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

a superlinearly convergent sequential quadratically constrained quadratic programming algorithm for degenerate nonlinear programming we present an algorithm that achieves superlinear convergence for nonlinear programs satisfying the mangasarian fromovitz constraint qualification and the quadratic growth condition this convergence result is obtained despite the potential lack of a locally convex augmented lagrangian the algorithm solves a succession of subproblems that have quadratic objectives and quadratic constraints both possibly nonconvex by the use of a trust region constraint we guarantee that any stationary point of the subproblem induces superlinear convergence which avoids the problem of computing a global minimum we compare this algorithm with sequential quadratic programming algorithms on several degenerate nonlinear programs

nonlinear semidefinite programming sensitivity convergence and an application in passive reduced order modeling we consider the solution of nonlinear programs with nonlinear semidefiniteness constraints the need for an efficient exploitation of the cone of positive semidefinite matrices makes the solution of such nonlinear semidefinite programs more complicated than the solution of standard nonlinear programs this paper studies a sequential semidefinite programming ssp method which is a generalization of the well known sequential quadratic programming method for standard nonlinear programs we present a sensitivity result for nonlinear semidefinite programs and then based on this result we give a self contained proof of local quadratic convergence of the ssp method we also describe a class of nonlinear semidefinite programs that arise in passive reduced order modeling and we report results of some numerical experiments with the ssp method applied to problems in that class

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

a global algorithm for nonlinear semidefinite programming in this paper we propose a global algorithm for solving nonlinear semidefinite programming problems this algorithm inspired by the classic sqp sequentially quadratic programming method modifies the s sdp sequentially semidefinite programming local method by using a nondifferentiable merit function combined with a line search strategy

sextic formalism in anisotropic elasticity for almost non semisimple matrix n abstract the sextic formalism of stroh for anisotropic elasticity leads to the eigen relation n p in which n is a 6 6 real matrix the orthogonality and closure relations as well as many other relations involving the eigenvalues p and the eigenvectors x are based on the assumption that n is simple or semisimple so that the six eigenvectors span a six dimensional space problems arise when n is non semisimple in fact there are problems even when n is almost non semisimple we present a modified formalism which is valid regardless of whether n is simple almost non semisimple or non semisimple the modified formalism does not apply when n is semisimple

nonlinear robust optimization via sequential convex bilevel programming in this paper we present a novel sequential convex bilevel programming algorithm for the numerical solution of structured nonlinear min max problems which arise in the context of semi infinite programming here our main motivation are nonlinear inequality constrained robust optimization problems in the first part of the paper we propose a conservative approximation strategy for such nonlinear and non convex robust optimization problems under the assumption that an upper bound for the curvature of the inequality constraints with respect to the uncertainty is given we show how to formulate a lower level concave min max problem which approximates the robust counterpart in a conservative way this approximation turns out to be exact in some relevant special cases and can be proven to be less conservative than existing approximation techniques that are based on linearization with respect to the uncertainties in the second part of the paper we review existing theory on optimality conditions for nonlinear lower level concave min max problems which arise in the context of semi infinite programming regarding the optimality conditions for the concave lower level maximization problems as a constraint of the upper level minimization problem we end up with a structured mathematical program with complementarity constraints mpcc the special hierarchical structure of this mpcc can be exploited in a novel sequential convex bilevel programming algorithm we discuss the surprisingly strong global and locally quadratic convergence properties of this method which can in this form neither be obtained with existing sqp methods nor with interior point relaxation techniques for general mpccs finally we discuss the application fields and implementation details of the new method and demonstrate the performance with a numerical example

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 relaxed constant positive linear dependence constraint qualification and applications in this work we introduce a relaxed version of the constant positive linear dependence constraint qualification cpld that we call rcpld this development is inspired by a recent generalization of the constant rank constraint qualification by minchenko and stakhovski that was called rcrcq we show that rcpld is enough to ensure the convergence of an augmented lagrangian algorithm and that it asserts the validity of an error bound we also provide proofs and counter examples that show the relations of rcrcq and rcpld with other known constraint qualifications in particular rcpld is strictly weaker than cpld and rcrcq while still stronger than abadie s constraint qualification we also verify that the second order necessary optimality condition holds under rcrcq

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

hamiltonian interpretation of the stroh formalism in anisotropic elasticity stroh s sextic formalism for static problems or steady motions in anisotropic elasticity is a formulation in which the equation of equilibrium motion is written as a system of first order differential equations for the displacement and traction in terms of one of the spatial variables the so called fundamental elasticity matrix n appearing in this formulation has the property that when partitioned as a 2 2 block matrix its 12 and 21 blocks are symmetric matrices and its 11 block is the transpose of its 22 block this property gives rise to a large number of orthogonality and closure relations and is fundamental to the success of the stroh formalism in solving a large variety of problems in general anisotropic elasticity first we show that the matrix n is guaranteed to have the above property by the fact that the stroh formulation is in fact a hamiltonian formulation with one of the spatial variables acting as the time like variable this interpretation provides a much desired guide in dealing with other problems for which the governing equations are different such as incompressible elasticity and problems associated with anisotropic elastic plates as described by the kirchhoff plate theory we show that for the last two problems the hamiltonian interpretation simplifies the derivations significantly leading to a stroh formulation in each case which is equivalent to but much simpler than what is available in the existing literature

optimality condition and complexity analysis for linearly constrained optimization without differentiability on the boundary in this paper we consider the minimization of a continuous function that is potentially not differentiable or not twice differentiable on the boundary of the feasible region by exploiting an interior point technique we present first and second order optimality conditions for this problem that reduces to classical ones when the derivative on the boundary is available for this type of problems existing necessary conditions often rely on the notion of subdifferential or become non trivially weaker than the kkt condition in the twice differentiable counterpart problems in contrast this paper presents a new set of first and second order necessary conditions that are derived without the use of subdifferential and reduce to exactly the kkt condition when twice differentiability holds as a result these conditions are stronger than some existing ones considered for the discussed minimization problem when only non negativity constraints are present to solve for these optimality conditions in the special but important case of linearly constrained problems we present two novel interior point trust region algorithms and show that their worst case computational efficiency in achieving the potentially stronger optimality conditions match the best known complexity bounds since this work considers a more general problem than those in the literature our results also indicate that best known existing complexity bounds are actually held for a wider class of nonlinear programming problems this new development is significant since optimality conditions play a fundamental role in computational optimization and more and more nonlinear and nonconvex problems need to be solved in practice

optimality conditions for nonconvex semidefinite programming this paper concerns nonlinear semidefinite programming problems for which no convexity assumptions can be made we derive first and second order optimality conditions analogous to those for nonlinear programming using techniques similar to those used in nonlinear programming we extend existing theory to cover situations where the constraint matrix is structurally sparse the discussion covers the case when strict complementarity does not hold the regularity conditions used are consistent with those of nonlinear programming in the sense that the conventional optimality conditions for nonlinear programming are obtained when the constraint matrix is diagonal

the rate of convergence of the augmented lagrangian method for nonlinear semidefinite programming we analyze the rate of local convergence of the augmented lagrangian method in nonlinear semidefinite optimization the presence of the positive semidefinite cone constraint requires extensive tools such as the singular value decomposition of matrices an implicit function theorem for semismooth functions and variational analysis on the projection operator in the symmetric matrix space without requiring strict complementarity we prove that under the constraint nondegeneracy condition and the strong second order sufficient condition the rate of convergence is linear and the ratio constant is proportional to 1 c where c is the penalty parameter that exceeds a threshold overline c 0

newton s method may fail to recognize proximity to optimal points in constrained optimization we will show examples in which the primal sequence generated by the newton lagrange method converges to a strict local minimizer of a constrained optimization problem but the gradient of the lagrangian does not tend to zero independently of the choice of the dual sequence

a new sequential optimality condition for constrained nonsmooth optimization we introduce a sequential optimality condition for locally lipschitz constrained nonsmooth optimization verifiable just using derivative information and which holds even in the absence of any con

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

two new weak constraint qualifications and applications we present two new constraint qualifications cqs that are weaker than the recently introduced relaxed constant positive linear dependence rcpld cq rcpld is based on the assumption that many subsets of the gradients of the active constraints preserve positive linear dependence locally a major open question was to identify the exact set of gradients whose properties had to be preserved locally and that would still work as a cq this is done in the first new cq which we call the constant rank of the subspace component crsc cq this new cq also preserves many of the good properties of rcpld such as local stability and the validity of an error bound we also introduce an even weaker cq called the constant positive generator cpg which can replace rcpld in the analysis of the global convergence of algorithms we close this work by extending convergence results of algorithms belonging to all the main classes of nonlinear optimization methods sequential quadratic programming augmented lagrangians

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

octet formalism for kirchhoff anisotropic plates an octet formalism is established for stretching and bending deformations of an anisotropic elastic plate based on the kirchhoff theory the plate is inhomogeneous in the thickness direction and thus includes the laminated plate as a special case by defining a 4 eigenvector associated with two in plane displacements and two rotational angles of a normal of the mid plane and a 4 eigenvector associated with four stress functions the three equilibrium equations of the plate reduce to a standard eight dimensional eigenrelation the fundamental elastic plate matrix defined in this work is found to place the two eigenvectors in reverse order to form the left and right 8 eigenvectors of the matrix the same capability has been observed for the fundamental elasticity matrix in stroh s sextic formalism for generalized plane strain elasticity thus our octet formalism symbolically preserves extensive elegant properties and identities pertaining to the capability e g the orthogonality and closure relations which have been established in the stroh sextic formalism having shown that the eigenvalues cannot be real we provide the eigenvalue representations for special cases of material and geometry the non semisimple case including an isotropic material is briefly discussed

properties of the log barrier function on degenerate nonlinear programs we examine the sequence of local minimizers of the log barrier function for a nonlinear program near a solution at which second order sufficient conditions and the mangasarian fromovitz constraint qualification are satisfied but the active constraint gradients are not necessarily linearly independent when a strict complementarity condition is satisfied we show uniqueness of the local minimizer of the barrier function in the vicinity of the nonlinear program solution and we obtain a semiexplicit characterization of this point when strict complementarity does not hold we obtain several other interesting characterizations in particular an estimate of the distance between the minimizers of the barrier function and the nonlinear program in terms of the barrier parameter and a result about the direction of approach of the sequence of minimizers of the barrier function to the nonlinear programming solution

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

an augmented lagrangian method for optimization problems in banach spaces we propose a variant of the classical augmented lagrangian method for constrained optimization problems in banach spaces our theoretical framework does not require any convexity or second order assumptions and it allows the treatment of inequality constraints with infinite dimensional image space moreover we discuss the convergence properties of our algorithm with regard to feasibility global optimality and kkt conditions some numerical results are given to illustrate the practical viability of the method

first and second order analysis of nonlinear semidefinite programs in this paper we study nonlinear semidefinite programming problems convexity duality and first order optimality conditions for such problems are presented a second order analysis is also given second order necessary and sufficient optimality conditions are derived finally sensitivity analysis of such programs is discussed

spectral projected gradient method with inexact restoration for minimization with nonconvex constraints this work takes advantage of the spectral projected gradient direction within the inexact restoration framework to address nonlinear optimization problems with nonconvex constraints the proposed strategy includes a convenient handling of the constraints together with nonmonotonic features to speed up convergence the numerical performance is assessed by experiments with hard spheres problems pointing out that the inexact restoration framework provides an adequate environment for the extension of the spectral projected gradient method for general nonlinearly constrained optimization

on high order model regularization for constrained optimization in two recent papers regularization methods based on taylor polynomial models for minimization were proposed that only rely on holder conditions on the higher order employed derivatives grapiglia and nesterov considered cubic regularization with a sufficient descent condition that uses the current gradient and resembles the classical armijo s criterion cartis gould and toint used taylor models with arbitrary order regularization and defined methods that tackle convex constraints employing the descent criterion that compares actual reduction with predicted reduction the methods presented in this paper consider general not necessarily taylor models of arbitrary order employ a very mild descent criterion and handle general nonnecessarily convex constraints complexity results are compatible with the ones presented in the papers mentioned above

strict constraint qualifications and sequential optimality conditions for constrained optimization sequential optimality conditions for constrained optimization are necessarily satisfied by local minimizers independently of the fulfillment of constraint qualifications these conditions support the employment of different stopping criteria for practical optimization algorithms on the other hand when an appropriate property on the constraints holds at a point that satisfies a sequential optimality condition such a point also satisfies the karush kuhn tucker conditions those properties will be called strict constraint qualifications in this paper as a consequence for each sequential optimality condition it is natural to ask for its weakest strict associated constraint qualification this problem has been solved in a recent paper for the approximate karush kuhn tucker sequential optimality condition in the present paper we characterize the weakest strict constraint qualifications associated with other sequential optimality conditions that are useful for defining stopping criteria of algorithms i

extended meshfree analysis of transverse and inplane loading of a laminated anisotropic plate of general planform geometry abstract an extended meshfree method is presented for the analysis of a laminated anisotropic plate under elastostatic loading the plate may be of any planform shape with its thickness profile composed of perfectly bonded uniform thickness layers of distinct anisotropic materials both transverse and inplane loads are considered using a first order shear deformation theory for flexural behavior and generalized plane stress for the membrane behavior in this extended meshfree method a rectangular domain is initially considered with the plate of arbitrary geometry inscribed within it a particular solution in the form of an analytic generalized navier solution a compound double fourier series is used to capture the response due to the loading within the rectangular domain then a homogeneous solution by meshfree analysis is added to treat the augmented boundary conditions on the actual contour of the plate these augmented conditions are composed of the prescribed values and that of the particular solution evaluated around the plate s contour concentrated transverse and inplane loads in the form of uniform loads over a very small patch are considered with this generalized navier solution representation when a meshfree portion is added to account for the boundary conditions such solutions constitute the green s functions for the plate the viability of these double fourier series representations is shown by the convergence rates for the kinematic and force moment fields an additional example of a two layer 30 angleply circular plate is given to illustrate the capability of this extended meshfree method

on the dependence of the reproducing kernel on the weight of integration abstract the space of admissible weights i e the weights for which the reproducing kernel of bergman type can be defined is introduced and a sufficient condition for a weight to be an admissible weight is proved it is shown that orthogonal projectors and corresponding reproducing kernels depend analytically on weights explicit formulas for taylor series in both cases are given

on the solution of large scale sdp problems by the modified barrier method using iterative solvers the limiting factors of second order methods for large scale semidefinite optimization are the storage and factorization of the newton matrix for a particular algorithm based on the modified barrier method we propose to use iterative solvers instead of the routinely used direct factorization techniques the preconditioned conjugate gradient method proves to be a viable alternative for problems with a large number of variables and modest size of the constrained matrix we further propose to avoid explicit calculation of the newton matrix either by an implicit scheme in the matrix vector product or using a finite difference formula this leads to huge savings in memory requirements and for certain problems to further speed up of the algorithm

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

the lp boundedness of the bergman projection for a class of bounded hartogs domains abstract we generalize the hartogs triangle to a class of bounded hartogs domains and we prove that the corresponding bergman projection is bounded on l p if and only if p is in the range 2 n n 1 2 n n 1

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

evaluation complexity for nonlinear constrained optimization using unscaled kkt conditions and high order models the evaluation complexity of general nonlinear possibly nonconvex constrained optimization is analyzed it is shown that under suitable smoothness conditions an epsilon approximate first order critical point of the problem can be computed in order o epsilon 1 2 p 1 p evaluations of the problem s functions and their first p derivatives this is achieved by using a two phase algorithm inspired by cartis gould and toint siam j optim 21 2011 pp 1721 1739 siam j optim 23 2013 pp 1553 1574 it is also shown that strong guarantees in terms of handling degeneracies on the possible limit points of the sequence of iterates generated by this algorithm can be obtained at the cost of increased complexity at variance with previous results the epsilon approximate first order criticality is defined by satisfying a version of the kkt conditions with an accuracy that does not depend on the size of the lagrange multipliers

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

effects of finite precision arithmetic on interior point methods for nonlinear programming we show that the effects of finite precision arithmetic in forming and solving the linear system that arises at each iteration of primal dual interior point algorithms for nonlinear programming are benign provided that the iterates satisfy centrality and feasibility conditions of the type usually associated with path following methods when we replace the standard assumption that the active constraint gradients are independent by the weaker mangasarian fromovitz constraint qualification rapid convergence usually is attainable even when cancellation and roundoff errors occur during the calculations in deriving our main results we prove a key technical result about the size of the exact primal dual step this result can be used to modify existing analysis of primal dual interior point methods for convex programming making it possible to extend the superlinear local convergence results to the nonconvex case

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

inexact restoration method for derivative free optimization with smooth constraints a new method is introduced for solving constrained optimization problems in which the derivatives of the constraints are available but the derivatives of the objective function are not the method is based on the inexact restoration framework by means of which each iteration is divided in two phases in the first phase one considers only the constraints in order to improve feasibility in the second phase one minimizes a suitable objective function subject to a linear approximation of the constraints the second phase must be solved using derivative free methods an algorithm introduced recently by kolda lewis and torczon for linearly constrained derivative free optimization is employed for this purpose under usual assumptions convergence to stationary points is proved a computer implementation is described and numerical experiments are presented

local and superlinear convergence of a primal dual interior point method for nonlinear semidefinite programming in this paper we consider a primal dual interior point method for solving nonlinear semidefinite programming problems we propose primal dual interior point methods based on the unscaled and scaled newton methods which correspond to the aho hrvw ksh m and nt search directions in linear sdp problems we analyze local behavior of our proposed methods and show their local and superlinear convergence properties

optimality conditions and global convergence for nonlinear semidefinite programming sequential optimality conditions have played a major role in unifying and extending global convergence results for several classes of algorithms for general nonlinear optimization in this paper we extend theses concepts for nonlinear semidefinite programming we define two sequential optimality conditions for nonlinear semidefinite programming the first is a natural extension of the so called approximate karush kuhn tucker akkt well known in nonlinear optimization the second one called trace akkt is more natural in the context of semidefinite programming as the computation of eigenvalues is avoided we propose an augmented lagrangian algorithm that generates these types of sequences and new constraint qualifications are proposed weaker than previously considered ones which are sufficient for the global convergence of the algorithm to a stationary point

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

a sequential convex semidefinite programming algorithm with an application to multiple load free material optimization a new method for the efficient solution of a class of convex semidefinite programming sdp problems is introduced the method extends the sequential convex programming scp concept to optimization problems with matrix variables the basic idea of the new method is to approximate the original optimization problem by a sequence of subproblems in which nonlinear functions defined in matrix variables are approximated by block separable convex functions the subproblems are semidefinite programs with a favorable structure which can be efficiently solved by existing sdp software the new method is shown to be globally convergent the article is concluded by a series of numerical experiments with free material optimization problems demonstrating the effectiveness of the generalized scp approach

a cone continuity constraint qualification and algorithmic consequences every local minimizer of a smooth constrained optimization problem satisfies the sequential approximate karush kuhn tucker akkt condition this optimality condition is used to define the stopping criteria of many practical nonlinear programming algorithms it is natural to ask for conditions on the constraints under which akkt implies kkt these conditions will be called strict constraint qualifications scqs in this paper we define a cone continuity property ccp that will be shown to be the weakest possible scq its relation to other constraint qualifications will also be clarified in particular it will be proved that ccp is strictly weaker than the constant positive generator constraint qualification

global minimization using an augmented lagrangian method with variable lower level constraints a novel global optimization method based on an augmented lagrangian framework is introduced for continuous constrained nonlinear optimization problems at each outer iteration k the method requires the varepsilon k global minimization of the augmented lagrangian with simple constraints where varepsilon k to varepsilon global convergence to an varepsilon global minimizer of the original problem is proved the subproblems are solved using the bb method numerical experiments are presented

on using the elastic mode in nonlinear programming approaches to mathematical programs with complementarity constraints we investigate the possibility of solving mathematical programs with complementarity constraints mpccs using algorithms and procedures of smooth nonlinear programming although mpccs do not satisfy a constraint qualification we establish sufficient conditions for their lagrange multiplier set to be nonempty mpccs that have nonempty lagrange multiplier sets and satisfy the quadratic growth condition can be approached by the elastic mode with a bounded penalty parameter in this context the elastic mode transforms mpcc into a nonlinear program with additional variables that has an isolated stationary point and local minimum at the solution of the original problem which in turn makes it approachable by sequential quadratic programming sqp algorithms one such algorithm is shown to achieve local linear convergence once the problem is relaxed under stronger conditions we also prove superlinear convergence to the solution of an mpcc using an adaptive elastic mode approach for an sqp algorithm recently analyzed in an mpcc context in r fletcher s leyffer s sholtes and d ralph local convergence of sqp methods for mathematical programs with equilibrium constraints tech report na 210 university of dundee dundee uk 2002 our assumptions are more general since we do not use a critical assumption from that reference in addition we show that the elastic parameter update rule will not interfere locally with the superlinear convergence once the penalty parameter is appropriately chosen

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

on uniqueness of kuhn tucker multipliers in nonlinear programming recently fujiwara han and mangasarian introduced a new constraint qualification which is a slight tightening of the well known mangasarian fromovitz constraint qualification we show that this new qualification is a necessary and sufficient condition for the uniqueness of kuhn tucker multipliers we also show that it implies the satisfaction of second order necessary optimality conditions at a local minimum

weighted bergman projections on the hartogs triangle abstract we prove the l p regularity of the weighted bergman projections on the hartogs triangle where the weights are powers of the distance to the singularity at the boundary the restricted range of p is proved to be sharp by using a two weight inequality on the upper half plane with muckenhoupt weights we can consider a slightly wider class of weights

new sequential optimality conditions for mathematical programs with complementarity constraints and algorithmic consequences in recent years the theoretical convergence of iterative methods for solving nonlinear constrained optimization problems has been addressed using sequential optimality conditions which are satisf

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

weighted and boundary l p estimates for solutions of the equation on lineally convex domains of finite type and applications we obtain sharp weighted estimates for solutions of the equation u f in a lineally convex domain of finite type precisely we obtain estimates in the spaces l p being the distance to the boundary with two different types of hypothesis on the form f first if the data f belongs to l p 1 we have a mixed gain on the index p and the exponent secondly we obtain a similar estimate when the data f satisfies an apropriate anisotropic l p estimate with weight 1 moreover we extend those results to 1 and obtain l p and bmo estimates these results allow us to extend the l p regularity results for weighted bergman projection obtained in cdm14b for convex domains to more general weights

stroh like formalism for the coupled stretching bending analysis of composite laminates due to the two dimensional nature of thin plates the lamination theory considering the composite laminates with in plane and plate bending problems coupling each other is treated in this paper by using complex variable formulation by following the steps of stroh formalism for two dimensional linear anisotropic elasticity a displacement complex variable formalism developed by the other researchers was introduced and re derived in a different but more stroh like way in addition a brand new mixed formalism mixed use of displacements and stresses as basic functions is established to compensate the displacement formalism in order to transfer all the related formulae and mathematical techniques of the stroh formalism to these two formalisms the general solutions for the basic equations of lamination theory and their associated eigenrelations have been purposely arranged in the form of stroh formalism moreover by using the presently developed mixed formalism the explicit expressions for the fundamental matrix and eigenvectors are obtained first time for the most general composite laminates furthermore letting the coupling stiffness vanish the formalism has been reduced to the case of symmetric laminates and checked by a recently developed stroh like formalism for the plate bending problems the comparison between stroh formalism for two dimensional problem stroh like formalism for plate bending problem displacement formalism and mixed formalism is then made at the end of this paper and through their connection some useful relations are obtained

a superlinearly convergent sqp method without boundedness assumptions on any of the iterative sequences this paper is aimed to present a new sequential quadratic programming sqp algorithm for finding a solution to nonlinear constrained programming problems with weak conditions where the improved direction can be yielded by solving one quadratic programming qp and the correction direction can be obtained by solving another qp the main characters of the proposed algorithm are as follows first by limiting infeasibility of sqp iterates the boundedness of the iteration sequence can be obtained in the case of the feasible set being nonempty and bounded as well as the constraint functions being convex second global convergence can be proved under slater constraint qualification cq furthermore superlinear convergence can be ensured under suitable conditions third the proposed algorithm is further improved with a bidirectional line search technique finally some numerical experiments are operated to test the proposed algorithms and the results demonstrate that they are promising

global convergence of augmented lagrangian methods applied to optimization problems with degenerate constraints including problems with complementarity constraints we consider global convergence properties of the augmented lagrangian methods on problems with degenerate constraints with a special emphasis on mathematical programs with complementarity constrai

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

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

an augmented lagrangian method for a class of lmi constrained problems in robust control theory we present a new approach to a class of non convex lmi constrained problems in robust control theory the problems we consider may be recast as the minimization of a linear objective subject to linear matrix inequality lmi constraints in tandem with non convex constraints related to rank deficiency conditions we solve these problems using an extension of the augmented lagrangian technique the lagrangian function combines a multiplier term and a penalty term governing the non convex constraints the lmi constraints due to their special structure are retained explicitly and not included in the lagrangian global and fast local convergence of our approach is then obtained either by an lmi constrained newton type method including line search or by a trust region strategy the method is conveniently implemented with available semi definite programming sdp interior point solvers we compare its performance to the wellknown d k iteration scheme in robust control two test problems are investigated and

weighted estimates for the bergman projection on the hartogs triangle abstract we apply modern techniques of dyadic harmonic analysis to obtain sharp estimates for the bergman projection in weighted bergman spaces our main theorem focuses on the bergman projection on the hartogs triangle the estimates of the operator norm are in terms of a bekolle bonami type constant as an application of the results obtained we give for example an upper bound for the l p norm of the bergman projection on the generalized hartogs triangle h m n in c 2

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

an exact three dimensional solution for normal loading of inhomogeneous and laminated anisotropic elastic plates of moderate thickness an exact three dimensional solution is presented for the deformation and stress distribution in an elliptical plate under uniform normal loading of the lateral surfaces and clamped along its edge the plate is assumed to be of constant moderate thickness and composed of anisotropic elastic material which is inhomogeneous in the through thickness direction but symmetric about the mid plane the only material symmetry assumed is that of reflectional symmetry in planes parallel to the mid plane a transfer matrix method is used which without making any further assumptions gives the exact solution at each point in the plate in terms of the stress and displacement at the mid plane the two dimensional differential equations governing these mid plane values are found to be the same as those for an equivalent homogeneous plate whose constant elastic moduli are determined by appropriate through thickness weighted averages of the inhomogeneous moduli the solution of the two dimensional problem is known for such a plate when subject to the specified surface and edge conditions and yields a closed form analytical solution that satisfies all the governing equations and surface conditions of the full three dimensional elasticity problem with edge displacement conditions satisfied on the mid plane the important special case of an anisotropic laminated plate is given by assuming piecewise constant properties through the thickness

a new sequential optimality condition for constrained optimization and algorithmic consequences necessary first order sequential optimality conditions provide adequate theoretical tools to justify stopping criteria for nonlinear programming solvers sequential optimality conditions are satisfied by local minimizers of optimization problems independently of the fulfillment of constraint qualifications a new condition of this type is introduced in the present paper it is proved that a well established augmented lagrangian algorithm produces sequences whose limits satisfy the new condition practical consequences are discussed

lp regularity of the bergman projection on domains covered by the polydisc abstract if a bounded domain can be covered by the polydisc through a rational proper holomorphic map then the bergman projection is l p bounded for p in a non trivial interval depending on the ramified rational covering this result can be applied to the symmetrized polydisc and to the hartogs triangle with exponent

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

some explicit expressions of stroh like formalism for coupled stretching bending analysis abstract since stroh like formalism for coupled stretching bending analysis preserves essential features of stroh formalism for two dimensional anisotropic elasticity it becomes important to get the corresponding explicit expressions of some important matrices frequently appeared in stroh like formalism in this paper explicit expressions are obtained for the fundamental elasticity matrix n the generalized fundamental elasticity matrix n the material eigenvector matrices a and b and barnett lothe tensors l s and h in these expressions n a and b are presented for the general laminates whereas the other matrices are presented only for the symmetric laminates to show the applicability to the degenerate laminates these expressions are further reduced to the isotropic plates finally they are applied to get the explicit real component form solutions of the green s functions for an infinite isotropic plate

boundary integral equations for general laminated plates with coupled stretching bending deformation the boundary integral equations for the stretching bending coupling analysis of general laminated plates are derived in this paper with the aid of the reciprocal theorem of betti and raleigh no restriction is placed on the location of the point load or moment and so it can be an internal or external point or a smooth or non smooth boundary point the fundamental solutions derived from green s function for an infinite laminated plate subjected to concentrated forces moments are presented in the complex matrix form through the use of some identities converting complex form to real form the free term coefficients which are important for the boundary integral equations are obtained explicitly and expressed in the real form alternative formulae for calculating the free term coefficients are also derived using five rigid body movements by using the explicit real expressions obtained recently for the stroh like formalism of coupled stretching bending analysis the free term coefficients are further reduced to the cases of isotropic plates and are then compared with known expressions published in the literature since stretching and bending decouple for isotropic plates comparison is made separately for the in plane problem and plate bending problem and it is shown that the boundary integral equations derived in this paper agree with previous results for these two special cases

some properties of the augmented lagrangian in cone constrained optimization a large class of optimization problems can be modeled as minimization of an objective function subject to constraints given in a form of set inclusions in this paper we discuss augmented lagrangian duality for such optimization problems we formulate the augmented lagrangian dual problems and study conditions ensuring existence of the corresponding augmented lagrange multipliers we also discuss sensitivity of optimal solutions to small perturbations of augmented lagrange multipliers

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

on the rate of convergence of sequential quadratic programming with nondifferentiable exact penalty function in the presence of constraint degeneracy we analyze the convergence of a sequential quadratic programming sqp method for nonlinear programming for the case in which the jacobian of the active constraints is rank deficient at the solution and or strict complementarity does not hold for some or any feasible lagrange multipliers we use a nondifferentiable exact penalty function and we prove that the sequence generated by an sqp using a line search is locally r linearly convergent if the matrix of the quadratic program is positive definite and constant over iterations provided that the mangasarian fromovitz constraint qualification and some second order sufficiency conditions hold