for the ssqp method for variational inequality and optimization problems the quasi newton strategy presented in this paper preserves one of the most important features of the stabilized sequential quadratic programming method the local convergence without constraint qualifications assumptions it is known that the primal dual sequence converges quadratically assuming only the second order sufficient condition in this work we show that if the matrices are updated by performing a minimization of a bregman distance which includes the classic updates the quasi newton version of the method converges superlinearly without introducing further assumptions also we show that even for an unbounded lagrange multipliers set the generated matrices satisfies a bounded deterioration property and the dennis more condition

multiplier convergence in trust region methods with application to convergence of decomposition methods for mpecs we study piecewise decomposition methods for mathematical programs with equilibrium constraints mpecs for which all constraint functions are linear at each iteration of a decomposition method one step of a nonlinear programming scheme is applied to one piece of the mpec to obtain the next iterate our goal is to understand global convergence to b stationary points of these methods when the embedded nonlinear programming solver is a trust region scheme and the selection of pieces is determined using multipliers generated by solving the trust region subproblem to this end we study global convergence of a linear trust region scheme for linearly constrained nlps that we call a trust search method the trust search has two features that are critical to global convergence of decomposition methods for mpecs a robustness property with respect to switching pieces and a multiplier convergence result that appears to be quite new for trust region methods these combine to clarify and strengthen global convergence of decomposition methods without resorting either to additional conditions such as eventual inactivity of the trust region constraint or more complex methods that require a separate subproblem for multiplier estimation

a globally convergent lp newton method we develop a globally convergent algorithm based on the lp newton method which has been recently proposed for solving constrained equations possibly nonsmooth and possibly with nonisolated solutions the new algorithm makes use of linesearch for the natural merit function and preserves the strong local convergence properties of the original lp newton scheme we also present computational experiments on a set of generalized nash equilibrium problems and a comparison of our approach with the previous hybrid globalization employing the potential reduction method

convergence analysis of a regularized interior point algorithm for the barrier problems with singular solutions abstract wachter and biegler 2006 25 presented and implemented an interior point filter line search method called ipopt for large scale nonlinear programming to make ipopt efficient and robust they add some safeguards in their implementation one crucial safeguard is to apply the inertia correction technique to modify the kkt matrix we focus on the degenerate problems in which the kkt matrix might be singular to complement theoretical foundation of this technique in the degenerate case we propose and analyze a regularized interior point method for the barrier problems with singular solutions which largely relies on ipopt based on the inertia correction technique the matrix arising in the kkt equation is corrected to be a nonsingular one so that the newton step can be calculated at each iteration even in the degenerate case we emphasize that the special inertia correction for the kkt matrix does not impose any negative impact on the fast convergence of our algorithm under some local error bound condition we prove that the rate of quadratic convergence is achieved when the second order condition or some constraint qualification condition fails to hold or both

a feasible active set qp free method for nonlinear programming we propose a monotone descent active set qp free method for inequality constrained optimization that ensures the feasibility of all iterates and allows for iterates on the boundary of the feasible set the study is motivated by the facchinei fischer kanzow active set identification technique for nonlinear programming and variational inequalities f facchinei a fischer and c kanzow siam j optim 9 1999 pp 14 32 distinguishing features of the proposed method compared with existing qp free methods include lower subproblem costs and a fast convergence rate under milder assumptions specifically four reduced linear systems with a common coefficient matrix involving only constraints in a working set are solved at each iteration to determine the working set the method makes use of multipliers from the last iteration eliminating the need to compute a new estimate and no additional linear systems are solved to select linearly independent constraint gradients a new technique is presented to avoid possible ill conditioned newton systems caused by dual degeneracy it is shown that the method converges globally to kkt points under the linear independence constraint qualification licq and the asymptotic rate of convergence is q superlinear under an additional strong second order sufficient condition ssosc without strict complementarity