a smoothing newton method for a type of inverse semi definite quadratic programming problem we consider an inverse problem arising from the semi definite quadratic programming sdqp problem we represent this problem as a cone constrained minimization problem and its dual denoted isdqd is a semismoothly differentiable sc 1 convex programming problem with fewer variables than the original one the karush kuhn tucker conditions of the dual problem isdqd can be formulated as a system of semismooth equations which involves the projection onto the cone of positive semi definite matrices a smoothing newton method is given for getting a karush kuhn tucker point of isdqd the proposed method needs to compute the directional derivative of the smoothing projector at the corresponding point and to solve one linear system per iteration the quadratic convergence of the smoothing newton method is proved under a suitable condition numerical experiments are reported to show that the smoothing newton method is very effective for solving this type of inverse quadratic programming problems

an interior point constrained trust region method for a special class of nonlinear semidefinite programming problems in this paper an interior point trust region algorithm for the solution of a class of nonlinear semidefinite programming sdp problems is described and analyzed such nonlinear and nonconvex programs arise e g in the design of optimal static or reduced order output feedback control laws and have the structure of abstract optimal control problems in a finite dimensional hilbert space the algorithm treats the abstract states and controls as independent variables in particular an algorithm for minimizing a nonlinear matrix objective functional subject to a nonlinear sdp condition a positive definiteness condition and a nonlinear matrix equation is considered the algorithm is designed to take advantage of the structure of the problem it is an extension of an interior point trust region method to nonlinear and nonconvex sdps with a special structure which applies sequential quadratic programming techniques to a sequence of barrier problems and uses trust regions to ensure robustness of the iteration some convergence results are given and finally several numerical examples demonstrate the applicability of the considered algorithm

cartesian p property and its applications to the semidefinite linear complementarity problem we introduce a cartesian p property for linear transformations between the space of symmetric matrices and present its applications to the semidefinite linear complementarity problem sdlcp with this cartesian p property we show that the sdlcp has gus property i e globally unique solvability and the solution map of the sdlcp is locally lipschitzian with respect to input data our cartesian p property strengthens the corresponding p properties of gowda and song 15 and allows us to extend several numerical approaches for monotone sdlcps to solve more general sdlcps namely sdlcps with the cartesian p property in particular we address important theoretical issues encountered in those numerical approaches such as issues related to the stationary points in the merit function approach and the existence of newton directions and boundedness of iterates in the non interior continuation method of chen and tseng 6

analysis of nonsmooth symmetric matrix valued functions with applications to semidefinite complementarity problems for any function f from mathbb r to mathbb r one can define a corresponding function on the space of n n block diagonal real symmetric matrices by applying f to the eigenvalues of the spectral decomposition we show that this matrix valued function inherits from f the properties of continuity local lipschitz continuity directional differentiability frechet differentiability continuous differentiability as well as rho order semismoothness our analysis uses results from nonsmooth analysis as well as perturbation theory for the spectral decomposition of symmetric matrices we also apply our results to the semidefinite complementarity problem addressing some basic issues in the analysis of smoothing semismooth newton methods for solving this problem

a potential reduction newton method for constrained equations extending our previous work t wang r d c monteiro and j s pang math programming 74 1996 pp 159 195 this paper presents a general potential reduction newton method for solving a constrained system of nonlinear equations a major convergence result for the method is established specializations of the method to a convex semidefinite program and a monotone complementarity problem in symmetric matrices are discussed strengthened convergence results are established in the context of these specializations

a modified alternating direction method for convex quadratically constrained quadratic semidefinite programs we propose a modified alternating direction method for solving convex quadratically constrained quadratic semidefinite optimization problems the method is a first order method therefore requires much less computational effort per iteration than the second order approaches such as the interior point methods or the smoothing newton methods in fact only a single inexact metric projection onto the positive semidefinite cone is required at each iteration we prove global convergence and provide numerical evidence to show the effectiveness of this method

on the coerciveness of some merit functions for complementarity problems over symmetric cones abstract one of the popular solution methods for the complementarity problem over symmetric cones is to reformulate it as the global minimization of a certain merit function an important question to be answered for this class of methods is under what conditions the level sets of the merit function are bounded the coerciveness of the merit function in this paper we introduce the generalized weak coerciveness of a continuous transformation under this condition we prove the coerciveness of some merit functions such as the natural residual function the normal map and the fukushima yamashita function for complementarity problems over symmetric cones we note that this is a much milder condition than strong monotonicity used in the current literature

strong semismoothness of eigenvalues of symmetric matrices and its application to inverse eigenvalue problems it is well known that the eigenvalues of a real symmetric matrix are not everywhere differentiable a classical result of ky fan states that each eigenvalue of a symmetric matrix is the difference of two convex functions which implies that the eigenvalues are semismooth functions based on a recent result of the authors it is further proved in this paper that the eigenvalues of a symmetric matrix are strongly semismooth everywhere as an application it is demonstrated how this result can be used to analyze the quadratic convergence of newton s method for solving inverse eigenvalue problems ieps and generalized ieps with multiple eigenvalues

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

lowner s operator and spectral functions in euclidean jordan algebras we study analyticity differentiability and semismoothness of lowner s operator and spectral functions under the framework of euclidean jordan algebras in particular we show that many optimization related classical results in the symmetric matrix space can be generalized within this framework for example the metric projection operator over any symmetric cone defined in a euclidean jordan algebra is shown to be strongly semismooth the research also raises several open questions whose answers would be of strong interest for optimization research

on two interior point mappings for nonlinear semidefinite complementarity problems extending our previous work monteiro and pang 1996 this paper studies properties of two fundamental mappings associated with the family of interior point methods for solving monotone nonlinear complementarity problems over the cone of symmetric positive semidefinite matrices the first of these maps lead to a family of new continuous trajectories which include the central trajectory as a special case these trajectories completely fill up the set of interior feasible points of the problem in the same way as the weighted central paths do the interior of the feasible region of a linear program unlike the approach based on the theory of maximal monotone maps taken by shida and shindoh 1996 and shida shindoh and kojima 1995 our approach is based on the theory of local homeomorphic maps in nonlinear analysis

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

constraint nondegeneracy strong regularity and nonsingularity in semidefinite programming it is known that the karush kuhn tucker kkt conditions of semidefinite programming can be reformulated as a nonsmooth system via the metric projector over the cone of symmetric and positive semidefinite matrices we show in this paper that the primal and dual constraint nondegeneracies the strong regularity the nonsingularity of the b subdifferential of this nonsmooth system and the nonsingularity of the corresponding clarke s generalized jacobian at a kkt point are all equivalent moreover we prove the equivalence between each of these conditions and the nonsingularity of clarke s generalized jacobian of the smoothed counterpart of this nonsmooth system used in several globally convergent smoothing newton methods in particular we establish the quadratic convergence of these methods under the primal and dual constraint nondegeneracies but without the strict complementarity

a partial parallel splitting augmented lagrangian method for solving constrained matrix optimization problems alternating directions methods adms are very effective for solving convex optimization problems with separable structure however when these methods are applied to solve convex optimization problems with three separable operators their convergence results have not been established as yet in this paper we consider a class of constrained matrix optimization problems the problem is first reformulated into a convex optimization problem with three separable operators then it is solved by a proposed partial parallel splitting method the proposed method combines the parallel splitting augmented lagrangian method psalm and the alternating directions method adm and it is referred to as padalm in short the main difference between padalm and psalm is that in padalm two operators are handled first by a parallel method then the third operator and the former two are dealt with by an alternating method finally the convergence result for padalm is established and numerical results are provided to show the efficacy of padalm and its superiority over psalm

complementarity and nondegeneracy in semidefinite programming primal and dual nondegeneracy conditions are defined for semidefinite programming given the existence of primal and dual solutions it is shown that primal nondegeneracy implies a unique dual solution and that dual nondegeneracy implies a unique primal solution the converses hold if strict complementarity is assumed primal and dual nondegeneracy assumptions do not imply strict complementarity as they do in lp the primal and dual nondegeneracy assumptions imply a range of possible ranks for primal and dual solutions x and z this is in contrast with lp where nondegeneracy assumptions exactly determine the number of variables which are zero it is shown that primal and dual nondegeneracy and strict complementarity all hold generically numerical experiments suggest probability distributions for the ranks of x and z which are consistent with the nondegeneracy conditions

on the nesterov todd direction in semidefinite programming we study different choices of search direction for primal dual interior point methods for semidefinite programming problems one particular choice we consider comes from a specialization of a class of algorithms developed by nesterov and todd for certain convex programming problems we discuss how the search directions for the nesterov todd nt method can be computed efficiently and demonstrate how they can be viewed as newton directions this last observation also leads to convenient computation of accelerated steps using the mehrotra predictor corrector approach in the nt framework we also provide an analytical and numerical comparison of several methods using different search directions and suggest that the method using the nt direction is more robust than alternative methods

complementarity forms of theorems of lyapunov and stein and related results abstract the well known lyapunov s theorem in matrix theory continuous dynamical systems asserts that a complex square matrix a is positive stable i e all eigenvalues lie in the open right half plane if and only if there exists a positive definite matrix x such that ax xa is positive definite in this paper we prove a complementarity form of this theorem a is positive stable if and only if for any hermitian matrix q there exists a positive semidefinite matrix x such that ax xa q is positive semidefinite and x ax xa q 0 by considering cone complementarity problems corresponding to linear transformations of the form i s we show that a complex matrix a has all eigenvalues in the open unit disk of the complex plane if and only if for every hermitian matrix q there exists a positive semidefinite matrix x such that x axa q is positive semidefinite and x x axa q 0 by specializing q to i we deduce the well known stein s theorem in discrete linear dynamical systems a has all eigenvalues in the open unit disk if and only if there exists a positive definite matrix x such that x axa is positive definite

robust control via sequential semidefinite programming this paper discusses nonlinear optimization techniques in robust control synthesis with special emphasis on design problems which may be cast as minimizing a linear objective function under linear matrix inequality lmi constraints in tandem with nonlinear matrix equality constraints the latter type of constraints renders the design numerically and algorithmically difficult we solve the optimization problem via sequential semidefinite programming ssdp a technique which expands on sequential quadratic programming sqp known in nonlinear optimization global and fast local convergence properties of ssdp are similar to those of sqp and ssdp is conveniently implemented with available semidefinite programming sdp solvers using two test examples we compare ssdp to the augmented lagrangian method another classical scheme in nonlinear optimization and to an approach using concave optimization

iterated tabu search for identifying community structure in complex networks this paper presents an iterated tabu search denoted by its algorithm for optimizing the modularity of community structure in complex networks the proposed algorithm follows a general framework composed of two phases basic optimization and postrefinement when the basic optimization cannot improve the modularity any more a postrefinement procedure is employed to further optimize the objective function with a global view for both these two phases iterated tabu search algorithm is employed to optimize the objective function computational results show the high effectiveness of the proposed its algorithm compared with six state of the art algorithms in the literature in particular our its algorithm improves the previous best known modularity for several small and medium size networks

semidefinite programs new search directions smoothing type methods and numerical results motivated by some results for linear programs and complementarity problems this paper gives some new characterizations of the central path conditions for semidefinite programs exploiting these characterizations some smoothing type methods for the solution of semidefinite programs are derived the search directions generated by these methods are automatically symmetric and the overall methods are shown to be globally and locally superlinearly convergent under suitable assumptions some numerical results are also included which indicate that the proposed methods are very promising and comparable to several interior point methods moreover the current method seems to be superior to the smoothing method recently proposed by chen and tseng non interior continuation methods for solving semidefinite complementarity problems technical report department of mathematics university of washington seattle 1999

solving variational inequality problems via smoothing nonsmooth reformulations it has long been known that variational inequality problems can be reformulated as nonsmooth equations recently locally high order convergent newton methods for nonsmooth equations have been well established via the concept of semismoothness when the constraint set of the variational inequality problem is a rectangle several locally convergent newton methods for the reformulated nonsmooth equations can also be globalized in this paper our main aim is to provide globally and locally high order convergent newton methods for solving variational inequality problems with general constraints to achieve this we first prove via convolution that these nonsmooth equations can be well approximated by smooth equations which have desirable properties for the design of newton methods we then reformulate the variational inequality problems as equivalent smoothing nonsmooth equations and apply newton type methods to solve the latter systems and so the variational inequality problems stronger convergence results have been obtained

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

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

interior point methods for the monotone semidefinite linear complementarity problem in symmetric matrices the sdlcp semidefinite linear complementarity problem in symmetric matrices introduced in this paper provides a unified mathematical model for various problems arising from systems and control theory and combinatorial optimization it is defined as the problem of finding a pair x y of n times n symmetric positive semidefinite matrices which lies in a given n n 1 2 dimensional affine subspace fc of sc 2 and satisfies the complementarity condition x bullet y 0 where sc denotes the n n 1 2 dimensional linear space of symmetric matrices and x bullet y the inner product of x and y the problem enjoys a close analogy with the lcp in the euclidean space in particular the central trajectory leading to a solution of the problem exists under the nonemptiness of the interior of the feasible region and a monotonicity assumption on the affine subspace fc the aim of this paper is to establish a theoretical basis of interior point methods with the use of newton directions toward the central trajectory for the monotone sdlcp

strong semismoothness of the fischer burmeister sdc and soc complementarity functions we show that the fischer burmeister complementarity functions associated to the semidefinite cone sdc and the second order cone soc respectively are strongly semismooth everywhere interestingly enough the proof relys on a relationship between the singular value decomposition of a nonsymmetric matrix and the spectral decomposition of a symmetric matrix

search directions in the sdp and the monotone sdlcp generalization and inexact computation we propose a family of directions that generalizes many directions proposed so far in feasible interior point methods for the sdp semidefinite programming and for the monotone sdlcp semidefinite linear complementarity problem we derive the family from the helmberg rendl vanderbei wolkowicz kojima shindoh hara monteiro direction by relaxing its centrality equation into a centrality inequality using this family we present a predictor corrector infeasible interior point method to provide a theoretical basis for inexact computation of directions in primal dual interior point methods for the sdp

differentiability and semismoothness properties of integral functions and their applications in this paper we study differentiability and semismoothness properties of functions defined as integrals of parameterized functions we also discuss applications of the developed theory to the problems of shape preserving interpolation option pricing and semi infinite programming

a note on the lipschitz continuity of the gradient of the squared norm of the matrix valued fischer burmeister function based on a formula of tseng we show that the squared norm of the matrix valued fischer burmeister function has a lipschitz continuous gradient

inexact non interior continuation method for solving large scale monotone sdcp for exact newton method for solving monotone semidefinite complementarity problems sdcp one needs to exactly solve a linear system of equations at each iteration for problems of large size solving the linear system of equations exactly can be very expensive in this paper we propose a new inexact smoothing continuation algorithm for solution of large scale monotone sdcp at each iteration the corresponding linear system of equations is solved only approximately under mild assumptions the algorithm is shown to be both globally and superlinearly convergent

quadratic convergence of a nonsmooth newton type method for semidefinite programs without strict complementarity we consider a newton type method for the solution of semidefinite programs this newton type method is based on a semismooth reformulation of the semidefinite program as a nonsmooth system of equations we establish local quadratic convergence of this method under a linear independence assumption and a weak nondegeneracy condition that implies uniqueness of the optimal solution but does not imply strict complementarity

analysis of nonsmooth vector valued functions associated with second order cones let open image in new window be the lorentz second order cone in open image in new window for any function f from open image in new window to open image in new window one can define a corresponding function fsoc x on open image in new window by applying f to the spectral values of the spectral decomposition of x open image in new window with respect to open image in new window we show that this vector valued function inherits from f the properties of continuity local lipschitz continuity directional differentiability frechet differentiability continuous differentiability as well as order semismoothness these results are useful for designing and analyzing smoothing methods and nonsmooth methods for solving second order cone programs and complementarity problems

semismooth matrix valued functions matrix valued functions play an important role in the development of algorithms for semidefinite programming problems this paper studies generalized differential properties of such functions related to nonsmooth smoothing newton methods the first part of this paper discusses basic properties such as the generalized derivative rademacher s theorem b derivative directional derivative and semismoothness the second part shows that the matrix absolute value function the matrix semidefinite projection function and the matrix projective residual function are strongly semismooth

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

a predictor corrector interior point algorithm for the semidefinite linear complementarity problem using the alizadeh haeberly overton search direction this paper proposes a globally convergent predictor corrector infeasible interior point algorithm for the monotone semidefinite linear complementarity problem using the alizadeh haeberly overton search direction and shows its quadratic local convergence under the strict complementarity condition

semismooth homeomorphisms and strong stability of semidefinite and lorentz complementarity problems based on an inverse function theorem for a system of semismooth equations this paper establishes several necessary and sufficient conditions for an isolated solution of a complementarity problem defined on the cone of symmetric positive semidefinite matrices to be strongly regular stable we show further that for a parametric complementarity problem of this kind if a solution corresponding to a base parameter is strongly stable then a semismooth implicit solution function exists whose directional derivatives can be computed by solving certain affine problems on the critical cone at the base solution similar results are also derived for a complementarity problem defined on the lorentz cone the analysis relies on some new properties of the directional derivatives of the projector onto the semidefinite cone and the lorentz cone

smoothing methods for nonsmooth nonconvex minimization we consider a class of smoothing methods for minimization problems where the feasible set is convex but the objective function is not convex not differentiable and perhaps not even locally lipschitz at the solutions such optimization problems arise from wide applications including image restoration signal reconstruction variable selection optimal control stochastic equilibrium and spherical approximations in this paper we focus on smoothing methods for solving such optimization problems which use the structure of the minimization problems and composition of smoothing functions for the plus function x many existing optimization algorithms and codes can be used in the inner iteration of the smoothing methods we present properties of the smoothing functions and the gradient consistency of subdifferential associated with a smoothing function moreover we describe how to update the smoothing parameter in the outer iteration of the smoothing methods to guarantee convergence of the smoothing methods to a stationary point of the original minimization problem

discriminant analysis when all variables are ordered determination of the equation that relates an ordered dependent variable to ordered independent variables is sought one solution non parametric discriminant analysis npd involves obtaining the best monotonic step function by means of a computer search procedure although one can use alternative selection criteria in obtaining the equation the illustrative examples use absolute distance this paper compares the prediction procedures obtained from npd with those from linear discriminant analysis linear regression with and without transformed variables and logistic regression we show that npd is analogous to regression tree analysis with incorporation of ordered variables and monotonicity we use various prediction functions to predict the example data the data using the leave one out technique and a verification set consistently non parametric discriminant analysis performs as good as or better than the tested alternatives