uncorrelated random networks we define a statistical ensemble of nondegenerate graphs i e graphs without multiple connections and self connections between nodes the node degree distribution is arbitrary but the nodes are assumed to be uncorrelated this completes our earlier publication phys rev 64 046118 2001 where trees and degenerate graphs were considered an efficient algorithm generating nondegenerate graphs is constructed the corresponding computer code is available on request finite size effects in scale free graphs i e those where the tail of the degree distribution falls like n beta are carefully studied we find that in the absence of dynamical internode correlations the degree distribution is cut at a degree value scaling like n gamma with gamma min 1 2 1 beta 1 where n is the total number of nodes the consequence is that independently of any specific model the internode correlations seem to be a necessary ingredient of the physics of scale free networks observed in nature

practical aspects of higher order numerical schemes for wave propagation phenomena this paper examines practical issues related to the use of compact difference based fourth and sixth order schemes for wave propagation phenomena with focus on maxwell s equations of electromagnetics an outline of the formulation and scheme optimization is followed by an assessment of the error accruing from application on stretched meshes with two approaches transformed plane method and physical space differencing in the first technique the truncation error expansion for the sixth order compact scheme confirms that the order of accuracy is preserved if a consistent mesh refinement strategy is followed and further that metrics should be evaluated numerically even if analytic expressions are available physical space differencing formulas are derived for the five point stencil by expressing the coefficients in terms of local spacing ratios the order of accuracy of the reconstruction operator is then verified with a numerical experiment on stretched meshes to ensure stability for a broad range of problems fourier analysis is employed to develop a single parameter family of up to tenth order tridiagonal based spatial filters the implementation of these filters is discussed in terms of their effect on the interior scheme as well as in a 1 d cavity where they are employed to suppress a late time instability the paper concludes after demonstrating the application of the scheme to several 3 d canonical problems utilizing cartesian as well as curvilinear meshes published in 1999 by john wiley sons ltd this article is a u s government work and is in the public domain in the united states

numerical analysis of mathcal v mathcal u decomposition mathcal u gradient and mathcal u hessian approximations advances in bundle methods for nonsmooth optimization have lead to the development of mathcal v mathcal u decompositions the mathcal u gradient and the mathcal u hessian these variational analysis constructs have proven extremely valuable and lead to the development of the superlinearly convergent mathcal v mathcal u algorithm for nonsmooth optimization in this paper we examine these constructs from the viewpoint of derivative free optimization we show that given a finite max function f x max i 0 1 ldots m f i x and a black box which returns function values for each f i it is possible to construct approximations of the mathcal v mathcal u decompositions mathcal u gradient and mathcal u hessian the approximations do not require excessive black box calls and the accuracy of the approximations is directly related to the accuracy of the approximate gradient and hessians for each f i

uneven order decentered shapiro filters for boundary filtering this paper addresses the use of shapiro filters for boundary filtering a new class of uneven order decentered shapiro filters is proposed and compared to classical shapiro filters and even order decentered shapiro filters the theoretical analysis shows that the proposed boundary filters are more accurate than the centered shapiro filters and more robust than the even order decentered boundary filters usable at the same distance to the boundary the benefit of the new boundary filters is assessed for computations using the compressible euler equations

second order characterizations of tilt stability with applications to nonlinear programming the paper is devoted to the study of tilt stable local minimizers of general optimization problems in finite dimensional spaces and its applications to classical nonlinear programs with twice continuously differentiable data the importance of tilt stability has been well recognized from both theoretical and numerical aspects of optimization and this notion has been extensively studied in the literature based on advanced tools of second order variational analysis and generalized differentiation we develop a new approach to tilt stability which allows us to derive not only qualitative but also quantitative characterizations of tilt stable minimizers with calculating the corresponding moduli the implementation of this approach and general results in the classical framework of nonlinear programming provides complete characterizations of tilt stable minimizers under new second order qualification and optimality conditions

dispersion and group velocity in numerical schemes for three dimensional hydrodynamic equations we investigate dispersion and group velocity relations of numerical schemes for the three dimensional hydrodynamic equations by using the transform function method the numerical methods are developed for two of arakawa s spatial grid types namely b grid and c grid using a spectral method in the vertical direction one of our results is that if the horizontal scale length is greater than the rossby radius of deformation then the b grid is more appropriate than the commonly used c grid

on the relation between u hessians and second order epi derivatives abstract we consider second order objects for a convex function defined as the maximum of a finite number of c2 functions variational analysis yields explicit formulae for the second order epi derivatives of such max functions another second order object can be defined by means of a space decomposition that allows us to identify a subspace on which a lagrangian related to a max function is smooth this decomposition yields an explicit expression for the so called u hessian of the function defined as the hessian of the related lagrangian we show that the second order epi derivative relative to the u subspace and the u hessian are equivalent second order objects thus the u lagrangian of a max function captures the function s second order epi differential behavior with ordinary second derivatives

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

priority diffusion model in lattices and complex networks we introduce a model for diffusion of two classes of particles a and b with priority where both species are present in the same site the motion of a s takes precedence over that of b s this describes realistic situations in wireless and communication networks in regular lattices the diffusion of the two species is normal but the b particles are significantly slower due to the presence of the a particles from the fraction of sites where the b particles can move freely which we compute analytically we derive the diffusion coefficients of the two species in heterogeneous networks the fraction of sites where b s are free decreases exponentially with the degree of the sites this coupled with accumulation of particles in high degree nodes leads to trapping of the low priority particles in scale free networks

differentiability of relations and differential stability of perturbed optimization problems several new concepts of differentiability of multifunctions are introduced special attention is paid to relations given by perturbed inequalities applications are given to the study of the value or marginal function of a perturbed nonlinear program

a reduced gravity primitive equation model of the upper equatorial ocean abstract this paper describes a fourth order finite difference model of the equatorial ocean that is designed to study dynamic and thermodynamic processes on time scales of a decade or less it is a primitive equation model employing the reduced gravity assumption so that the deep ocean is at rest below the active upper ocean the model consists of a surface mixed layer and an active layer below which is divided into an arbitrary number of numerical layers by means of a sigma coordinate the model can be used in an unstratified version when temperature acts as a passive tracer as well as in the full stratified version the numerical formulation of the model is described in detail experiments comparing three different horizontal smoothers shapiro filter laplacian friction and biharmonic friction are presented it is concluded that at the level needed to maintain computational stability the shapiro filter damps the fields least in addition it is faster and easier to implement when the horizontal finite difference grid is stretched

an approximate u lagrangian and algorithm to uv decomposition in this paper an concept and some properties of approximate u lagrangian are presented they can be reviewed as an extension to the ones of u lagrangian an algorithm for solving an unconstrained minimization problem and the convergence analysis are given when the uv decomposition exists

high order low dispersive and low dissipative explicit schemes for multiple scale and boundary problems explicit high order numerical schemes are proposed for the accurate computation of multiple scale problems and for the implementation of boundary conditions specific high order node centered finite differences and selective filters removing grid to grid oscillations are first designed for the discretization of the buffer region between a x grid domain and 2 x grid domain the coefficients of these matching schemes are chosen so that the maximum order of accuracy is reached non centered finite differences and selective filters are then developed with the aim of accurately computing boundary conditions they are constructed by minimizing the dispersion and the dissipation errors in the wave number space for waves down to four points per wavelength the dispersion and dissipation properties of the matching and the boundary schemes are described in detail and their accuracy limits are determined to show that these schemes calculate accurately waves with at least five points per wavelength test problems including linear convection wall reflection and acoustic scattering around a cylinder are finally solved to illustrate the accuracy of the schemes

random walk with priorities in communicationlike networks we study a model for a random walk of two classes of particles a and b where both species are present in the same site the motion of a s takes precedence over that of b s the model was originally proposed and analyzed in maragakis et al phys rev e 77 020103 r 2008 here we provide additional results we solve analytically the diffusion coefficients of the two species in lattices for a number of protocols in networks we find that the probability of a b particle to be free decreases exponentially with the node degree in scale free networks this leads to localization of the b s at the hubs and arrest of their motion to remedy this we investigate several strategies to avoid trapping of the b s including moving an a instead of the hindered b allowing a trapped b to hop with a small probability biased walk toward non hub nodes and limiting the capacity of nodes we obtain analytic results for lattices and networks and we discuss the advantages and shortcomings of the possible strategies

a semi spectral primitive equation ocean circulation model using vertical sigma and orthogonal curvilinear horizontal coordinates abstract we describe a new diabatic primitive equation model for studying regional and basin scale ocean circulation processes the model features coordinate transformations that efficiently incorporate moderately irregular basin geometries and large variations in bottom topography and permits the inclusion of both thermal and wind forcing a novel semi spectral solution procedure in which the vertical structure of the model variables is represented as a finite sum of user specifiable structure functions e g chebyshev polynomials provides faster than algebraic convergence of the vertical approximation scheme model performance is assessed on a variety of test problems drawn from coastal and large scale oceanography including unforced linear wave propagation in both regular and irregular geometries non linear flow over rough bottom topography and eddy mean flow interaction in a wind driven midlatitude ocean basin computational efficiency of the model is found to be comparable to other existing primitive equation ocean models despite the utilization of the higher order spectral methods

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

composite proximal bundle method we consider minimization of nonsmooth functions which can be represented as the composition of a positively homogeneous convex function and a smooth mapping this is a sufficiently rich class that includes max functions largest eigenvalue functions and norm 1 regularized functions the bundle method uses an oracle that is able to compute separately the function and subgradient information for the convex function and the function and derivatives for the smooth mapping with this information it is possible to solve approximately certain proximal linearized subproblems in which the smooth mapping is replaced by its taylor series linearization around the current serious step our numerical results show the good performance of the composite bundle method for a large class of problems

necessary and sufficient optimality conditions for a class of nonsmooth minimization problems the purpose of this paper is to derive in a unified way second order necessary and sufficient optimality criteria for four types of nonsmooth minimization problems thediscrete minimax problem thediscrete l1 approximation the minimization of theexact penalty function and the minimization of theclassical exterior penalty function our results correct and supplement conditions obtained by various authors in recent papers

tilt stability uniform quadratic growth and strong metric regularity of the subdifferential we prove that uniform second order growth tilt stability and strong metric regularity of the subdifferential three notions that have appeared in entirely different settings are all essentially equivalent for any lower semicontinuous extended real valued function

adaptive grid refinement for numerical weather prediction abstract an adaptive atmospheric flow model is described and results of integrations with this model are presented the adaptive technique employed is that of berger and oliger the technique uses a finite difference method to integrate the dynamical equations first on a coarse grid and then on finer grids which have been placed based on a richardson type estimate of the truncation error in the coarse grid solution by correctly coupling the integrations on the various grids periodically re estimating the error and recreating the finer grids uniformily accurate solutions are economically produced the primitive hydrostatic equations of meteorology are solved for the advection of a barotropic cyclone and for the development of a baroclinic disturbance which results from the perturbation of an unstable jet these integrations demonstrate the feasibility of using multiple rotated overlapping fine grids direct computations of the truncation error are used to confirm the accuracy of the richardson type truncation error estimates

a family of low dispersive and low dissipative explicit schemes for flow and noise computations explicit numerical methods for spatial derivation filtering and time integration are proposed they are developed with the aim of computing flow and noise with high accuracy and fidelity all the methods are constructed in the same way by minimizing the dispersion and the dissipation errors in the wavenumber space up to k x 2 corresponding to four points per wavelength they are shown to be more accurate and also more efficient numerically than most of the standard explicit high order methods for uniform and slowly non uniform grids two problems involving long range sound propagation are resolved to illustrate their respective precisions remarks about their practical applications are then made especially about the connection with the boundary conditions finally their relevance for the simulation of turbulent flows is emphasized

active sets nonsmoothness and sensitivity nonsmoothness pervades optimization but the way it typically arises is highly structured nonsmooth behavior of an objective function is usually associated locally with an active manifold on this manifold the function is smooth whereas in normal directions it is vee shaped active set ideas in optimization depend heavily on this structure important examples of such functions include the pointwise maximum of some smooth functions and the maximum eigenvalue of a parametrized symmetric matrix among possible foundations for practical nonsmooth optimization this broad class of partly smooth functions seems a promising candidate enjoying a powerful calculus and sensitivity theory in particular we show under a natural regularity condition that critical points of partly smooth functions are stable small perturbations to the function cause small movements of the critical point on the active manifold

the u lagrangian of the maximum eigenvalue function in this paper we apply the u lagrangian theory to the maximum eigenvalue function la and to its precomposition with affine matrix valued mappings we first give geometrical interpretations of the u objects that we introduce we also show that the u lagrangian of la has a hessian which can be explicitly computed the second order development of the u lagrangian provides a second order development of la along a characteristic smooth manifold the set of symmetric matrices whose maximal eigenvalues have a fixed multiplicity the same results can be obtained when we precompose la with an affine matrix valued mapping a provided that this mapping satisfies a regularity condition em transversality condition we show that the hessian of the u lagrangian of la circ a coincides with the reduced hessian encountered in sequential quadratic programming finally we use the u lagrangian to derive second order algorithms for minimizing la circ a

some sufficient optimality conditions in nonsmooth analysis in this paper we study some second order conditions that may be added to the first order nessessary optimality condition 0 in partial p f bar x with partial p f denoting the proximal subdifferential in order to obtain a sufficient condition for a strict local minimum for extended real valued nonsmooth functions three different types of second order conditions are investigated all based on a different second order subdifferential namely the subhessian the graphical derivative and the contingent coderivative to the proximal subdifferential

partial smoothness tilt stability and generalized hessians we compare two recent variational analytic approaches to second order conditions and sensitivity analysis for nonsmooth optimization we describe a broad setting where computing the generalized hessian of mordukhovich is easy in this setting the idea of tilt stability introduced by poliquin and rockafellar is equivalent to a classical smooth second order condition

identifiable surfaces in constrained optimization the concept of a class c p identifiable surface of a convex set in euclidean space is introduced the paper shows how the smoothness of these surfaces is related to the smoothness of the projection operator and presents finite identification results for certain algorithms for minimization of a function over this set the work uses a partially geometric view of constrained optimization to generalize previous finite identification results

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

primal dual gradient structured functions second order results links to epi derivatives and partly smooth functions we give second order expansions for quite general nonsmooth functions from the cal v cal u space decomposition point of view the results depend on primal dual gradient structure which we relate to general concepts of second order epi derivatives and partly smooth functions expressions for the associated second order objects are given in terms of cal u subspace hessians

active set newton methods and partial smoothness diverse optimization algorithms correctly identify in finite time intrinsic constraints that must be active at optimality analogous behavior extends beyond optimization to systems involving part

limiting subhessians limiting subjets and their calculus we study calculus rules for limiting subjets of order two these subjets are obtained as limits of sequences of subjets a subjet of a function f at some point x being the taylor expansion of a twice differentiable function which minorizes f and coincides with f at x these calculus rules are deduced from approximate or fuzzy calculus rules for subjets of order two in turn these rules are consequences of delicate results of crandall ishii lions we point out the similarities and the differences with the case of first order limiting subdifferentials

newton methods for nonsmooth convex minimization connections among lagrangian riemannian newton and sqp methods this paper studies newton type methods for minimization of partly smooth convex functions sequential newton methods are provided using local parameterizations obtained from lagrangian theory and from riemannian geometry the hessian based on the lagrangian depends on the selection of a dual parameter g by revealing the connection to riemannian geometry a natural choice of g emerges for which the two newton directions coincide this choice of g is also shown to be related to the least squares multiplier estimate from a sequential quadratic programming sqp approach and with this multiplier sqp gives the same search direction as the newton methods

on a class of nonsmooth composite functions we discuss in this paper a class of nonsmooth functions which can be represented in a neighborhood of a considered point as a composition of a positively homogeneous convex function and a smooth mapping which maps the considered point into the null vector we argue that this is a sufficiently rich class of functions and that such functions have various properties useful for purposes of optimization

a generalized second order derivative in nonsmooth optimization in this work a new notion of generalized second order directional derivative and generalized hessian for nonsmooth real valued functions is studied the general properties of these mathematical objects are investigated together with some calculus rules that may facilitate their practical computation two applications of these derivatives in optimization theory are considered first to obtaining necessary and sufficient second order optimality conditions for problems with or without constraints and second to extending the newton method for the minimization of a mathcal c 1 1 function

a multiscale pressure splitting of the shallow water equations abstract direct representation of the free surface in ocean circulation models leads to a number of computational difficulties that are due to the fast time scales associated with free surface waves these fast time scales generally result in severe time step restrictions when the free surface is advanced using an explicit scheme and may result in large phase errors when the free surface is treated implicitly with a large time step a multiple scale analysis of the shallow water equations is used to analyze this stiffness and to guide the construction of a computational methodology that overcomes the associated difficulties specifically we explore a class of fractional step methods that utilize coarsened grids in the propagation of long wave data the behavior of the corresponding schemes is examined in detail in light of one dimensional model problems based on finite difference or spectral element discretizations

exact encounter times for many random walkers on regular and complex networks the exact mean time between encounters of a given particle in a system consisting of many particles undergoing random walks in discrete time is calculated on both regular and complex networks analytical results are obtained both for independent walkers where any number of walkers can occupy the same site and for walkers with an exclusion interaction when no site can contain more than one walker these analytical results are then compared with numerical simulations showing very good agreement

a comparison of two numerical methods for integrating a quasi geostrophic multilayer model of ocean circulations to study the feasibility of using the finite elements method for ocean modelling a quasi geostrophic eddy resolving general circulation model has been driven using either a classical finite differences technique or a finite elements technique this feasibility has been clearly demonstrated and a comparison of the models performances shows that for simulations of realistic basins the finite elements model cpu cost is only four times the finite difference model cost

biased random walks in complex networks the role of local navigation rules we study the biased random walk process in random uncorrelated networks with arbitrary degree distributions in our model the bias is defined by the preferential transition probability which in recent years has been commonly used to study the efficiency of different routing protocols in communication networks we derive exact expressions for the stationary occupation probability and for the mean transit time between two nodes the effect of the cyclic search on transit times is also explored results presented in this paper provide the basis for a theoretical treatment of transport related problems in complex networks including quantitative estimation of the critical value of the packet generation rate

the quadratic sub lagrangian of a prox regular function given a finite valued convex function f the u lagrangian as defined by lemarechal oustry and sagastizabal provides an envelope of f that maintains these properties by selecting the appropriate uv decomposition on one can also ensure that the u lagrangian is not only continuous at the origin but differentiable there this thesis extends these properties to proxregular functions by creating a new envelope which we call the quadratic sublagrangian in further exploration of this envelope it will be demonstrated how to use a quadratic expansion of the quadratic sub lagrangian to create a quadratic expansion for the original function finally some properties that guarantee such an expansion for the quadratic sub lagrangian will be developed acknowledgements and dedication to dr h bell for making me think it would be possible to dr r kerman for making me think it would be interesting and to dr r poliquin for making both these things true i would also like to express my deep gratitude to the natural sciences and engineering research council of canada and the university of alberta for the financial support they have given me throughout my studies finally i would like to thank dr tuszynski and dr van roessel who dedicated much of their time and energy to the improvement of this thesis

robustness of interdependent and interconnected clustered networks abstract in real world most systems show significant clustering and it is more practical to investigate the behaviors of clustered network previous studies are mostly focused on single clustered network and coupled clustered networks with dependency links here we study two clustered networks coupled with both interdependent and interconnected links by introducing generating function of the joint degree distribution when the networks are fully dependent we obtain the analytical solution of giant component p we show rich phase transition phenomena and analyze their behaviors we find that as dependency coupling strength increases the system changes from second order phase transition through hybrid transition to first order phase transition for weak dependency coupling strength q a corresponding to second order phase transition we find that clustering has almost no effect on the robustness of network but for strong dependency coupling strength q a corresponding to first order transition the more clustered system is more vulnerable at the same time we notice that when the system is more clustered the hybrid order region is almost unchangeable the first order region becomes smaller and the second order region is larger additionally we can see that the bigger the clustering coefficient c is the bigger the second order region becomes for the same c the density of connectivity links between networks is higher the second order region becomes smaller and the density of connectivity links within each network is higher the second order region becomes bigger

book review implicit functions and solution mappings a view from variational analysis second edition by a l dontchev and r t rockafellar springer new york 2014

a study of tilt stable optimality and sufficient conditions abstract recent results by eberhard et al 2006 4 and eberhard and wenczel 2009 3 on the interaction of single and double envelope operations of nonsmooth functions and their interaction with second order derivations have been used to study tilt stability of local minima this continues the study begun by poliquin and rockafellar 1998 1 but now armed with new tools we are able to make some new observations we observe that tilt stability entails a local density within the graph of the proximal subderivative of strict local minima order two of the tilted function indeed it also entails the strict local minimality order two of the tilt stable local minimum itself for prox regular subdifferentially continuous functions this density property characterises tilt stability

random graphs with arbitrary degree distributions and their applications recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the poisson degree distributions that have been widely studied in the past in this paper we develop in detail the theory of random graphs with arbitrary degree distributions in addition to simple undirected unipartite graphs we examine the properties of directed and bipartite graphs among other results we derive exact expressions for the position of the phase transition at which a giant component first forms the mean component size the size of the giant component if there is one the mean number of vertices a certain distance away from a randomly chosen vertex and the average vertex vertex distance within a graph we apply our theory to some real world graphs including the worldwide web and collaboration graphs of scientists and fortune 1000 company directors we demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world while in others there is a measurable discrepancy between theory and reality perhaps indicating the presence of additional social structure in the network that is not captured by the random graph

conjugation and second order properties of convex functions abstract we single out some second order properties of convex functions that are well behaved with respect to the conjugacy operator as an application we prove that if a convex lower semicontinuous function f r n r has a second order taylor expansion at the origin f x 1 2 ax x o x 2 as x 0 and the matrix a is symmetric and positive definite then the conjugate function f has a second order taylor expansion at the origin too given by f y 1 2 a 1 y y o y 2 as y 0

on the use of higher order finite difference schemes on curvilinear and deforming meshes this study enables the use of very high order finite difference schemes for the solution of conservation laws on stretched curvilinear and deforming meshes to illustrate these procedures we focus on up to 6th order pade type spatial discretizations coupled with up to 10th order low pass filters these are combined with explicit and implicit time integration methods to examine wave propagation and wall bounded flows described by the navier stokes equations it is shown that without the incorporation of the filter application of the high order compact scheme to nonsmooth meshes results in spurious oscillations which inhibit their applicability inclusion of the discriminating low pass high order filter restores the advantages of high order approach even in the presence of large grid discontinuities when three dimensional curvilinear meshes are employed the use of standard metric evaluation procedures significantly degrades accuracy since freestream preservation is violated to overcome this problem a simple technique is adopted which ensures metric cancellation and thus ensures freestream preservation even on highly distorted curvilinear meshes for dynamically deforming grids an effective numerical treatment is described to evaluate expressions containing the time varying transformation metrics with these techniques metric cancellation is guaranteed regardless of the manner in which grid speeds are defined the efficacy of the new procedures is demonstrated by solving several model problems as well as by application to flow past a rapidly pitching airfoil and past a flexible panel

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

necessary and sufficient conditions for isolated local minima of nonsmooth functions we consider the mathematical programming problem find inf f x mid x in c where f is an arbitrary extended real valued function and c a subset of a finite dimensional space we give necessary and sufficient optimality conditions for this problem generalizing previous results of a auslendex this journal 22 1984 pp 239 254

nonlinear programming via an exact penalty function asymptotic analysis in this paper we consider the final stage of a global method to solve the nonlinear programming problem we prove 2 step superlinear convergence in the process of analyzing this asymptotic behavior we compare our method theoretically to the popular successive quadratic programming approach

spectral bundle methods for non convex maximum eigenvalue functions second order methods we study constrained and unconstrained optimization programs for nonconvex maximum eigenvalue functions we show how second order techniques may be introduced as soon as it is possible to reliably guess the multiplicity of the maximum eigenvalue at a limit point we examine in which way standard and projected newton steps may be combined with a nonsmooth first order method to obtain a globally convergent algorithm with a fair chance to local superlinear or quadratic convergence

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

on partial smoothness tilt stability and the mathcal vu decomposition under the assumption of prox regularity and the presence of a tilt stable local minimum we are able to show that a mathcal vu like decomposition gives rise to the existence of a smooth manifold on which the function in question coincides locally with a smooth function

chebyshev like generalized shapiro filters for high accuracy flow computations this paper presents chebyshev like generalized shapiro cs filters with improved spectral like resolution compared to existing generalized shapiro filters these new filters combine the advantages of shapiro filters i e arbitrary accuracy order no dispersion full damping of 2 waves and the advantages of chebyshev filters i e purely dissipative response function with equal ripples satisfying an arbitrary chebyshev criterion in passband thanks to the formalism of generalized shapiro filters general formulas are derived for arbitrary accuracy orders and arbitrary chebyshev criterion a python script is provided in appendix to compute cs filter coefficients computations based on the euler equations assess the benefit of cs filters compared to the standard shapiro filters since cs filters differ from shapiro filters only by their coefficients they can easily and advantageously be implemented in computational solvers already making use of generalized shapiro filters