nondegenerate solutions and related concepts in affine variational inequalities the notion of a strictly complementary solution for complementarity problems is extended to that of a nondegenerate solution of variational inequalities several equivalent formulations of nondegeneracy are given in the affine case an existence theorem for a nondegenerate solution is given in terms of several related concepts which are shown to be equivalent in this context these include a weak sharp minimum the minimum principle sufficiency and error bounds the gap function associated with the variational inequality plays a central role in this existence theorem

sufficient matrices and the linear complementarity problem abstract we pose and answer two questions about solutions of the linear complementarity problem lcp the first question is concerned with the conditions on a matrix m which guarantee that for every vector q the solutions of the lcp q m are identical to the karush kuhn tucker points of the natural quadratic program associated with q m in answering this question we introduce the class of row sufficient matrices the transpose of such a matrix is what we call column sufficient the latter matrices turn out to furnish the answer to our second question which asks for the conditions on m under which the solution set of q m is convex for every q in addition to these two main results we discuss the connections of these twonew matrix classes with other well known matrix classes in linear complementarity theory

on homogeneous and self dual algorithms for lcp we present some generalizations of a homogeneous and self dual linear programming lp algorithm to solving the monotone linear complementarity problem lcp again while it achieves the best known interior point iteration complexity the algorithm does not need to use any big m number and it detects lcp infeasibility by generating a certificate to our knowledge this is the first interior point and infeasible starting algorithm for the lcp with these desired features

on the pervasiveness of difference convexity in optimization and statistics with the increasing interest in applying the methodology of difference of convex dc optimization to diverse problems in engineering and statistics this paper establishes the dc property of many functions in various areas of applications not previously known to be of this class motivated by a quadratic programming based recourse function in two stage stochastic programming we show that the optimal value function of a copositive thus not necessarily convex quadratic program is dc on the domain of finiteness of the program when the matrix in the objective function s quadratic term and the constraint matrix are fixed the proof of this result is based on a dc decomposition of a piecewise hbox lc 1 function i e functions with lipschitz gradients armed with these new results and known properties of dc functions existed in the literature we show that many composite statistical functions in risk analysis including the value at risk var conditional value at risk cvar optimized certainty equivalent and the expectation based var based and cvar based random deviation functionals are all dc adding the known class of dc surrogate sparsity functions that are employed as approximations of the ell 0 function in statistical learning our work significantly expands the classes of dc functions and positions them for fruitful applications

lipschitz continuity of solutions of linear inequalities programs and complementarity problems it is shown that solutions of linear inequalities linear programs and certain linear complementarity problems e g those with p matrices or z matrices but not semidefinite matrices are lipschitz continuous with respect to changes in the right hand side data of the problem solutions of linear programs are not lipschitz continuous with respect to the coefficients of the objective function the lipschitz constant given here is a generalization of the role played by the norm of the inverse of a nonsingular matrix in bounding the perturbation of the solution of a system of equations in terms of a right hand side perturbation

iterative methods for large convex quadratic programs a survey in this paper we give a state of the art review of many iterative methods for solving large convex quadratic programs we attempt to classify several of the more basic methods in two categories within each of which a unified iterative scheme will be introduced and its convergence analyzed hybrid iterative methods such as the proximal point algorithm and a diagonalization scheme that make use of the more basic schemes will also be described the results of an extensive computer experimentation which is aimed at comparing the relative performance of the various methods will be reported and discussed finally several important topics which require future research will be highlighted

an algorithm for the minimization of mixed l sub 1 and l sub 2 norms with application to bayesian estimation the regularizing functional approach is widely used in many estimation problems in practice the solution is defined as one minimum point of a suitable functional the main part of which accounts for the underlying physical model whereas the regularizing part represents some prior information about the unknowns in the bayesian interpretation one has a maximum a posteriori map estimator in which the main and regularizing parts are represented respectively by likelihood and prior distributions when either the prior or likelihood is a laplace distribution and the other is a gaussian distribution one is led to consider functionals that include both absolute and square norms the authors present a characterization of the minimum points of such functionals together with a descent type algorithm for numerical computations the results of monte carlo simulations are also reported

wavelet based image estimation an empirical bayes approach using jeffrey s noninformative prior the sparseness and decorrelation properties of the discrete wavelet transform have been exploited to develop powerful denoising methods however most of these methods have free parameters which have to be adjusted or estimated in this paper we propose a wavelet based denoising technique without any free parameters it is in this sense a universal method our approach uses empirical bayes estimation based on a jeffreys noninformative prior it is a step toward objective bayesian wavelet based denoising the result is a remarkably simple fixed nonlinear shrinkage thresholding rule which performs better than other more computationally demanding methods

a parallel relaxation method for quadratic programming problems with interval constraints abstract optimization problems with interval constraints are encountered in various fields such as network flows and computer tomography as these problems are usually very large they are not easy to solve without taking their sparsity into account recently row action methods which originate from the classical hildreth s method for quadratic programming problems have drawn much attention since they are particularly useful for large and sparse problems various row action methods have already been proposed for optimization problems with interval constraints but they mostly belong to the class of sequential methods based on the gauss seidel and sor methods in this paper we propose a highly parallelizable method for solving those problems which may be regarded as an application of the jacobi method to the dual of the original problems we prove convergence of the algorithm and report some computational results to demonstrate its effectiveness

block monotone iterative methods for elliptic variational inequalities this paper discusses some important block iterative methods for solving variational inequalities with a nonlinear operator and proves their monotone convergence properties and comparison theorems these results extend c v pao s research work in numer math 72 1995 239 to variational inequalities

an lpcc approach to nonconvex quadratic programs filling a gap in nonconvex quadratic programming this paper shows that the global resolution of a feasible quadratic program qp which is not known a priori to be bounded or unbounded below can be accomplished in finite time by solving two linear programs with linear complementarity constraints i e lpccs specifically this task can be divided into two lpccs the first confirms whether the qp is bounded below on the feasible set and if not computes a feasible ray on which the qp is unbounded the second lpcc computes a globally optimal solution if it exists by identifying a stationary point that yields the best quadratic objective value in turn the global resolution of these lpccs can be accomplished by a parameter free mixed integer programming based finitely terminating algorithm developed recently by the authors which can be enhanced in this context by a new kind of valid cut derived from the second order conditions of the qp and by exploiting the special structure of the lpccs throughout our treatment makes no boundedness assumption of the qp this is a significant departure from much of the existing literature which consistently employs the boundedness of the feasible set as a blanket assumption the general theory is illustrated by 3 classes of indefinite problems qps with simple upper and lower bounds existence of optimal solutions is guaranteed same qps with an additional inequality constraint extending the case of simple bound constraints and nonnegatively constrained copositive qps no guarantee of the existence of an optimal solution we also present numerical results to support the special cuts obtained due to the qp connection

a new algorithm for state constrained separated continuous linear programs during the last few decades significant progress has been made in solving large scale finite dimensional and semi infinite linear programming problems in contrast little progress has been made in solving linear programs in infinite dimensional spaces despite their importance as models in manufacturing and communication systems inspired by the research on separated continuous linear programs we propose a new class of continuous linear programming problems that has a variety of important applications in communications manufacturing and urban traffic control this class of continuous linear programs contains the separated continuous linear programs as a subclass using ideas from quadratic programming we propose an efficient algorithm for solving large scale problems in this new class under mild assumptions on the form of the problem data we prove algorithmically the absence of a duality gap for this class of problems without any boundedness assumptions on the solution set we show this class of problems admits piecewise constant optimal control when the optimal solution exists we give conditions for the existence of an optimal solution we also report computational results which illustrate that the new algorithm is effective in solving large scale realistic problems with several hundred continuous variables arising in manufacturing systems

error bound and convergence analysis of matrix splitting algorithms for the affine variational inequality problem consider the affine variational inequality problem it is shown that the distance to the solution set from a feasible point near the solution set can be bounded by the norm of a natural residual at that point this bound is then used to prove linear convergence of a matrix splitting algorithm for solving the symmetric case of the problem this latter result improves upon a recent result of luo and tseng that further assumes the problem to be monotone

non monotone trust region algorithms for nonlinear optimization subject to convex constraints this paper presents two new trust region methods for solving nonlinear optimization problems over convex feasible domains these methods are distinguished by the fact that they do not enforce strict monotonicity of the objective function values at successive iterates the algorithms are proved to be convergent to critical points of the problem from any starting point extensive numerical experiments show that this approach is competitive with the lancelot package

error bounds for monotone linear complementarity problems we give a bound on the distance between an arbitrary point and the solution set of a monotone linear complementarity problem in terms of a condition constant that depends on the problem data only and a residual function of the violations of the complementary problem conditions by the point considered when the point satisfies the linear inequalities of the complementarity problem the residual consists of the complementarity condition plus its square root this latter term is essential and without it the error bound cannot hold we also show that another natural residual that has been employed to bound errors for strictly monotone linear complementarity problems fails to bound errors for the monotone case considered here

a family of inexact sqa methods for non smooth convex minimization with provable convergence guarantees based on the luo tseng error bound property we propose a new family of inexact sequential quadratic approximation sqa methods which we call the inexact regularized proximal newton irpn method for minimizing the sum of two closed proper convex functions one of which is smooth and the other is possibly non smooth our proposed method features strong convergence guarantees even when applied to problems with degenerate solutions while allowing the inner minimization to be solved inexactly specifically we prove that when the problem possesses the so called luo tseng error bound eb property irpn converges globally to an optimal solution and the local convergence rate of the sequence of iterates generated by irpn is linear superlinear or even quadratic depending on the choice of parameters of the algorithm prior to this work such eb property has been extensively used to establish the linear convergence of various first order methods however to the best of our knowledge this work is the first to use the luo tseng eb property to establish the superlinear convergence of sqa type methods for non smooth convex minimization as a consequence of our result irpn is capable of solving regularized regression or classification problems under the high dimensional setting with provable convergence guarantees we compare our proposed irpn with several empirically efficient algorithms by applying them to the ell 1 regularized logistic regression problem experiment results show the competitiveness of our proposed method

convergence rates in forward backward splitting forward backward splitting methods provide a range of approaches to solving large scale optimization problems and variational inequalities in which structures conducive to decomposition can be utilized apart from special cases where the forward step is absent and a version of the proximal point algorithm comes out efforts at evaluating the convergence potential of such methods have so far relied on lipschitz properties and strong monotonicity or inverse strong monotonicity of the mapping involved in the forward step the perspective mainly being that of projection algorithms here convergence is analyzed by a technique that allows properties of the mapping in the backward step to be brought in as well for the first time in such a general setting global and local contraction rates are derived moreover they are derived in a form which makes it possible to determine the optimal step size relative to certain constants associated with the given problem insights are thereby gained into the effects of shifting strong monotonicity between the forward and backward mappings when a splitting is selected

descent methods with linesearch in the presence of perturbations we consider the class of descent algorithms for unconstrained optimization with an armijo type stepsize rule in the case when the gradient of the objective function is computed inexactly an important novel feature in our theoretical analysis is that perturbations associated with the gradient are not assumed to be relatively small or to tend to zero in the limit as a practical matter we expect them to be reasonably small so that a meaningful approximate solution can be obtained this feature makes our analysis applicable to various difficult problems encounted in practice we propose a modified armijo type rule for computing the stepsize which guarantees that the algorithm obtains a reasonable approximate solution furthermore if perturbations are small relative to the size of the gradient then our algorithm retains all the standard convergence properties of descent methods

new improved error bounds for the linear complementarity problem new local and global error bounds are given for both nonmonotone and monotone linear complementarity problems comparisons of various residuals used in these error bounds are given a possible candidate for a best error bound emerges from our comparisons as the sum of two natural residuals

on the identification of zero variables in an interior point framework we consider column sufficient linear complementarity problems and study the problem of identifying those variables that are zero at a solution to this end we propose a new computationally inexpensive technique that is based on growth functions we analyze in detail the theoretical properties of the identification technique and test it numerically the identification technique is particularly suited to interior point methods but can be applied to a wider class of methods

linear convergence of a type of iterative sequences in nonconvex quadratic programming abstract by using error bounds for affine variational inequalities we prove that any iterative sequence generated by the projection dc difference of convex functions decomposition algorithm in quadratic programming is r linearly convergent provided that the original problem has solutions our result solves in the affirmative the first part of the conjecture stated by le thi pham dinh and yen in their recent paper 8 p 489

on the rate of convergence of a partially asynchronous gradient projection algorithm recently bertsekas and tsitsiklis proposed a partially asynchronous implementation of the gradient projection algorithm of goldstein and levitin and polyak for the problem of minimizing a differentiable function over a closed convex set in this paper the rate of convergence of this algorithm is analyzed it is shown that if the standard assumptions hold that is the solution set is nonempty and the gradient of the function is lipschitz continuous and i the isocost surfaces of the objective function restricted to the solution set are properly separated and ii a certain multifunction associated with the problem is locally upper lipschitzian then this algorithm attains a linear rate of convergence

error estimates and lipschitz constants for best approximation in continuous function spaces abstract we use a structural characterization of the metric projection pg f from the continuous function space to its one dimensional subspace g to derive a lower bound of the hausdorff strong unicity constant or weak sharp minimum constant for pg and then show this lower bound can be attained then the exact value of lipschitz constant for pg is computed the process is a quantitative analysis based on the g teaux derivative of pg a representation of local lipschitz constants the equivalence of local and global lipschitz constants for lower semicontinuous mappings and construction of functions

interior gradient and epsilon subgradient descent methods for constrained convex minimization we extend epsilon subgradient descent methods for unconstrained nonsmooth convex minimization to constrained problems over polyhedral sets in particular over r p this is done by replacing the usual squared quadratic regularization term used in subgradient schemes by the logarithmic quadratic distancelike function recently introduced by the authors we then obtain interior e subgradient descent methods which allow us to provide a natural extension of bundle methods and polyak s subgradient projection methods for nonsmooth convex minimization furthermore similar extensions are considered for smooth constrained minimization to produce interior gradient descent methods global convergence as well as an improved global efficiency estimate are established for these methods within a unifying principle and minimal assumptions on the problem s data

prox method with rate of convergence o 1 t for variational inequalities with lipschitz continuous monotone operators and smooth convex concave saddle point problems we propose a prox type method with efficiency estimate o epsilon 1 for approximating saddle points of convex concave c 1 1 functions and solutions of variational inequalities with monotone lipschitz continuous operators application examples include matrix games eigenvalue minimization and computing the lovasz capacity number of a graph and these are illustrated by numerical experiments with large scale matrix games and lovasz capacity problems

a class of dantzig wolfe type decomposition methods for variational inequality problems we consider a class of decomposition methods for variational inequalities which is related to the classical dantzig wolfe decomposition of linear programs our approach is rather general in that it can be used with certain types of set valued or nonmonotone operators as well as with various kinds of approximations in the subproblems of the functions and derivatives in the single valued case also subproblems may be solved approximately convergence is established under reasonable assumptions we also report numerical experiments for computing variational equilibria of the game theoretic models of electricity markets our numerical results illustrate that the decomposition approach allows to solve large scale problem instances otherwise intractable if the widely used path solver is applied directly without decomposition

a robust sequential quadratic programming method the sequential quadratic programming method developed by wilson han and powell may fail if the quadratic programming subproblems become infeasible or if the associated sequence of search directions is unbounded this paper considers techniques which circumvent these difficulties by modifying the structure of the constraint region in the quadratic programming subproblems furthermore questions concerning the occurrence of an unbounded sequence of multipliers and problem feasibility are also addressed

a posteriori error bounds for the linearly constrained varitional inequality problem in this paper we derive a posteriori error bound of an approximate solution to a linearly constrained variational inequality problem three measures of error are used with each depending on a particular characterization of an exact solution and related to a certain class of iterative methods

weighted max norm estimate of additive schwarz iteration scheme for solving linear complementarity problems in this paper we consider an algebraic additive schwarz iteration scheme for solving the finite dimensional linear complementarity problem that involves an m matrix the scheme contains some existing algorithms as special cases we establish monotone convergence of the iteration scheme under appropriate conditions moreover using the concept of weak regular splitting we estimate weighted max norm bounds for iteration errors thereby we show that the sequence generated by the iteration scheme converges to the unique solution of the problem without any restriction on the initial point

on the linear convergence of descent methods for convex essentially smooth minimization consider the problem of minimizing over a polyhedral set the composition of an affine mapping with a strictly convex essentially smooth function a general result on the linear convergence of descent methods for solving this problem is presented by applying this result the linear convergence of both the gradient projection algorithm of goldstein and levitin and polyak and a matrix splitting algorithm using regular splitting is established the results do not require that the cost function be strongly convex or that the optimal solution set be bounded the key to the analysis lies in a new error bound for estimating the distance from a feasible point to the optimal solution set

maximum likelihood reconstruction for emission tomography previous models for emission tomography et do not distinguish the physics of et from that of transmission tomography we give a more accurate general mathematical model for et where an unknown emission density x y z generates and is to be reconstructed from the number of counts n d in each of d detector units d within the model we give an algorithm for determining an estimate of which maximizes the probability p n of observing the actual detector count data n over all possible densities let independent poisson variables n b with unknown means b b 1 b represent the number of unobserved emissions in each of b boxes pixels partitioning an object containing an emitter suppose each emission in box b is detected in detector unit d with probability p b d d 1 d with p b d a one step transition matrix assumed known we observe the total number n n d of emissions in each detector unit d and want to estimate the unknown b b 1 b for each the observed data n has probability or likelihood p n the em algorithm of mathematical statistics starts with an initial estimate 0 and gives the following simple iterative procedure for obtaining a new estimate new from an old estimate old to obtain k k 1 2 new b old b dd 1 n d p b d old b p b d b 1 b

a class of methods for projection on the intersection of several ellipsoids the paper is concerned with the basic optimization problem of projecting a point onto the intersection of several ellipsoids we present a class of methods where the problem is reduced to a sequence of projections onto the intersection of several balls the subproblems are simpler and more tractable but the main advantage is that in so doing we can avoid solving linear systems completely and thus the methods are very suitable for large scale problems the methods have been shown to have nice convergence properties under slater s constraint qualification and to be very reliable and efficient in our testing

platelets a multiscale approach for recovering edges and surfaces in photon limited medical imaging the nonparametric multiscale platelet algorithms presented in this paper unlike traditional wavelet based methods are both well suited to photon limited medical imaging applications involving poisson data and capable of better approximating edge contours this paper introduces platelets localized functions at various scales locations and orientations that produce piecewise linear image approximations and a new multiscale image decomposition based on these functions platelets are well suited for approximating images consisting of smooth regions separated by smooth boundaries for smoothness measured in certain holder classes it is shown that the error of m term platelet approximations can decay significantly faster than that of m term approximations in terms of sinusoids wavelets or wedgelets this suggests that platelets may outperform existing techniques for image denoising and reconstruction fast platelet based maximum penalized likelihood methods for photon limited image denoising deblurring and tomographic reconstruction problems are developed because platelet decompositions of poisson distributed images are tractable and computationally efficient existing image reconstruction methods based on expectation maximization type algorithms can be easily enhanced with platelet techniques experimental results suggest that platelet based methods can outperform standard reconstruction methods currently in use in confocal microscopy image restoration and emission tomography

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

global and asymptotic convergence rate estimates for a class of projected gradient processes projected gradient processes of the goldstein levitin polyak type are considered for constrained minimization problems min omega f with omega a convex set in a hilbert space x and f x to mathbb r 1 a differentiable functional global and local convergence theorems are established for a large class of these processes including those generated with implicit step length rules proposed by bertsekas and goldstein in this analysis traditional uniform strong positivity conditions on the hessian nabla 2 f are replaced by weaker pseudoconvexity conditions and growth conditions on f when f has a unique minimizes in omega convergence rates are shown to depend on how rapidly the function gamma sigma inf r f x f xi mid x in omega x xi geqq sigma grows with increasing sigma 0 if gamma sigma geqq b sigma nu for some b 0 the processes f n in question converge to inf f like o n nu nu 2 linear

on conic qpccs conic qcqps and completely positive programs this paper studies several classes of nonconvex optimization problems defined over convex cones establishing connections between them and demonstrating that they can be equivalently formulated as convex completely positive programs the problems being studied include a conic quadratically constrained quadratic program qcqp a conic quadratic program with complementarity constraints qpcc and rank constrained semidefinite programs our results do not make any boundedness assumptions on the feasible regions of the various problems considered the first stage in the reformulation is to cast the problem as a conic qcqp with just one nonconvex constraint q x le 0 q x 0 where q x is nonnegative over the convex conic and linear constraints so the problem fails the slater constraint qualification a conic qpcc has such a structure we prove the converse namely that any conic qcqp satisfying a constraint qualification can be expressed as an equivalent conic qpcc the second stage of the reformulation lifts the problem to a completely positive program and exploits and generalizes a result of burer we also show that a frank wolfe type result holds for a subclass of this class of conic qcqps further we derive necessary and sufficient optimality conditions for nonlinear programs where the only nonconvex constraint is a quadratic constraint of the structure considered elsewhere in the paper

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

global error bounds for monotone affine variational inequality problems abstract a global gradient projection type error bound is given for monotone affine variational inequality problems this new bound is based on a previously obtained global error bound for monotone linear complementarity problems and requires no assumptions on the monotone affine variational inequality other than solvability the error bound can be used in conjunction with recent results of tseng and luo to establish global linear convergence of any regular splitting method for solvable symmetric monotone affine variational inequality problems

convergence of iterates of an inexact matrix splitting algorithm for the symmetric monotone linear complementarity problem convergence of iterates is established for a symmetric regular matrix splitting algorithm for the solution of the symmetric monotone linear complementarity problem where the subproblems are solved inexactly the notable iterate convergence recently established by luo and tseng for exact subproblem solution is extended here to inexact subproblem solution for a symmetric matrix splitting a principal application of the present result is to iterate convergence for the inexact block jacobi method for which pang and yang established convergence of a subsequence of the iterates

on some steplength approaches for proximal algorithms we discuss a number of novel steplength selection schemes for proximal based convex optimization algorithms in particular we consider the problem where the lipschitz constant of the gradient of the smooth part of the objective function is unknown we generalize two optimization algorithms of khobotov type and prove convergence we also take into account possible inaccurate computation of the proximal operator of the non smooth part of the objective function secondly we show convergence of an iterative algorithm with armijo type steplength rule and discuss its use with an approximate computation of the proximal operator numerical experiments show the efficiency of the methods in comparison to some existing schemes

newton s method for large bound constrained optimization problems we analyze a trust region version of newton s method for bound constrained problems our approach relies on the geometry of the feasible set not on the particular representation in terms of constraints the convergence theory holds for linearly constrained problems and yields global and superlinear convergence without assuming either strict complementarity or linear independence of the active constraints we also show that the convergence theory leads to an efficient implementation for large bound constrained problems

feasible descent algorithms for mixed complementarity problems in this paper we consider a general algorithmic framework for solving nonlinear mixed complementarity problems the main features of this framework are a it is well defined for an arbitrary mixed complementarity problem b it generates only feasible iterates c it has a strong global convergence theory and d it is locally fast convergent under standard regularity assumptions this framework is applied to the path solver in order to show viability of the approach numerical results for an appropriate modification of the path solver indicate that this framework leads to substantial computational improvements

a conic duality frank wolfe type theorem via exact penalization in quadratic optimization the famous frank wolfe theorem ensures attainability of the optimal value for quadratic objective functions over a possibly unbounded polyhedron if the feasible values are bounded this theorem does not hold in general for conic programs where linear constraints are replaced by more general convex constraints like positive semidefiniteness or copositivity conditions despite the fact that the objective can be even linear this paper studies exact penalizations of classical quadratic programs i e optimization of quadratic functions over a polyhedron and applies the results to establish a frank wolfe type theorem for the primal dual pair of a class of conic programs that frequently arises in applications one result is that uniqueness of the solution of the primal ensures dual attainability i e existence of the solution of the dual

this is spiral tap sparse poisson intensity reconstruction algorithms theory and practice observations in many applications consist of counts of discrete events such as photons hitting a detector which cannot be effectively modeled using an additive bounded or gaussian noise model and instead require a poisson noise model as a result accurate reconstruction of a spatially or temporally distributed phenomenon f from poisson data y cannot be effectively accomplished by minimizing a conventional penalized least squares objective function the problem addressed in this paper is the estimation of f from y in an inverse problem setting where the number of unknowns may potentially be larger than the number of observations and f admits sparse approximation the optimization formulation considered in this paper uses a penalized negative poisson log likelihood objective function with nonnegativity constraints since poisson intensities are naturally nonnegative in particular the proposed approach incorporates key ideas of using separable quadratic approximations to the objective function at each iteration and penalization terms related to l1 norms of coefficient vectors total variation seminorms and partition based multiscale estimation methods

new error bounds for the linear complementarity problem recently mangasarian and solodov 1993 showed that every nonlinear complementarity problem ncp is equivalent to the unconstrained minimization of a certain implicit lagrangian in particular it was shown that this implicit lagrangian is nonnegative everywhere and its set of zeros coincides with the solution set of the original ncp in this paper we consider the linear complementarity problem lcp and show that the distance to the solution set of the lcp from any point sufficiently close to the set can be bounded above by the square root of the implicit lagrangian for the lcp in other words the square root of the implicit lagrangian is a local error bound for the lcp our proof is based on showing that the square root of the implicit lagrangian is equivalent to the residual function used in a known local error bound robinson 1981 luo and tseng 1992 when the matrix associated with the lcp is nondegenerate the new error bound is in fact global this extends the error bound result of mathias and

level rings and algebras with straightening laws abstract stanley 13 defined the concept of level rings intermediate between cohen macaulay and gorenstein which is of interest from viewpoints of both commutative algebra and combinatorics via the theory of canonical modules the study of level rings turns out to be one of the most important foundations of the theory of algebras with straightening laws the purpose of this paper is to state some basic results on level rings which are either stanley reisner rings or algebras with straightening laws

the dilworth number of artin gorenstein rings 1let a be an artin local ring the dilworth number of a is defined to be the supremum of p z for ideals z in a where p is the minimal number of generators in my paper 4 i developed some methods to compute this number when a is homogeneously graded over a field k and especially if a is gorenstein and ch k 0 i showed that it is the dominating case that the dilworth number is attained by a power of the maximal ideal in such a case we shall say that a has the sperner property in the present paper we consider artin rings a m not necessarily graded and compare the dilworth number of a and that of gr a and it will be shown that the main result of 4 is valid without the assumption that the rings are homogeneous our main result is theorem 7 where we prove that most artin gorenstein rings have the sperner property actually we prove a stronger result which says that it has the weak stanley property to explain the term most we represent an artin gorenstein ring a in the

on the associated graded modules of canonical modules the canonical modules of cohen macaulay rings give a useful technical tool in commutative algebra algebraic geometry and singularity theory via various duality theorems but not much is known about the canonical modules themselves in this paper we investigate the structure of the canonical moduies expecially their associated graded modules we ask the following two questions let k be a canonical module over a cohenmacaulay local ring r then

on a global projection type error bound for the linear complementarity problem abstract we consider a projection type error bound for the linear complementarity problem involving a matrix m and vector q first we show that the mangasarian ren sufficient condition on m for this error bound to hold globally for all q such that the problem is solvable is also necessary second we derive necessary and sufficient conditions on m and q for this error bound to hold globally thus extending a result of luo and tseng which in addition assumes m is psd plus third we derive necessary and sufficient conditions on m for this error bound to hold globally independent of q these conditions are closely related to m being lipschitzian we also discuss application of our result to the affine variational inequality problem

on the convergence of a matrix splitting algorithm for the symmetric monotone linear complementarity problem a matrix splitting algorithm for the linear complementarily problem is considered where the matrix is symmetric positive semidefinite it is shown that if the splitting is regular then the iterates generated by the algorithm are well defined and converge to a solution this result resolves in the affirmative a long standing question about the convergence of the point successive overrelaxation sor method for solving this problem this result is also extended to related iterative methods as direct consequences convergence of the methods of respectively aganagic cottle et al mangasarian pang and others is obtained without making any additional assumptions on the problem

new error bounds and their applications to convergence analysis of iterative algorithms we present two new error bounds for optimization problems over a convex set whose objective function f is either semianalytic or strictly convex with 1 we then apply these error bounds to analyze the rate of convergence of a wide class of iterative descent algorithms for the aforementioned optimization problem our analysis shows that the function sequence f xk converges at least at the sublinear rate of k for some positive constant where k is the iteration index moreover the distances from the iterate sequence xk to the set of stationary points of the optimization problem converge to zero at least sublinearly

a nonsmooth regularization approach based on shearlets for poisson noise removal in roi tomography abstract due to its potential to lower exposure to x ray radiation and reduce the scanning time region of interest roi computed tomography ct is particularly appealing for a wide range of biomedical applications to overcome the severe ill posedness caused by the truncation of projection measurements ad hoc strategies are required since traditional ct reconstruction algorithms result in instability to noise and may give inaccurate results for small roi to handle this difficulty we propose a nonsmooth convex optimization model based on l1 shearlet regularization whose solution is addressed by means of the variable metric inexact line search algorithm vmila a proximal gradient method that enables the inexact computation of the proximal point defining the descent direction we compare the reconstruction performance of our strategy against a smooth total variation stv approach by using both poisson noisy simulated data and real data from fan beam ct geometry the results show that while for synthetic data both shearets and stv perform well for real data the proposed nonsmooth shearlet based approach outperforms stv since the localization and directional properties of shearlets allow to detect finer structures of a textured image finally our approach appears to be insensitive to the roi size and location

on linear convergence of iterative methods for the variational inequality problem abstract we present new linear convergence results for iterative methods for solving the variational inequality problem the methods include the extragradient method the proximal point method a matrix splitting method and a certain feasible descent method the proofs of the results are based on certain error bounds related to the algorithmic mappings moreover we show that all these error bounds hold if a projection type error bound holds

new projection type methods for monotone lcp with finite termination in this paper we establish two new projection type methods for the solution of monotone linear complementarity problem lcp the methods are a combination of the extragradient method and the newton method in which the active set strategy is used and only one linear system of equations with lower dimension is solved at each iteration it is shown that under the assumption of monotonicity these two methods are globally and linearly convergent furthermore under a nondegeneracy condition they have a finite termination property at last the methods are extended to solving monotone affine variational inequality problem

local linear convergence of the alternating direction method of multipliers for quadratic programs the douglas rachford alternating direction method of multipliers admm has been widely used in various areas the global convergence of admm is well known while research on its convergence rate

a linearly convergent dual based gradient projection algorithm for quadratically constrained convex minimization this paper presents a new dual formulation for quadratically constrained convex programs the special structure of the derived dual problem allows us to apply the gradient projection algorithm to produce a simple explicit method involving only elementary vector matrix operations proven to converge at a linear rate

on the identification of active constraints nondegeneracy conditions that guarantee that the optimal active constraints are identified in a finite number of iterations are studied results of this type have only been established for a few algorithms and then under restrictive hypothesis the main result is a characterization of those algorithms that identify the optimal constraints in a finite number of iterations this result is obtained with a nondegeneracy assumption which is equivalent in the standard nonlinear programming problem to the assumption that there is a set of strictly complementary lagrange multipliers as an important consequence of the authors results the way that this characterization applies to gradient projection and sequential quadratic programming algorithms is shown

the ordered subsets mirror descent optimization method with applications to tomography we describe an optimization problem arising in reconstructing three dimensional medical images from positron emission tomography pet a mathematical model of the problem based on the maximum likelihood principle is posed as a problem of minimizing a convex function of several million variables over the standard simplex to solve a problem of these characteristics we develop and implement a new algorithm ordered subsets mirror descent and demonstrate theoretically and computationally that it is well suited for solving the pet reconstruction problem

global lipschitzian error bounds for semidefinite complementarity problems with emphasis on ncps error bound theory play an important role in many mathematical programming problems in this paper we propose a new assumption condition under which we establish a global lipshitzian error bound for the semidefinite complementarity problem sdcp as corollaries two results involving the assumptions of strong monotonicity and bd regularity are given when the sdcp reduces to the nonlinear complementarity problem ncp our results strictly generalize the well known error bound results which were established under the assumptions of strong monotonicity and uniform p property we also establish a new error bound for the ncp under bd regularity and some r 0 type condition

free steering relaxation methods for problems with strictly convex costs and linear constraints we consider dual coordinate ascent methods for minimizing a strictly convex possibly nondifferentiable function subject to linear constraints such methods are useful in large scale applications e g entropy maximization quadratic programming network flow because they are simply can exploit sparsity and in certain cases are highly parallelizable we establish their global convergence under weak conditions and a free steering order of relaxation previous comparable results were restricted to special problems with separable costs and equality constraints our convergence framework unifies to a certain extent the approaches of bregman censor and lent de pierro and iusem and luo and tseng and complements that of bertsekas and tseng

parallel constraint distribution in convex quadratic programming we consider convex quadratic programs with large numbers of constraints we distribute these constraints among several parallel processors and modify the objective function for each of these subproblems with lagrange multiplier information from the other processors new lagrange multiplier information is aggregated in a master processor and the whole process is repeated linear convergence is established for strongly convex quadratic programs by formulating the algorithm in an appropriate dual space the algorithm corresponds to a step of an iterative matrix splitting algorithm for a symmetric linear complementarity problem followed by a projection onto a subspace

convergence rate analysis of iteractive algorithms for solving variational inequality problems we present a unified convergence rate analysis of iterative methods for solving the variational inequality problem our results are based on certain error bounds they subsume and extend the linear and sublinear rates of convergence established in several previous studies we also derive a new error bound for gamma strictly monotone variational inequalities the class of algorithms covered by our analysis in fairly broad it includes some classical methods for variational inequalities e g the extragradient matrix splitting and proximal point methods for these methods our analysis gives estimates not only for linear convergence which had been studied extensively but also sublinear depending on the properties of the solution in addition our framework includes a number of algorithms to which previous studies are not applicable such as the infeasible projection methods a separation projection method inexact hybrid proximal point methods and some splitting techniques finally our analysis covers certain feasible descent methods of optimization for which similar convergence rate estimates have been recently obtained by luo 14

an implementable active set algorithm for computing a b stationary point of a mathematical program with linear complementarity constraints erratum in m fukushima and p tseng siam j optim 12 2002 pp 724 739 an active set algorithm was proposed for solving a mathematical program with a smooth objective function and linear inequality complementarity constraints it is asserted therein that under a uniform licq on the feasible set this algorithm generates iterates whose cluster points are b stationary points of the problem however the proof has a gap and shows only that each cluster point is an m stationary point we discuss this gap and show that b stationarity can be achieved if the algorithm is modified and an additional error bound condition holds

on the convergence of constrained parallel variable distribution algorithms we consider the parallel variable distribution pvd approach proposed by ferris and mangasarian siam j optim 4 1994 pp 815 832 for solving optimization problems the problem variables are distributed among p processors with each processor having the primary responsibility for updating its block of variables while allowing the remaining secondary variables to change in a restricted fashion along some easily computable directions for constrained nonlinear programs convergence in m c ferris and o l mangasarian siam j optim 4 1994 pp 815 832 was established in the special case of convex block separable constraints for general inseparable constraints it was suggested that a dual differentiable exact penalty function reformulation of the problem be used we propose to apply the pvd approach to problems with general convex constraints directly and show that the algorithm converges provided certain conditions are imposed on the change of secondary variables these conditions are both natural and practically implementable we also show that the original requirement of exact global solution of the parallel subproblems can be replaced by a less stringent sufficient descent condition the first rate of convergence result for the class of constrained pvd algorithms is also given

a new error bound for linear complementarity problems with weakly chained diagonally dominant b matrices abstract recently some error bounds for the linear complementarity problem with a weakly chained diagonally dominant b matrix have been given under certain conditions in this paper we provide a new and even optimal error bound for such linear complementarity problems under weaker conditions than those there greatly improving the existing ones some numerical examples are performed to illustrate the sharpness and optimality of our new bound

new properties of forward backward splitting and a practical proximal descent algorithm abstract in this paper we discuss a proximal descent algorithm for finding a zero of the sum of two maximal monotone operators in a real hilbert space some new properties of forward backward splitting are given which extend the well known properties of the usual projection then they are used to analyze the weak convergence of the proximal descent algorithm without assuming lipschitz continuity of the forward operator we also give a new technique of choosing trial values of the step length involved in an armijo like condition which returns the not necessarily decreasing step length self adaptively rudimentary numerical experiments show that it is effective in practical implementations

error bounds for non polyhedral convex optimization and applications to linear convergence of fdm and pgm abstract this paper is concerned with the composite convex minimization problem where the cost function consists of a smooth convex function with lipschitz gradient and a closed proper convex function with non polyhedral epigraph for this class of non polyhedral convex optimization problems we establish the locally lipschitzian type error bounds for estimating the distance to the solution set from any feasible point near the solution set and the globally lipschitzian type error bounds for the smooth function with a special structure with the help of the local weak sharpness of minima or the calmness of appropriate multifunctions and also provide verifiable regularity conditions to guarantee them to hold although the derived local error bounds are weaker than the one used in luo and tseng 1992 1993 and wei and jen 2014 when applying them to an inexact feasible descent method fdm and proximal gradient method pgm respectively we still achieve the asymptotic q linear convergence and r linear convergence of the objective value sequence and the iterate sequence respectively as an illustration of these results we obtain the linear convergence of the pgm for the trace norm regularized least squares problem under a regularity condition of the linear operator

a limited memory quasi newton trust region method for box constrained optimization by means of wolfe conditions strategy we propose a quasi newton trust region method to solve box constrained optimization problems this method is an adequate combination of the compact limited memory bfgs and the trust region direction while the generated point satisfies the wolfe conditions and therefore maintains a positive definite approximation to the hessian of the objective function the global convergence and the quadratic convergence of this method are established under suitable conditions finally we compare our algorithms iwtral and ibwtral with an active set trust region algorithm astral xu and burke 2007 on the cutest box constrained test problems presented by gould et al 2015 numerical results show that the presented method is competitive and totally interesting for solving box constrained optimization