the primal dual active set strategy as a semismooth newton method this paper addresses complementarity problems motivated by constrained optimal control problems it is shown that the primal dual active set strategy which is known to be extremely efficient for this class of problems and a specific semismooth newton method lead to identical algorithms the notion of slant differentiability is recalled and it is argued that the max function is slantly differentiable in lp spaces when appropriately combined with a two norm concept this leads to new local convergence results of the primal dual active set strategy global unconditional convergence results are obtained by means of appropriate merit functions

the application of the differential operator method to the blume emery griffiths model abstract within an effective field approximation general expressions for evaluating the second order phase transition line and the tricritical point of the anisotropic blume emery griffiths model are obtained by the use of the differential operator technique the phase diagrams and the behavior of the tricritical point are investigated numerically for the honeycomb lattice z 3 and square lattice z 4 we find a new disordered phase which may correspond to the staggered quadrupolar phase predicted in the monte carlo simulation when some conditions are satisfied the phase diagrams for z 3 and z 4 systems exhibit a reentrant behavior for positive values of the uniaxial anisotropy parameter the change of the tricritical point with the value of the reduced biquadratic parameter is also studied for the system with z 3

a godunov type method for the shallow water equations with discontinuous topography in the resonant regime abstract we investigate the riemann problem for the shallow water equations with variable and possibly discontinuous topography and provide a complete description of the properties of its solutions existence uniqueness in the non resonant regime multiple solutions in the resonant regime this analysis leads us to a numerical algorithm that provides one with a riemann solver next we introduce a godunov type scheme based on this riemann solver which is well balanced and of quasi conservative form finally we present numerical experiments which demonstrate the convergence of the proposed scheme even in the resonance regime except in the limiting situation when riemann data precisely belong to the resonance hypersurface

a semismooth newton cg based dual ppa for matrix spectral norm approximation problems we consider a class of matrix spectral norm approximation problems for finding an affine combination of given matrices having the minimal spectral norm subject to some prescribed linear equality and inequality constraints these problems arise often in numerical algebra engineering and other areas such as finding chebyshev polynomials of matrices and fastest mixing markov chain models based on classical analysis of proximal point algorithms ppas and recent developments on semismooth analysis of nonseparable spectral operators we propose a semismooth newton cg based dual ppa for solving the matrix norm approximation problems furthermore when the primal constraint nondegeneracy condition holds for the subproblems our semismooth newton cg method is proven to have at least a superlinear convergence rate we also design efficient implementations for our proposed algorithm to solve a variety of instances and compare its performance with the nowadays popular first order alternating direction method of multipliers admm the results show that our algorithm which is warm started with an initial point obtained from the admm substantially outperforms the pure admm especially for the constrained cases and it is able to solve the problems robustly and efficiently to a relatively high accuracy

up and downgrading the 1 center in a network abstract we study budget constrained network improvement and degrading problems based on the vertex 1 center problem on graphs given a graph with vertex weights and edge lengths the task is to decrease and increase the vertex weights within certain limits such that the optimal 1 center objective value with respect to the new weights is minimized and maximized respectively the upgrading improvement problem is shown to be solvable in o n 2 time provided that the distance matrix is given the downgrading 1 center problem is shown to be strongly np hard on general graphs but can be solved in o n 2 time on trees as byproduct we suggest an algorithm that solves the problem of minimizing over the upper envelope of n piecewise linear functions in o k n time where k is the total number of breakpoints

exact results of a general spin 1 model on 2d decorated lattices by the decoration iteration transformation we find a general relation between interaction parameters of a general spin 1 model on any lattice with unique coordination number and on its decorated lattice we obtain three exact phase transition points of the general spin 1 model on two dimensional decorated lattices in terms of the exact results of the model on corresponding two dimensional regular lattices including the triangle square kagome and honeycomb lattices we find the critical temperatures have no exact relation with the average coordination number and the type of coordination number for both the ising model and the general spin 1 model

a globally and superlinearly convergent algorithm for nonsmooth convex minimization it is well known that a possibly nondifferentiable convex minimization problem can be transformed into a differentiable convex minimization problem by way of the moreau yosida regularization this paper presents a globally convergent algorithm that is designed to solve the latter problem under additional semismoothness and regularity assumptions the proposed algorithm is shown to have a q superlinear rate of convergence

attractor and traveling waves of a fluid with nonlinear diffusion and dispersion abstract this work completes the description of the method of estimating attraction domain for traveling waves for several systems of conservation laws with viscosity and capillarity effects proposed in our earlier work thanh 2010 26 precisely we establish the global existence of traveling waves for an isentropic fluid with nonlinear diffusion and dispersion coefficients the shock wave can be classical or nonclassical interestingly we in particular show the existence of a traveling wave for a given lax shock but rather nonclassical when the straight line connecting the two left hand and right hand states crosses the graph of the pressure function two more times in the middle region furthermore we also discuss all the possibilities of saddle stable saddle saddle or stable stable connections

a class of node based bottleneck improvement problems abstract in this paper we consider a class of node based bottleneck improvement problems with respect to the system v e f where v is a node set e is an edge set and f 2 e is a nonempty family of subsets of e in the node based upgrading model a node v can be upgraded at the expense of c v and such an upgrade reduces weights on all edges incident to v for a given budget b the objective is to upgrade a subset of nodes s v such that the bottleneck weight of the family f is minimum under the constraint that the overall upgrade cost c s does not exceed b the problem includes some np hard cases for the special tree path system v e f in which t v e is a rooted tree and f is a set of paths from the root to the leaves we present an o n log n algorithm where n v we also consider some variations of this problem some related complexity results and algorithms are also presented

inverse polynomial optimization we consider the inverse optimization problem associated with the polynomial program f min f bx colon bx in k and a given current feasible solution y in k we provide a systematic numerical scheme to compute an inverse optimal solution that is we compute a polynomial tilde f which may be of the same degree as f if desired with the following properties a y is a global minimizer of tilde f on k with a putinar s certificate with an a priori degree bound d fixed and b tilde f minimizes vert f tilde f vert which can be the l1 l2 or l norm of the coefficients over all polynomials with such properties computing tilde f d reduces to solving a semidefinite program whose optimal value also provides a bound on how far fy is from the unknown optimal value f the size of the semidefinite program can be adapted to the available computational capabilities moreover if one uses the l1 norm then tilde f takes a simple and explicit canonical form some variations are also discussed

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

the inverse fermat weber problem abstract given n points in the plane with nonnegative weights the inverse fermat weber problem consists in changing the weights at minimum cost such that a prespecified point in the plane becomes the euclidean 1 median the cost is proportional to the increase or decrease of the corresponding weight in case that the prespecified point does not coincide with one of the given n points the inverse fermat weber problem can be formulated as linear program we derive a purely combinatorial algorithm which solves the inverse fermat weber problem with unit cost using o n greedy like iterations where each of them can be done in constant time if the points are sorted according to their slopes if the prespecified point coincides with one of the given n points it is shown that the corresponding inverse problem can be written as convex problem and hence is solvable in polynomial time to any fixed precision

data driven inverse optimization with imperfect information in data driven inverse optimization an observer aims to learn the preferences of an agent who solves a parametric optimization problem depending on an exogenous signal thus the observer seeks the agent s objective function that best explains a historical sequence of signals and corresponding optimal actions we formalize this inverse optimization problem as a distributionally robust program minimizing the worst case risk that the estimated decision i e the decision implied by a particular candidate objective differs from the agent s actual response to a random signal we show that our framework offers attractive out of sample performance guarantees for different prediction errors and that the emerging inverse optimization problems can be reformulated as or approximated by tractable convex programs when the prediction error is measured in the space of objective values a main strength of the proposed approach is that it naturally generalizes to situations where the observer has imperfect information e g when the agent s true objective function is not contained in the space of candidate objectives when the agent suffers from bounded rationality or implementation errors or when the observed signal response pairs are corrupted by measurement noise

global existence of diffusive dispersive traveling waves for general flux functions abstract we establish a global existence of traveling waves for diffusive dispersive conservation laws for locally lipschitz flux functions using lyapunov stability techniques we reduce the global problem of finding traveling waves to considering local behaviors of a stable trajectory of the saddle point

a numerically stable dual method for solving strictly convex quadratic programs an efficient and numerically stable dual algorithm for positive definite quadratic programming is described which takes advantage of the fact that the unconstrained minimum of the objective function can be used as a starting point its implementation utilizes the cholesky and qr factorizations and procedures for updating them the performance of the dual algorithm is compared against that of primal algorithms when used to solve randomly generated test problems and quadratic programs generated in the course of solving nonlinear programming problems by a successive quadratic programming code the principal motivation for the development of the algorithm these computational results indicate that the dual algorithm is superior to primal algorithms when a primal feasible point is not readily available the algorithm is also compared theoretically to the modified simplex type dual methods of lemke and van de panne and whinston and it is illustrated by a numerical example

generalized inverse multiobjective optimization with application to cancer therapy we generalize the standard method of solving inverse optimization problems to allow for the solution of inverse problems that would otherwise be ill posed or infeasible in multiobjective linear optimization given a solution that is not a weakly efficient solution to the forward problem our method generates objective function weights that make the given solution a near weakly efficient solution our generalized inverse optimization model specializes to the standard model when the given solution is weakly efficient and retains the complexity of the underlying forward problem we provide a novel interpretation of our inverse formulation as the dual of the well known benson s method and by doing so develop a new connection between inverse optimization and pareto surface approximation techniques we apply our method to prostate cancer data obtained from princess margaret cancer centre in toronto canada we demonstrate that clinically acceptable treatments can be generated using a small number of objective

an efficient hessian based algorithm for solving large scale sparse group lasso problems the sparse group lasso is a widely used statistical model which encourages the sparsity both on a group and within the group level in this paper we develop an efficient augmented lagrangian method for large scale non overlapping sparse group lasso problems with each subproblem being solved by a superlinearly convergent inexact semismooth newton method theoretically we prove that if the penalty parameter is chosen sufficiently large the augmented lagrangian method converges globally at an arbitrarily fast linear rate for the primal iterative sequence the dual infeasibility and the duality gap of the primal and dual objective functions computationally we derive explicitly the generalized jacobian of the proximal mapping associated with the sparse group lasso regularizer and exploit fully the underlying second order sparsity through the semismooth newton method the efficiency and robustness of our proposed algorithm are demonstrated by numerical experiments on both the synthetic and real data sets

an augmented lagrangian dual optimization approach to the h weighted model updating problem abstract model updating for the quadratic eigenvalue problem aims to update the model q 2 m c k by given eigendata in this paper we consider the h weighted model updating problem which can not only preserve the symmetry and definiteness of the original model but also express our confidence in the original model through assigning different confidence weights we propose an augmented lagrangian dual method for the h weighted model updating problem under some mild assumptions our method is shown to converge at least linearly numerical results illustrate the effectiveness of our method in addition we compare our method with the semi definite programming sdp method numerical results illustrate that when the scale of the model becomes large our method still works but the sdp method failed to converge

the resonant cases and the riemann problem for a model of two phase flows abstract the resonant cases in the riemann problem for a model of two phase flows with resonance are investigated which complete the construction of riemann solutions the model is given by a nonconservative hyperbolic system of balance laws the phase decomposition method is developed to construct solutions in each phase in a relatively separated way but still glued to each other via the contact discontinuities composite wave curves can be built and intersections of wave curves in each phase plane help to determine various configurations of riemann solutions in particular solutions containing multiple waves propagating with the same shock speed are constructed

correlated effective field treatment of spin one ising models abstract a new version of correlated effective field theory is applied to a study of the spin one ising models namely the blume capel and blume emery griffiths modesl using the honmura kaneyoshi differential operator technique this method is illustrated in a honeycomb lattice by investigating the phase diagram the magnetization the correlated effective field parameter and the short range order parameter the theory correctly accounts for many physical quantities and leads to results that are quite superior and in some aspects different to those of the previous works

a newton cg augmented lagrangian method for semidefinite programming we consider a newton cg augmented lagrangian method for solving semidefinite programming sdp problems from the perspective of approximate semismooth newton methods in order to analyze the rate of convergence of our proposed method we characterize the lipschitz continuity of the corresponding solution mapping at the origin for the inner problems we show that the positive definiteness of the generalized hessian of the objective function in these inner problems a key property for ensuring the efficiency of using an inexact semismooth newton cg method to solve the inner problems is equivalent to the constraint nondegeneracy of the corresponding dual problems numerical experiments on a variety of large scale sdp problems with the matrix dimension n up to 4 110 and the number of equality constraints m up to 2 156 544 show that the proposed method is very efficient we are also able to solve the sdp problem fap36 with n 4 110 and m 1 154 467 in the seventh dimacs implementation challenge much more accurately than in previous attempts

the riemann problem for a nonisentropic fluid in a nozzle with discontinuous cross sectional area we present a full investigation of the riemann problem for a nonisentropic polytropic fluid in a nozzle with piecewise constant cross section first we introduce the concept of elementary waves which turn out to make up riemann solutions second we study a procedure to select admissible stationary waves relying on the monotone criterion by projecting all the wave curves in the p u plane we construct riemann solutions existence of riemann solutions can be obtained for large initial data furthermore we establish the uniqueness of riemann solutions in strictly hyperbolic domains our argument can lead to estimate regions where the riemann problem admits a unique solution

relating homogeneous cones and positive definite cones via t algebras t algebras are nonassociative algebras defined by vinberg in the early 1960s for the purpose of studying homogeneous cones vinberg defined a cone k cal a for each t algebra cal a and proved that every homogeneous cone is isomorphic to one such k cal a we relate each t algebra cal a with a space of linear operators in such a way that k cal a is isomorphic to the cone of positive definite self adjoint operators together with vinberg s result we conclude that every homogeneous cone is isomorphic to a slice of a cone of positive definite matrices

a projected newton method for l p norm location problems this paper is concerned with the numerical solution of continuous minisum multifacility location problems involving thel p norm where 1 p x this class of problems is potentially difficult to solve because the objective function is not everywhere diflerentiable after developing conditions that characterize the minimum of the problems under consideration a second order algorithm is presented this algorithm is based on the solution of a finite sequence of linearly constrained subproblems descent directions for these subproblems are obtained by projecting the newton direction onto the corresponding constraint manifold univariate minimization is achieved via a specialized linesearch which recognizes the possibility of first derivative discontinuity and second derivative unboundedness at points along the search direction the algorithm motivated by earlier works of calamai and conn and related to methods recently described by overton and dax is shown to possess both global and quadratic convergence properties degeneracy can complicate the numerical solution of the subproblems this degeneracy is identified and a method for handling it is outlined an implementation of the algorithm that exploits the intrinsic structure of the location problem formulation is then described along with a discussion of numerical results

on an instance of the inverse shortest paths problem the inverse shortest paths problem in a graph is considered that is the problem of recovering the arc costs given some information about the shortest paths in the graph the problem is first motivated by some practical examples arising from applications an algorithm based on the goldfarb idnani method for convex quadratic programming is then proposed and analyzed for one of the instances of the problem preliminary numerical results are reported

weight reduction problems with certain bottleneck objectives abstract this paper is concerned with bottleneck weight reduction problems wrps stated as follows we are given a finite set e a class f of nonempty subsets of e a weight w e r and a cost c e r for each e e c e stands for the cost of reducing weight w e by one unit for each subset f f the bottleneck weight of f is w f min e f w e the weight of the family f is the maximum of w f for all f in f the problem is to determine new weights x e w e such that the weight of f is minimized under the constraint that the overall reduction cost does not exceed a given budget b similarly to capacity expansion problems wrps include np hard problems a wrp can be formulated as a parametric optimization problem over all transversal sets t of the class f this leads to strongly polynomial solution procedures for special systems f in particular we outline a polynomial algorithm in the case when f is the class of all spanning trees in an undirected graph

some reverse location problems abstract the general location problem in a network is to find locations of facilities in a given network such that the total cost is minimum it is very often to have its reverse case in which the facilities have already been located on the network and the problem is how to improve the network within a given budget so that the improved network is as efficient as possible in this paper we present a strongly polynomial algorithm for a reverse location problem in tree networks the method can be extended to solve reverse two or more location problems

inverse chromatic number problems in interval and permutation graphs abstract given a graph g and a positive integer k the inverse chromatic number problem consists in modifying the graph as little as possible so that it admits a chromatic number not greater than k in this paper we focus on the inverse chromatic number problem for certain classes of graphs first we discuss diverse possible versions and then focus on two application frameworks which motivate this problem in interval and permutation graphs the inverse booking problem and the inverse track assignment problem the inverse booking problem is closely related to some previously known scheduling problems we propose new hardness results and polynomial cases the inverse track assignment problem motivates our study of the inverse chromatic number problem in permutation graphs we show how to solve in polynomial time a generalization of the problem with a bounded number of colors

solving log determinant optimization problems by a newton cg primal proximal point algorithm we propose a newton cg primal proximal point algorithm ppa for solving large scale log determinant optimization problems our algorithm employs the essential ideas of ppa the newton method and the preconditioned cg solver when applying the newton method to solve the inner subproblem we find that the log determinant term plays the role of a smoothing term as in the traditional smoothing newton technique focusing on the problem of maximum likelihood sparse estimation of a gaussian graphical model we demonstrate that our algorithm performs favorably compared to existing state of the art algorithms and is much preferred when a high quality solution is required for problems with many equality constraints

a phase decomposition approach and the riemann problem for a model of two phase flows abstract we present a phase decomposition approach to deal with the generalized rankine hugoniot relations and then the riemann problem for a model of two phase flows by investigating separately the jump relations for equations in conservative form in the solid phase we show that the volume fractions can change only across contact discontinuities then we prove that the generalized rankine hugoniot relations are reduced to the usual form it turns out that shock waves and rarefaction waves remain on one phase only and the contact waves serve as a bridge between the two phases by decomposing riemann solutions into each phase we show that riemann solutions can be constructed for large initial data furthermore the riemann problem admits a unique solution for an appropriate choice of initial data

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

some inverse optimization problems under the hamming distance given a feasible solution to a particular combinatorial optimization problem defined on a graph and a cost vector defined on the arcs of the graph the corresponding inverse problem is to disturb the cost vector such that the feasible solution becomes optimal the aim is to optimize the difference between the initial cost vector and the disturbed one this difference can be measured in several ways we consider the hamming distance measuring in how many components two vectors are different where weights are associated to the components general algorithms for the bottleneck or minimax criterion are described and after modification applied to the inverse minimum spanning tree problem the inverse shortest path tree problem and the linear assignment problem

nonnegative inverse eigenvalue problems with partial eigendata in this paper we consider the inverse problem of constructing an n n real nonnegative matrix a from the prescribed partial eigendata we first give the solvability conditions for the inverse problem without the nonnegative constraint and then discuss the associated best approximation problem to find a nonnegative solution we reformulate the inverse problem as a monotone complementarity problem and propose a nonsmooth newton type method for solving its equivalent nonsmooth equation under some mild assumptions the global and quadratic convergence of our method is established we also apply our method to the symmetric nonnegative inverse problem and to the cases of prescribed lower bounds and of prescribed entries numerical tests demonstrate the efficiency of the proposed method and support our theoretical findings

a robust numerical method for approximating solutions of a model of two phase flows and its properties the objective of the present paper is to extend our earlier works on simpler systems of balance laws in nonconservative form such as the model of fluid flows in a nozzle with variable cross section to a more complicated system consisting of seven equations which has applications in the modeling of deflagration to detonation transition in granular materials first we transform the system into an equivalent one which can be regarded as a composition of three subsystems then depending on the characterization of each subsystem we propose a convenient numerical treatment of the subsystem separately precisely in the first subsystem of the governing equations in the gas phase stationary waves are used to absorb the nonconservative terms into an underlying numerical scheme in the second subsystem of conservation laws of the mixture we can take a suitable scheme for conservation laws for the third subsystem of the compaction dynamics equation the fact that the velocities remain constant across solid contacts suggests us to employ the technique of engquist osher s scheme then we prove that our method possesses some interesting properties it preserves the positivity of the volume fractions in both phases and in the gas phase our scheme is capable of capturing equilibrium states preserves the positivity of the density and satisfies the numerical minimum entropy principle numerical tests show that our scheme can provide reasonable approximations for data the supersonic regions but the results are not satisfactory in the subsonic region however the scheme is numerically stable and robust

a dual optimization approach to inverse quadratic eigenvalue problems with partial eigenstructure the inverse quadratic eigenvalue problem iqep arises in the field of structural dynamics it aims to find three symmetric matrices known as the mass the damping and the stiffness matrices such that they are closest to the given analytical matrices and satisfy the measured data the difficulty of this problem lies in the fact that in applications the mass matrix should be positive definite and the stiffness matrix positive semidefinite based on an equivalent dual optimization version of the iqep we present a quadratically convergent newton type method our numerical experiments confirm the high efficiency of the proposed method

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

what shape is your conjugate a survey of computational convex analysis and its applications computational convex analysis algorithms have been rediscovered several times in the past by researchers from different fields to further communications between practitioners we review the field of computational convex analysis which focuses on the numerical computation of fundamental transforms arising from convex analysis current models use symbolic numeric and hybrid symbolic numeric algorithms our objective is to disseminate widely the most efficient numerical algorithms useful for applications in image processing computing the distance transform the generalized distance transform and mathematical morphology operators partial differential equations solving hamilton jacobi equations and using differential equations numerical schemes to compute the convex envelope max plus algebra computing the equivalent of the fast fourier transform multifractal analysis etc the fields of applications include among others computer vision robot navigation thermodynamics electrical networks medical imaging and network communication

a simple min cut algorithm we present an algorithm for finding the minimum cut of an undirected edge weighted graph it is simple in every respect it has a short and compact description is easy to implement and has a surprisingly simple proof of correctness its runtime matches that of the fastest algorithm known the runtime analysis is straightforward in contrast to nearly all approaches so far the algorithm uses no flow techniques roughly speaking the algorithm consists of about v nearly identical phases each of which is a maximum adjacency search

a well balanced approximate riemann solver for compressible flows in variable cross section ducts a well balanced approximate riemann solver is introduced in this paper in order to compute approximations of one dimensional euler equations in variable cross section ducts the interface riemann solver is grounded on the vfroe ncv scheme and it enforces the preservation of riemann invariants of the steady wave the main properties of the scheme are detailed we provide numerical results to assess the validity of the scheme even when the cross section is discontinuous a first series is devoted to analytical test cases and the last results correspond to the simulation of a bubble collapse

convergence rate of newton s method for l 2 spectral estimation in the paper we prove the holder continuous property of the jacobian of the function generated from the dual of the power spectrum estimation problem it follows that the convergence of the newton method for the problem is at least of order where m is the order of the trigonometric bases this result theoretically confirms the numerical observation by potter 1990 and cole and goodrich 1993

semismoothness of solutions to generalized equations and the moreau yosida regularization we show that a locally lipschitz homeomorphism function is semismooth at a given point if and only if its inverse function is semismooth at its image point we present a sufficient condition for the semismoothness of solutions to generalized equations over cone reducible nonpolyhedral convex sets we prove that the semismoothness of solutions to the moreau yosida regularization of a lower semicontinuous proper convex function is implied by the semismoothness of the metric projector over the epigraph of the convex function

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

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

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

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

smoothing methods and semismooth methods for nondifferentiable operator equations we consider superlinearly convergent analogues of newton methods for nondifferentiable operator equations in function spaces the superlinear convergence analysis of semismooth methods for nondifferentiable equations described by a locally lipschitzian operator in rn is based on rademacher s theorem which does not hold in function spaces we introduce a concept of slant differentiability and use it to study superlinear convergence of smoothing methods and semismooth methods in a unified framework we show that a function is slantly differentiable at a point if and only if it is lipschitz continuous at that point an application to the dirichlet problems for a simple class of nonsmooth elliptic partial differential equations is discussed

a general theory of spin one ising models in the correlated effective field approximation abstract a new correlated effective field theory of spin one ising systems namely the blume capel and blume emery griffiths models is developed using exact ising spin identities the honmura kaneyoshi differential operator technique and the concept of correlated effective field in particular the general expressions for evaluating the second order phase transition and the tricritical point of both models are obtained the present method can be applied to a wide class of spin one ising systems

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

solving inverse spanning tree problems through network flow techniques given a solution x and an a priori estimated cost vector c the inverse optimization problem is to identify another cost vector d so that x is optimal with respect to the cost vector d and its deviation from c is minimum in this paper we consider the inverse spanning tree problem on an undirected graph g n a with n nodes and m arcs and where the deviation between c and d is defined by the rectilinear distance between the two vectors that is l1 norm we show that the inverse spanning tree problem can be formulated as the dual of an assignment problem on a bipartite network g0 n0 a0 with math not displayed and math not displayed the bipartite ne twork satisfies the property that n1 n 1 n2 m n 1 and a0 o nm in general n1 n2 using this special structure of the assignment problem we develop a specific implementation of the successive shortest path algorithm that runs in o n3 time we also consider the weighted version of the inverse spanning tree problem in which the objective function is to minimize the sum of the weighted deviations of arcs the weighted inverse spanning tree can be formulated as the dual of the transportation problem using a cost scaling algorithm this transportation problem can be solved in o n2m log nc time where c denotes the largest arc cost in the data finally we consider a minimax version of the inverse spanning tree problem and show that it can be solved in o n2 time

full stability of locally optimal solutions in second order cone programs the paper presents complete characterizations of lipschitzian full stability of locally optimal solutions to second order cone programs socps expressed entirely in terms of their initial data these characterizations are obtained via appropriate versions of the quadratic growth and strong second order sufficient conditions under the corresponding constraint qualifications we also establish close relationships between full stability of local minimizers for socps and strong regularity of the associated generalized equations at nondegenerate points our approach is mainly based on advanced tools of second order variational analysis and generalized differentiation

on the use of an inverse shortest paths algorithm for recovering linearly correlated costs this paper considers the inverse shortest paths problem where arc costs are subject to correlation constraints the motivation for this research arises from applications in traffic modelling and seismic tomography a new method is proposed for solving this class of problems it is constructed as a generalization of the algorithm presented in burton and toint mathematical programming 53 1992 for uncorrelated inverse shortest paths preliminary numerical experience with the new method is presented and discussed

inverse optimization for linearly constrained convex separable programming problems abstract in this paper we study inverse optimization for linearly constrained convex separable programming problems that have wide applications in industrial and managerial areas for a given feasible point of a convex separable program the inverse optimization is to determine whether the feasible point can be made optimal by adjusting the parameter values in the problem and when the answer is positive find the parameter values that have the smallest adjustments a sufficient and necessary condition is given for a feasible point to be able to become optimal by adjusting parameter values inverse optimization formulations are presented with l 1 and l 2 norms these inverse optimization problems are either linear programming when l 1 norm is used in the formulation or convex quadratic separable programming when l 2 norm is used

computational aspects of the inverse single facility location problem on trees under lk norm abstract we consider in this paper the inverse versions of the two popular problems in location theory say the 1 median and the 1 center problems on trees the cost for modifying vertex weights is measured under l k norm for a positive integer k and hence the cost function is nonlinear for the inverse 1 median problem we develop a linear time algorithm based on the optimal solution of the induced unconstrained problem for the inverse 1 center problem we prove that the problem can be decomposed into linearly many subproblems and the objective function in each subproblem is piecewise convex furthermore we also discuss an o n log n time algorithm for the inverse 1 center problem under l 2 norm based on the convexity of the cost function of each subproblem

traveling waves of an elliptic hyperbolic model of phase transitions via varying viscosity capillarity abstract we consider an elliptic hyperbolic model of phase transitions and we show that any lax shock can be approximated by a traveling wave with a suitable choice of viscosity and capillarity by varying viscosity and capillarity coefficients we can cover any lax shock which either remains in the same phase or admits a phase transition the argument used in this paper extends the one in our earlier works the method relies on lasalle s invariance principle and on estimating attraction region of the asymptotically stable of the associated autonomous system of differential equations we will show that the saddle point of this system of differential equations lies on the boundary of the attraction region and that there is a trajectory leaving the saddle point and entering the attraction region this gives us a traveling wave connecting the two states of the lax shock we also present numerical illustrations of traveling waves

algorithm for constraint partial inverse matroid problem with weight increase forbidden in a partial inverse matroid problem given a matroid m s i a real valued weight function w on s and an independent set i 0 i the goal is to modify the weight w as small as possible to a new weight w such that there exists a w maximum base containing i 0 in this paper we study a constraint version of the partial inverse matroid problem in which the weight can only be decreased a polynomial time algorithm is presented under l norm

the inverse optimal value problem this paper considers the following inverse optimization problem given a linear program a desired optimal objective value and a set of feasible cost vectors determine a cost vector such that the corresponding optimal objective value of the linear program is closest to the desired value the above problem referred here as the inverse optimal value problem is significantly different from standard inverse optimization problems that involve determining a cost vector for a linear program such that a pre specified solution vector is optimal in this paper we show that the inverse optimal value problem is np hard in general we identify conditions under which the problem reduces to a concave maximization or a concave minimization problem we provide sufficient conditions under which the associated concave minimization problem and correspondingly the inverse optimal value problem is polynomially solvable for the case when the set of feasible cost vectors is polyhedral we describe an algorithm for the inverse optimal value problem based on solving linear and bilinear programming problems some preliminary computational experience is reported

a godunov type scheme for the isentropic model of a fluid flow in a nozzle with variable cross section we present a godunov type scheme for the isentropic model of a fluid flow in a nozzle with variable cross section the model is of nonconservative form making it hard for standard numerical discretizations of the nonconservative term in particular the error for a standard numerical scheme with a usual discretization of the nonconservative term may become larger as the mesh size gets smaller we first re investigate the riemann problem of the model pointing out several interesting properties of the wave curves and establishing specific existence domain for each type of solutions then we incorporate local riemann solutions to build a godunov type scheme for the model the scheme is constructed in subsonic and supersonic regions where the system is strictly hyperbolic tests show that our scheme can capture standing waves so that it is well balanced furthermore tests also show that our godunov type scheme can give a good accuracy for numerical approximations of exact solutions our godunov type scheme can resolve the difficulty of other existing schemes for similar models of fluid flows with nonconservative terms

a further study on inverse linear programming problems abstract in this paper we continue our previous study zhang and liu j comput appl math 72 1996 261 273 on inverse linear programming problems which requires us to adjust the cost coefficients of a given lp problem as less as possible so that a known feasible solution becomes the optimal one in particular we consider the cases in which the given feasible solution and one optimal solution of the lp problem are 0 1 vectors which often occur in network programming and combinatorial optimization and give very simple methods for solving this type of inverse lp problems besides instead of the commonly used l1 measure we also consider the inverse lp problems under l measure and propose solution methods

lagrangian dual functions and moreau yosida regularization in this paper we consider the lagrangian dual problem of a class of convex optimization problems we first discuss the semismoothness of the lagrangian dual function varphi this property is then used to investigate the second order properties of the moreau yosida regularization eta of the function varphi e g the semismoothness of the gradient g of the regularized function eta we show that varphi and g are piecewise c 2 and semismooth respectively for certain instances of the optimization problem we establish a relationship between the original problem and the fenchel conjugate of the regularization of the corresponding lagrangian dual problem we also find some instances of the optimization problem whose lagrangian dual function varphi is not piecewise smooth however its regularized function still possesses nice second order properties finally we provide an alternative way to study the semismoothness of the gradient under the structure of the epigraph of the dual function

generalized transformation for decorated spin models abstract the paper discusses the transformation of decorated ising models into an effective undecorated spin model using the most general hamiltonian for interacting ising models including a long range and high order interactions the inverse of a vandermonde matrix with equidistant nodes s s is used to obtain an analytical expression of the transformation this kind of transformation is very useful to obtain the partition function of decorated systems the method presented by fisher is also extended in order to obtain the correlation functions of the decorated ising models transforming into an effective undecorated ising model we apply this transformation to a particular mixed spin 1 2 1 and 1 2 2 square lattice with only nearest site interaction this model could be transformed into an effective uniform spin s square lattice with nearest and next nearest interaction furthermore the effective hamiltonian also includes combinations of three body and four body interactions in particular we considered spin 1 and 2

inverse 1 median problem on trees under mixed rectilinear and chebyshev norms abstract we consider in this paper the inverse 1 median problem on trees with variable vertex weights which are partitioned into groups while the modifications in each group are measured by chebyshev norm rectilinear norm is applied for computing the cost of groups and vice versa as a result it yields the sum of max and the max of sum objective functions for the sum of max objective we develop an o m log m time algorithm based on a parameterization approach where m is the number of vertices in the tree on the other hand the problem under the max of sum objective can be solved in linear time by an algorithm that prunes a half of indices in each iteration

constructing a godunov type scheme for the model of a general fluid flow in a nozzle with variable cross section a godunov type scheme for the model of a general fluid flow in a nozzle with variable cross section is presented the model has the form of a nonconservative system of balance laws which poses many challenging questions for study first exact riemann solvers in computing form are described in both subsonic and supersonic regions second the computable exact solutions of local riemann problem are incorporated into a godunov type scheme third the scheme is shown to be well balanced in the sense that it can capture exactly stationary waves finally numerical tests for data belong to both subsonic and supersonic regions are presented these tests show that the scheme has a very fine accuracy especially the scheme can give very good approximations to the exact solutions even in the resonant phenomenon where the solution contains three waves of the same speed

multicritical points and reentrant phenomenon in the beg model abstract the blume emery griffiths model is investigated by use of the cluster variation method in the pair approximation we determine the regions of the phase space where the reentrant phenomenon takes place two regions are found depending on the sign of the reduced quadrupole quadrupole coupling strength for negative we find para ferro para and ferro para ferro para transition sequences for positive a para ferro para sequence order parameters correlation function and specific heat are given in some typical cases by products of this work are the equations for the critical and tricritical lines

on the inverse problem of linear programming and its application to minimum weight perfect k matching we first introduce the inverse problem of linear programming we then apply it to the inverse problem of minimum weight perfect k matching of bipartite graphs we show that there is a strongly polynomial algorithm for solving the problem

calculating some inverse linear programming problems in this paper we consider some inverse lp problems in which we need to adjust the cost coefficients of a given lp problem as less as possible so that a known feasible solution becomes the optimal one a method for solving general inverse lp problem including upper and lower bound constraints is suggested which is based on the optimality conditions for lp problems it is found that when the method is applied to inverse minimum cost flow problem or inverse assignment problem we are able to obtain strongly polynomial algorithms

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

a penalty function method for solving inverse optimal value problem in order to consider the inverse optimal value problem under more general conditions we transform the inverse optimal value problem into a corresponding nonlinear bilevel programming problem equivalently using the kuhn tucker optimality condition of the lower level problem we transform the nonlinear bilevel programming into a normal nonlinear programming the complementary and slackness condition of the lower level problem is appended to the upper level objective with a penalty then we give via an exact penalty method an existence theorem of solutions and propose an algorithm for the inverse optimal value problem also analysis the convergence of the proposed algorithm the numerical result shows that the algorithm can solve a wider class of inverse optimal value problem

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

properties of the moreau yosida regularization of a piecewise c2 convex function in this paper we discuss second order properties of the moreau yosida regularization f of a piecewise twice continuously differentiable convex function f we introduce a new constraint qualification in order to prove that the gradient of f is piecewise continuously differentiable in addition we discuss conditions depending on the hessians of the pieces that guarantee positive definiteness of the generalized jacobians of the gradient of f

a version of the bundle idea for minimizing a nonsmooth function conceptual idea convergence analysis numerical results during recent years various proposals for the minimization of a nonsmooth functional have been made amongst these the bundle concept turned out to be an especially fruitful idea based on this concept a number of authors have developed codes that can successfully deal with nonsmooth problems the aim of the paper is to show that by adding some features of the trust region philosophy to the bundle concept the end result is a distinguished member of the bundle family with a more stable behaviour than some other bundle versions the reliability and efficiency of this code is demonstrated on the standard academic test examples and on some real life problems

new effective field theory for the blume capel model abstract a new type of effective field theory that has recently been used with success for many applications concerning the usual two state ising model is herein extended to the blume capel model the method which can explicitly and systematically include correlation effects is illustrated in several lattice structures by employing its simplest approximate version in which spin spin correlations are neglected it is shown that this approximative procedure although analytically simple provides in particular a vanishing critical temperature for one dimensional systems and leads to results quite superior to those currently obtained within the molecular field approximation for higher dimensional systems within this framework we discuss as functions of temperature and for any values of the anisotropy parameter d the order parameter the thermal average of the square of the single site spin variable the internal energy and the specific heat as well as the ferromagnetic phase stability limit whenever comparison is possible a satisfactory qualitative and to a certain extent quantitative agreement is observed with results available in the literature in which more sophisticated treatments are used in particular our result for the tricritical point at which the system exhibits a first order phase transition is in quite good agreement with those obtained by using series expansion methods

a dual active set algorithm for positive semi definite quadratic programming because of the many important applications of quadratic programming fast and efficient methods for solving quadratic programming problems are valued goldfarb and idnani 1983 describe one such method well known to be efficient and numerically stable the goldfarb and idnani method suffers only from the restriction that in its original form it cannot be applied to problems which are positive semi definite rather than positive definite in this paper we present a generalization of the goldfarb and idnani method to the positive semi definite case and prove finite termination of the generalized algorithm in our generalization we preserve the spirit of the goldfarb and idnani method and extend their numerically stable implementation in a natural way

primal dual interior point algorithm for convex quadratic semi definite optimization abstract in this paper we present a new primal dual interior point algorithm for solving a special case of convex quadratic semi definite optimization based on a parametric kernel function the proposed parametric kernel function is used both for determining the search directions and for measuring the distance between the given iterate and the center for the algorithm these properties enable us to derive the currently best known iteration bounds for the algorithm with large and small update methods namely o n log n log n e and o n log n e respectively which reduce the gap between the practical behavior of the algorithm and its theoretical performance results

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

convergence of newton s method for convex best interpolation in this paper we consider the problem of finding a convex function which interpolates given points and has a minimal l2 norm of the second derivative this problem reduces to a system of equations involving semismooth functions we study a newton type method utilizing clarke s generalized jacobian and prove that its local convergence is superlinear for a special choice of a matrix in the generalized jacobian we obtain the newton method proposed by irvine et al and settle the question of its convergence by using a line search strategy we present a global extension of the newton method considered the efficiency of the proposed global strategy is confirmed with numerical experiments

the strong second order sufficient condition and constraint nondegeneracy in nonlinear semidefinite programming and their implications for a locally optimal solution to the nonlinear semidefinite programming problem under robinson s constraint qualification the following conditions are proved to be equivalent the strong second order sufficient condition and constraint nondegeneracy the nonsingularity of clarke s jacobian of the karush kuhn tucker system the strong regularity of the karush kuhn tucker point and others

a sequential semismooth newton method for the nearest low rank correlation matrix problem based on the well known result that the sum of the largest eigenvalues of a symmetric matrix can be represented as a semidefinite programming problem sdp we formulate the nearest low rank correlation matrix problem as a nonconvex sdp and propose a numerical method that solves a sequence of least square problems each of the least square problems can be solved by a specifically designed semismooth newton method which is shown to be quadratically convergent the sequential method is guaranteed to produce a stationary point of the nonconvex sdp our numerical results demonstrate the high efficiency of the proposed method on large scale problems

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