global weak sharp minima for convex semi infinite optimization problems we mainly consider global weak sharp minima for convex infinite and semi infinite optimization problems cip in terms of the normal cone subdifferential and directional derivative we provide several characterizations for cip to have global weak sharp minimum property

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

subsmooth semi infinite and infinite optimization problems we first consider subsmoothness for a function family and provide formulas of the subdifferential of the pointwise supremum of a family of subsmooth functions next we consider subsmooth infinite and semi infinite optimization problems in particular we provide several dual and primal characterizations for a point to be a sharp minimum or a weak sharp minimum for such optimization problems

traveling wave solutions for some factorized nonlinear pdes in this work some new special traveling wave solutions of the convective fisher equation the time delayed burgers fisher equation the burgers fisher equation and a nonlinear dispersive dissipative equation kakutani and kawahara 1970 j phys soc japan 29 1068 are obtained through the factorization technique all of them share the same type of factorization scheme which reduces the original equation to a riccati equation of the same kind whose general solution is given in terms of bessel and neumann functions in addition some novel particular solutions of the nonlinear dispersive dissipative equation are provided

isolated calmness of solution mappings in convex semi infinite optimization this paper is concerned with isolated calmness of the solution mapping of a parameterized convex semi infinite optimization problem subject to canonical perturbations we provide a sufficient condition for isolated calmness of this mapping this sufficient condition characterizes the strong uniqueness of minimizers under the slater constraint qualification moreover on the assumption that the objective function and the constraints are linear we show that this condition is also necessary for isolated calmness

perturbation analysis of a condition number for convex inequality systems and global error bounds for analytic systems in this paper several types of perturbations on a convex inequality system are considered and conditions are obtained for the system to be well conditioned under these types of perturbations where the well conditionedness of a convex inequality system is defined in terms of the uniform boundedness of condition numbers under a set of perturbations it is shown that certain types of perturbations can be used to characterize the well conditionedness of a convex inequality system in which either the system has a bounded solution set and satisfies the slater condition or an associated convex inequality system which defines the recession cone of the solution set for the system satisfies the slater condition finally sufficient conditions are given for the existence of a global error bound for an analytic system it is shown that such a global error bound always holds for any inequality system defined by finitely many convex analytic functions when the zero vector is in the relative interior of the domain of an associated convex conjugate function

optimization problems with perturbations a guided tour this paper presents an overview of some recent and significant progress in the theory of optimization problems with perturbations we put the emphasis on methods based on upper and lower estimates of the objective function of the perturbed problems these methods allow one to compute expansions of the optimal value function and approximate optimal solutions in situations where the set of lagrange multipliers is not a singleton may be unbounded or is even empty we give rather complete results for nonlinear programming problems and describe some extensions of the method to more general problems we illustrate the results by computing the equilibrium position of a chain that is almost vertical or horizontal

a comparison of computational color constancy algorithms ii experiments with image data for pt i see ibid vol 11 no 9 p 972 84 2002 we test a number of the leading computational color constancy algorithms using a comprehensive set of images these were of 33 different scenes under 11 different sources representative of common illumination conditions the algorithms studied include two gray world methods a version of the retinex method several variants of forsyth s 1990 gamut mapping method cardei et al s 2000 neural net method and finlayson et al s color by correlation method finlayson et al 1997 2001 hubel and finlayson 2000 we discuss a number of issues in applying color constancy ideas to image data and study in depth the effect of different preprocessing strategies we compare the performance of the algorithms on image data with their performance on synthesized data all data used for this study are available online at http www cs sfu ca spl sim color data and implementations for most of the algorithms are also available http www cs sfu ca spl sim color code experiments with synthesized data part one of this paper suggested that the methods which emphasize the use of the input data statistics specifically color by correlation and the neural net algorithm are potentially the most effective at estimating the chromaticity of the scene illuminant unfortunately we were unable to realize comparable performance on real images here exploiting pixel intensity proved to be more beneficial than exploiting the details of image chromaticity statistics and the three dimensional 3 d gamut mapping algorithms gave the best performance

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

global error bounds for convex inequality systems in banach spaces we study conditions under which a global error bound in terms of a natural residual exists for a convex inequality system specifically we obtain an error bound result which unifies many existing results assuming a slater condition we also derive two characterizations for a convex inequality system to possess a global error bound one is in terms of metric regularity and the other is in terms of an associated convex inequality system as a consequence we show that in font bbbabstract msbm10 at 8pt mbox bbbabstract r n a global error bound holds for such a system under the assumption of the zero vector in the relative interior of the domain of an associated conjugate function along with metric regularity at every point of the feasible set defined by the system finally we discuss some applications of these results to convex programs

a comparison of computational color constancy algorithms i methodology and experiments with synthesized data we introduce a context for testing computational color constancy specify our approach to the implementation of a number of the leading algorithms and report the results of three experiments using synthesized data experiments using synthesized data are important because the ground truth is known possible confounds due to camera characterization and pre processing are absent and various factors affecting color constancy can be efficiently investigated because they can be manipulated individually and precisely the algorithms chosen for close study include two gray world methods a limiting case of a version of the retinex method a number of variants of forsyth s 1990 gamut mapping method cardei et al s 2000 neural net method and finlayson et al s color by correlation method finlayson et al 1997 2001 hubel and finlayson 2000 we investigate the ability of these algorithms to make estimates of three different color constancy quantities the chromaticity of the scene illuminant the overall magnitude of that illuminant and a corrected illumination invariant image we consider algorithm performance as a function of the number of surfaces in scenes generated from reflectance spectra the relative effect on the algorithms of added specularities and the effect of subsequent clipping of the data all data is available on line at http www cs sfu ca spl sim color data and implementations for most of the algorithms are also available http www cs sfu ca spl sim color code

the optimal value and optimal solutions of the proximal average of convex functions abstract the proximal average of a finite collection of convex functions is a parameterized convex function that provides a continuous transformation between the convex functions in the collection this paper analyzes the dependence of the optimal value and the minimizers of the proximal average on the weighting parameter concavity of the optimal value is established and implies further regularity properties of the optimal value boundedness outer semicontinuity single valuedness continuity and lipschitz continuity of the minimizer mapping are concluded under various assumptions sharp minimizers are given further attention several examples are given to illustrate our results

a class of gap functions for variational inequalities recently auchmuty 1989 has introduced a new class of merit functions or optimization formulations for variational inequalities in finite dimensional space we develop and generalize auchmuty s results and relate his class of merit functions to other works done in this field especially we investigate differentiability and convexity properties and present characterizations of the set of solutions to variational inequalities we then present new descent algorithms for variational inequalities within this framework including approximate solutions of the direction finding and line search problems the new class of merit functions include the primal and dual gap functions introduced by zuhovickii et al 1969a 1969b and the differentiable merit function recently presented by fukushima 1992 also the descent algorithm proposed by fukushima is a special case from the class of descent methods developed in this paper through a generalization of auchmuty s class of merit functions we extend those inherent in the works of dafermos 1983 cohen 1988 and wu et al 1991 new algorithmic equivalence results relating these algorithm classes to each other and to auchmuty s framework are also given

on the rate of convergence of optimal solutions of monte carlo approximations of stochastic programs in this paper we discuss monte carlo simulation based approximations of a stochastic programming problem we show that if the corresponding random functions are convex piecewise linear and the distribution is discrete then an optimal solution of the approximating problem provides an exact optimal solution of the true problem with probability one for sufficiently large sample size moreover by using the theory of large deviations we show that the probability of such an event approaches one exponentially fast with increase of the sample size in particular this happens in the case of linear two or multi stage stochastic programming with recourse if the corresponding distributions are discrete the obtained results suggest that in such cases monte carlo simulation based methods could be very efficient we present some numerical examples to illustrate the ideas involved

weak sharp solutions of variational inequalities xuexiang huang pointed out that the statement on line 5 of page 184 of p marcotte and d zhu siam j optim 9 1999 pp 179 189 is invalid and supplied an alternative proof of the inequality that appears on line 10 here is a simple proof of this result due to an anonymous referee the substitution starts at line 4 and ends at line 11

on finite convergence of proximal point algorithms for variational inequalities in this paper we first characterize finite convergence of an arbitrary iterative algorithm for solving the variational inequality problem vip where the finite convergence means that the algorithm can find an exact solution of the problem in a finite number of iterations by using this result we obtain that the well known proximal point algorithm possesses finite convergence if the solution set of vip is weakly sharp as an extension we show finite convergence of the inertial proximal method for solving the general variational inequality problem under the condition of weak g sharpness

weak sharp minima for set valued vector variational inequalities with an application abstract in this paper the notion of weak sharp minima is employed to the investigation of set valued vector variational inequalities the gap function t for set valued strong vector variational inequalities for short svvi is proved to be less than the gap function t for set valued weak vector variational inequalities for short wvvi under certain conditions which implies that the solution set of svvi is equivalent to the solution set of wvvi moreover it is shown that weak sharp minima for the solution sets of svvi and wvvi hold for min 1 i n p t i and for gap functions t and t under the assumption of strong pseudomonotonicity where p t i is a gap function for i th component of svvi and wvvi as an application the weak pareto solution set of vector optimization problems for short vop is proved to be weak sharp minimum for min 1 i n p g i when each component g i of objective function is strongly convex

lagrange multipliers exact regularization and error bounds for monotone variational inequalities we examine two central regularization strategies for monotone variational inequalities the first a direct regularization of the operative monotone mapping and the second via regularization of the associated dual gap function a key link in the relationship between the solution sets to these various regularized problems is the idea of exact regularization which in turn is fundamentally associated with the existence of lagrange multipliers for the regularized variational inequality a regularization is said to be exact if a solution to the regularized problem is a solution to the unregularized problem for all parameters beyond a certain value the lagrange multipliers corresponding to a particular regularization of a variational inequality on the other hand are defined via the dual gap function our analysis suggests various conceptual iteratively regularized numerical schemes for which we provide error bounds and hence stopping criteria under the additional assumption that the solution set to the unregularized problem is what we call weakly sharp of order greater than one

on convergence of the gauss newton method for convex composite optimization the local quadratic convergence of the gauss newton method for convex composite optimization f h f is established for any convex function h with the minima set c extending burke and ferris results in the case when c is a set of weak sharp minima for h

a gauss newton method for convex composite optimization an extension of the gauss newton method for nonlinear equations to convex composite optimization is described and analyzed local quadratic convergence is established for the minimization ofh f under two conditions namelyh has a set of weak sharp minima c and there is a regular point of the inclusionf x c this result extends a similar convergence result due to womersley this journal 1985 which employs the assumption of a strongly unique solution of the composite functionh f a backtracking line search is proposed as a globalization strategy for this algorithm a global convergence result is established with a quadratic rate under the regularity assumption

quantitative stability in stochastic programming in this paper we study stability of optimal solutions of stochastic programming problems with fixed recourse an upper bound for the rate of convergence is given in terms of the objective functions of the associated deterministic problems as an example it is shown how it can be applied to derivation of the law of iterated logarithm for the optimal solutions it is also shown that in the case of simple recourse this upper bound implies upper lipschitz continuity of the optimal solutions with respect to the kolmogorov smirnov distance between the corresponding cumulative probability distribution functions

weak sharp minima in mathematical programming the notion of a sharp or strongly unique minimum is extended to include the possibility of a nonunique solution set these minima will be called weak sharp minima conditions necessary for the solution set of a minimization problem to be a set of weak sharp minima are developed in both the unconstrained and constrained cases these conditions are also shown to be sufficient under the appropriate convexity hypotheses the existence of weak sharp minima is characterized in the cases of linear and quadratic convex programming and for the linear complementarity problem in particular a result of mangasarian and meyer is reproduced that shows that the solution set of a linear program is always a set of weak sharp minima whenever it is nonempty consequences for the convergence theory of algorithms are also examined especially conditions yielding finite termination

weak sharp minima for semi infinite optimization problems with applications we study local weak sharp minima and sharp minima for smooth semi infinite optimization problems sip we provide several dual and primal characterizations for a point to be a sharp minimum or a weak sharp minimum of sip as applications we present several sufficient and necessary conditions of calmness for infinitely many smooth inequalities in particular we improve some calmness results in r henrion and j outrata math program 104 2005 pp 437 464

weak sharpness for gap functions in vector variational inequalities abstract in this paper characterizations of the set of solutions for vvi are presented by using scalarization approaches the set of solutions of vvi is shown to be the set of weak sharpness for gap functions of some scalarization of vvi and for gap functions of vvi under semi strong monotonicity some examples are given to illustrate these results

random algorithms for convex minimization problems this paper deals with iterative gradient and subgradient methods with random feasibility steps for solving constrained convex minimization problems where the constraint set is specified as the intersection of possibly infinitely many constraint sets each constraint set is assumed to be given as a level set of a convex but not necessarily differentiable function the proposed algorithms are applicable to the situation where the whole constraint set of the problem is not known in advance but it is rather learned in time through observations also the algorithms are of interest for constrained optimization problems where the constraints are known but the number of constraints is either large or not finite we analyze the proposed algorithm for the case when the objective function is differentiable with lipschitz gradients and the case when the objective function is not necessarily differentiable the behavior of the algorithm is investigated both for diminishing and non diminishing stepsize values the almost sure convergence to an optimal solution is established for diminishing stepsize for non diminishing stepsize the error bounds are established for the expected distances of the weighted averages of the iterates from the constraint set as well as for the expected sub optimality of the function values along the weighted averages

faster subgradient methods for functions with holderian growth the purpose of this manuscript is to derive new convergence results for several subgradient methods applied to minimizing nonsmooth convex functions with holderian growth the growth condition is satisfied in many applications and includes functions with quadratic growth and weakly sharp minima as special cases to this end there are three main contributions first for a constant and sufficiently small stepsize we show that the subgradient method achieves linear convergence up to a certain region including the optimal set with error of the order of the stepsize second if appropriate problem parameters are known we derive a decaying stepsize which obtains a much faster convergence rate than is suggested by the classical o 1 sqrt k result for the subgradient method thirdly we develop a novel descending stairs stepsize which obtains this faster convergence rate and also obtains linear convergence for the special case of weakly sharp functions we also develop an adaptive variant of the descending stairs stepsize which achieves the same convergence rate without requiring an error bound constant which is difficult to estimate in practice

convergence study of decentralized min cost subgraph algorithms for multicast in coded networks the problem of establishing minimum cost multicast connections in coded networks can be viewed as an optimization problem and decentralized algorithms were proposed by lun to compute the optimal subgraph using the dual subgradient method however the convergence rate problem for these algorithms remains open there are limited results in the literature which bound the amount of infeasibility of the primal solution recovered after each iterations or the convergence rate however due to the special structure of the network coding problem we have an algorithm that generates a feasible solution after each iterations in addition the convergence rate of the primal problem is o 1 n to a neighborhood of the optimal solution we also propose heuristics to further improve our algorithm and demonstrate through simulations that the distributed algorithm converges to the optimal subgraph quickly and is robust against network topology changes

weak sharp minima for convex infinite optimization problems in normed linear spaces in the present paper we study the characterization issue of the weak sharp minima properties for convex infinite optimization problems in normed linear spaces we develop a new approach to establi

a novel algorithm for color constancy color constancy is the skill by which it is possible to tell the color of an object even under a colored light i interpret the color of an object as its color under a fixed canonical light rather than as a surface reflectance function this leads to an analysis that shows two distinct sets of circumstances under which color constancy is possible in this framework color constancy requires estimating the illuminant under which the image was taken the estimate is then used to choose one of a set of linear maps which is applied to the image to yield a color descriptor at each point this set of maps is computed in advance the illuminant can be estimated using image measurements alone because given a number of weak assumptions detailed in the text the color of the illuminant is constrained by the colors observed in the image this constraint arises from the fact that surfaces can reflect no more light than is cast on them for example if one observes a patch that excites the red receptor strongly the illuminant cannot have been deep blue two algorithms are possible using this constraint corresponding to different assumptions about the world the first algorithm crule will work for any surface reflectance crule corresponds to a form of coefficient rule but obtains the coefficients by using constraints on illuminant color the set of illuminants for which crule will be successful depends strongly on the choice of photoreceptors for narrowband photoreceptors crule will work in an unrestricted world the second algorithm mwext requires that both surface reflectances and illuminants be chosen from finite dimensional spaces but under these restrictive conditions it can recover a large number of parameters in the illuminant and is not an attractive model of human color constancy crule has been tested on real images of mondriaans and works well i show results for crule and for the retinex algorithm of land land 1971 land 1983 land 1985 operating on a number of real images the experimental work shows that for good constancy a color constancy system will need to adjust the gain of the receptors it employs in a fashion analagous to adaptation in humans

inexact subgradient methods for quasi convex optimization problems in this paper we consider a generic inexact subgradient algorithm to solve a nondifferentiable quasi convex constrained optimization problem the inexactness stems from computation errors and noise which come from practical considerations and applications assuming that the computational errors and noise are deterministic and bounded we study the effect of the inexactness on the subgradient method when the constraint set is compact or the objective function has a set of generalized weak sharp minima in both cases using the constant and diminishing stepsize rules we describe convergence results in both objective values and iterates and finite convergence to approximate optimality we also investigate efficiency estimates of iterates and apply the inexact subgradient algorithm to solve the cobb douglas production efficiency problem the numerical results verify our theoretical analysis and show the high efficiency of our proposed algorithm especially for the large scale problems

weak sharp minima on riemannian manifolds this is the first paper dealing with the study of weak sharp minima for constrained optimization problems on riemannian manifolds which are important in many applications we consider the notions of local weak sharp minima boundedly weak sharp minima and global weak sharp minima for such problems and establish their complete characterizations in the case of convex problems on finite dimensional riemannian manifolds and hadamard manifolds a number of the results obtained in this paper are also new for the case of conventional problems in finite dimensional euclidean spaces our methods involve appropriate tools of variational analysis and generalized differentiation on riemannian and hadamard manifolds developed and efficiently implemented in this paper

weak sharp solutions for variational inequalities in banach spaces abstract in this paper we introduce the notion of a weak sharp set of solutions to a variational inequality problem vip in a reflexive strictly convex and smooth banach space and present its several equivalent conditions we also prove under some continuity and monotonicity assumptions that if any sequence generated by an algorithm for solving vip converges to a weak sharp solution then we can obtain solutions for vip by solving a finite number of convex optimization subproblems with linear objective moreover in order to characterize finite convergence of an iterative algorithm we introduce the notion of a weak subsharp set of solutions to a variational inequality problem vip which is more general than that of weak sharp solutions in hilbert spaces we establish a sufficient and necessary condition for the finite convergence of an algorithm for solving vip which satisfies that the sequence generated by which converges to a weak subsharp solution of vip and show that the proximal point algorithm satisfies this condition as a consequence we prove that the proximal point algorithm possesses finite convergence whenever the sequence generated by which converges to a weak subsharp solution of vip

riccati parameter solutions of nonlinear second order odes it has been proven by rosu and cornejo perez rosu and cornejo perez 2005 phys rev e 71 046607 cornejo perez and rosu 2005 prog theor phys 114 533 that for some nonlinear second order odes it is a very simple task to find one particular solution once the nonlinear equation is factorized with the use of two first order differential operators here it is shown that an interesting class of parametric solutions is easy to obtain if the proposed factorization has a particular form which happily turns out to be the case in many problems of physical interest the method that we exemplify with a few explicitly solved cases consists in using the general solution of the riccati equation which contributes with one parameter to this class of parametric solutions for these nonlinear cases the riccati parameter serves as a growth parameter from the trivial null solution up to the particular solution found through the factorization procedure

weak sharp solutions of variational inequalities in hilbert spaces under some new conditions we present several equivalent and sufficient conditions for weak sharp solutions of variational inequalities in hilbert spaces and give a finite convergence result for a class of algorithms for solving variational inequalities

degrees of efficiency and degrees of minimality in this work we characterize different types of solutions of a vector optimization problem by means of a scalarization procedure usually different scalarizing functions are used in order to obtain the various solutions of the vector problem here we consider different kinds of solutions of the same scalarized problem our results allow us to establish a parallelism between the solutions of the scalarized problem and the various efficient frontiers stronger solution concepts of the scalar problem correspond to more restrictive notions of efficiency besides the usual notions of weakly efficient and efficient points which are characterized as global and strict global solutions of the scalarized problem we also consider some restricted notions of efficiency such as strict and proper efficiency which are characterized as tikhonov well posed minima and sharp minima for the scalarized problem

holder weak sharp minimizers and holder tilt stability abstract in this paper we introduce and study the notions of holder weak sharp minimizers stable holder weak sharp minimizers and holder tilt stable weak minimizers for a proper lower semicontinuous function f on a banach space in terms of the holder metric subregularity regularity of f we consider optimality conditions for holder weak sharp minimizers and stable holder weak sharp minimizers we prove that x is a stable holder weak sharp minimizer resp a holder tilt stable weak minimizer of f if and only if it is a stable holder sharp minimizer resp a holder tilt stable minimizer of f

the linear l 1 estimator and the huber m estimator relationships between a linear l1 estimation problem and the huber m estimator problem can be easily established by their dual formulations the least norm solution of a linear programming problem studied by mangasarian and meyer siam j control optim 17 1979 pp 745 752 provides a key link between the dual problems based on the dual formulations we establish a local linearity property of the huber m estimators with respect to the tuning parameter gamma and prove that the solution set of the huber m estimator problem is lipschitz continuous with respect to perturbations of the tuning parameter gamma as a consequence the set of the linear l1 estimators is the limit of the set of the huber m estimators as gamma to 0 thus the huber m estimator problem has many solutions for small tuning parameter gamma if the linear l1 estimation problem has multiple solutions a recursive version of madsen and nielsen s algorithm siam j optim 3 1993 pp 223 235 based on computation of the huber m estimator is proposed for finding a linear l1 estimator

phase separation dynamics in driven diffusive systems we study phase separation dynamics in a driven diffusive system our simulations are based on the cahn hilliard equation with an additional flux term due to an external field we study the dynamical scaling parallel and perpendicular to the field a crossover is observed from isotropic domains at early times to extremely anisotropic domains at later times we find that the inverse interfacial density an isotropic measure of the domain size increases ast with 1 3 from early times independent of the field strength even though we do not observe dynamical scaling during these times our results indicate that a growth exponent 1 3 may be more universal than previously expected we analyze the dynamics in terms of surface driven instabilities and one dimensional solitary waves

anomalous biennial oscillations in a fisher equation with a discretized verhulst term the dynamics of a biological population governed by a modified fisher equation is studied by means of monte carlo simulations reproduction of the population occurs at discrete times while transport caused by diffusion and conduction takes place on shorter time scales the discrete reproduction modeled with a set of coupled logistic maps exhibits phenomena which are not evident in the usual continuum version of the fisher equation several mechanisms for biennial oscillations of the total population are investigated one of these shows an ordered coupling between random diffusive motion and the chaotic attractor of the logistic map

conditional subgradient optimization theory and applications abstract we generalize the subgradient optimization method for nondifferentiable convex programming to utilize conditional subgradients firstly we derive the new method and establish its convergence by generalizing convergence results for traditional subgradient optimization secondly we consider a particular choice of conditional subgradients obtained by projections which leads to an easily implementable modification of traditional subgradient optimization schemes to evaluate the subgradient projection method we consider its use in three applications uncapacitated facility location two person zero sum matrix games and multicommodity network flows computational experiments show that the subgradient projection method performs better than traditional subgradient optimization in some cases the difference is considerable these results suggest that our simply modification may improve subgradient optimization schemes significantly this finding is important as such schemes are very popular especially in the context of lagrangean relaxation

conditions ensuring the applicability of cutting plane methods for solving variational inequalities let vip f c denote the variational inequality problem associated with the mapping f and the closed convex set c in this paper we introduce weak conditions on the mapping f that allow the development of a convergent cutting plane framework for solving vip f c in the process we introduce in a natural way new and useful notions of generalized monotonicity for which first order characterizations are presented

convergence properties of nonmonotone spectral projected gradient methods in a recent paper a nonmonotone spectral projected gradient spg method was introduced by birgin et al for the minimization of differentiable functions on closed convex sets and extensive presented results showed that this method was very efficient in this paper we give a more comprehensive theoretical analysis of the spg method in doing so we remove various boundedness conditions that are assumed in existing results such as boundedness from below of f boundedness of xk or existence of accumulation point of xk if f is uniformly continuous we establish the convergence theory of this method and prove that the spg method forces the sequence of projected gradients to zero moreover we show under appropriate conditions that the spg method has some encouraging convergence properties such as the global convergence of the sequence of iterates generated by this method and the finite termination etc therefore these results show that the spg method is attractive in theory

weak sharp minima in multicriteria linear programming in this short note we study the existence of weak sharp minima in multicriteria linear programming it is shown that weak sharp minimality holds for certain residual functions and gap functions

on convergence rates of linearized proximal algorithms for convex composite optimization with applications in the present paper we investigate a linearized proximal algorithm lpa for solving a convex composite optimization problem each iteration of the lpa is a proximal minimization of the convex composite function with the inner function being linearized at the current iterate the lpa has the attractive computational advantage that the solution of each subproblem is a singleton which avoids the difficulty as in the gauss newton method gnm of finding a solution with minimum norm among the set of minima of its subproblem while still maintaining the same local convergence rate as that of the gnm under the assumptions of local weak sharp minima of order p p in 1 2 and a quasi regularity condition we establish a local superlinear convergence rate for the lpa we also propose a globalization strategy for the lpa based on a backtracking line search and an inexact version of the lpa we further apply the lpa to solve a possibly nonconvex feasibility problem as well as a sensor network localiza

set regularities and feasibility problems we synthesize and unify notions of regularity both of individual sets and of collections of sets as they appear in the convergence theory of projection methods for consistent feasibility problems several new characterizations of regularities are presented which shed light on the relations between seemingly different ideas and point to possible necessary conditions for local linear convergence of fundamental algorithms

minimizing and stationary sequences of constrained optimization problems we study some fundamental asymptotic properties associated with the well posedness of constrained optimization problems emphasis is placed on the relation between minimizing and stationary sequences and their characterizations in terms of a set of asymptotic karush kuhn tucker kkt optimality conditions unlike the subdifferential approach used by auslender cominetti and crouzeix we use a residual function approach that is closely tied to the theory of error bounds this approach handles constraints explicitly and allows the effective treatment of infeasible stationary sequences the asymptotic kkt conditions provide an asymptotic optimality certificate for inequality constrained programs specializations of the results to convex quadratically constrained convex quadratic spline minimization problems and convex programs with holderian minima are discussed an application of the results to the convergence of a family of newton type iterative descent methods involving singular quasi newton matrices for solving a constrained minimization problem is also presented

nonlinear bias and the convective fisher equation we combine random walks growth and decay and convection in a monte carlo simulation to model 1d interface dynamics with fluctuations the continuum limit corresponds to the deterministic fisher equation with convection we find qualitatively the same type of asymmetry as well as velocity difference for interface profiles moving in opposite directions however a transition apparent in the mean field continuum limit is not found in the monte carlo simulation

characterizations of weak sharp minima for lower c1 functions abstract we derive characterizations of weak sharp minimizers of order one for the class of lower c 1 functions the characterizations of such minimizers are obtained for a nonlinear programming problem with an abstract set constraint the first characterization is formulated in terms of the proximal normals to a given set relative to the abstract set constraint and the directional derivative of the objective function two examples are given to illustrate this characterization the other characterizations are extensions of the main characterizations of weak sharp minima for convex functions recently proved by burke and deng 2002 13 theorems 2 2 and 2 3 to the class of lower c 1 functions

global weak sharp minima on banach spaces we consider a proper lower semicontinuous function f on a banach space x with lambda inf f x x in x infty let alpha geq lambda and s alpha x in x f x leq alpha we define the lower derivative of f at the set s alpha by underline d f s alpha liminf x rightarrow s alpha frac f x alpha dist x s alpha where x rightarrow s alpha can be interpreted in various ways we show that when f is convex and alpha lambda it is equal to the largest weak sharp minima constant in terms of these derivatives and subdifferentials we present several characterizations for convex f to have global weak sharp minima some of these results are also shown to be valid for nonconvex f as applications we give error bound results for abstract linear inequality systems

error bounds for eigenvalue and semidefinite matrix inequality systems in this paper we give sufficient conditions for existence of error bounds for systems expressed in terms of eigenvalue functions such as in eigenvalue optimization or positive semidefiniteness such as in semidefinite programming

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

the effect of deterministic noise in subgradient methods in this paper we study the influence of noise on subgradient methods for convex constrained optimization the noise may be due to various sources and is manifested in inexact computation of the subgradients and function values assuming that the noise is deterministic and bounded we discuss the convergence properties for two cases the case where the constraint set is compact and the case where this set need not be compact but the objective function has a sharp set of minima for example the function is polyhedral in both cases using several different stepsize rules we prove convergence to the optimal value within some tolerance that is given explicitly in terms of the errors in the first case the tolerance is nonzero but in the second case the optimal value can be obtained exactly provided the size of the error in the subgradient computation is below some threshold we then extend these results to objective functions that are the sum of a large number of convex functions in which case an incremental subgradient method can be used

weak sharp minima for piecewise linear multiobjective optimization in normed spaces abstract in a general normed space we consider a piecewise linear multiobjective optimization problem we prove that a cone convex piecewise linear multiobjective optimization problem always has a global weak sharp minimum property by a counter example we show that the weak sharp minimum property does not necessarily hold if the cone convexity assumption is dropped moreover under the assumption that the ordering cone is polyhedral we prove that a not necessarily cone convex piecewise linear multiobjective optimization problem always has a bounded weak sharp minimum property

color by correlation a simple unifying framework for color constancy the paper considers the problem of illuminant estimation how given an image of a scene recorded under an unknown light we can recover an estimate of that light obtaining such an estimate is a central part of solving the color constancy problem thus the work presented will have applications in fields such as color based object recognition and digital photography rather than attempting to recover a single estimate of the illuminant we instead set out to recover a measure of the likelihood that each of a set of possible illuminants was the scene illuminant we begin by determining which image colors can occur and how these colors are distributed under each of a set of possible lights we discuss how for a given camera we can obtain this knowledge we then correlate this information with the colors in a particular image to obtain a measure of the likelihood that each of the possible lights was the scene illuminant finally we use this likelihood information to choose a single light as an estimate of the scene illuminant computation is expressed and performed in a generic correlation framework which we develop we propose a new probabilistic instantiation of this correlation framework and show that it delivers very good color constancy on both synthetic and real images we further show that the proposed framework is rich enough to allow many existing algorithms to be expressed within it the gray world and gamut mapping algorithms are presented in this framework and we also explore the relationship of these algorithms to other probabilistic and neural network approaches to color constancy

weak sharp minima revisited part iii error bounds for differentiable convex inclusions the notion of weak sharp minima unifies a number of important ideas in optimization part i of this work provides the foundation for the theory of weak sharp minima in the infinite dimensional setting part ii discusses applications of these results to linear regularity and error bounds for nondifferentiable convex inequalities this work applies the results of part i to error bounds for differentiable convex inclusions a number of standard constraint qualifications for such inclusions are also examined

a multilinear constraint on dichromatic planes for illumination estimation a new multilinear constraint on the color of the scene illuminant based on the dichromatic reflection model is proposed the formulation avoids the problem common to previous dichromatic methods of having to first identify pixels corresponding to the same surface material once pixels from two or more materials have been identified their corresponding dichromatic planes can be intersected to yield the illuminant color however it is not always easy to determine which pixels from an arbitrary region of an image belong to which dichromatic plane the image region may cover an area of the scene encompassing several different materials and hence pixels from several different dichromatic planes the new multilinear constraint accounts for this multiplicity of materials and provides a mechanism for choosing the most plausible illuminant from a finite set of candidate illuminants the performance of this new method is tested on a database of real images

complete characterizations of local weak sharp minima with applications to semi infinite optimization and complementarity abstract in this paper we identify a favorable class of nonsmooth functions for which local weak sharp minima can be completely characterized in terms of normal cones and subdifferentials or tangent cones and subderivatives or their mixture in finite dimensional spaces the results obtained not only extend previous ones in the literature but also allow us to provide new types of criteria for local weak sharpness applications of the developed theory are given to semi infinite programming and to a new class of semi infinite complementarity problems

weak sharp minima revisited part ii application to linear regularity and error bounds the notion of weak sharp minima is an important tool in the analysis of the perturbation behavior of certain classes of optimization problems as well as in the convergence analysis of algorithms designed to solve these problems it has been studied extensively by several authors this paper is the second of a series on this subject where the basic results on weak sharp minima in part i are applied to a number of important problems in convex programming in part ii we study applications to the linear regularity and bounded linear regularity of a finite collection of convex sets as well as global error bounds in convex programming we obtain both new results and reproduce several existing results from a fresh perspective

color constancy from mutual reflection mutual reflection occurs when light reflected from one surface illuminates a second surface in this situation the color of one or both surfaces can be modified by a color bleeding effect in this article we examine how sensor values e g rgb values are modified in the mutual reflection region and show that a good approximation of the surface spectral reflectance function for each surface can be recovered by using the extra information from mutual reflection thus color constancy results from an examination of mutual reflection use is made of finite dimensional linear models for ambient illumination and for surface spectral reflectance if m and n are the number of basis functions required to model illumination and surface spectral reflectance respectively then we find that the number of different sensor classes p must satisfy the condition p 2 n m 3 if we use three basis functions to model illumination and three basis functions to model surface spectral reflectance then only three classes of sensors are required to carry out the algorithm results are presented showing a small increase in error over the error inherent in the underlying finite dimension models

gâteaux differentiability of the dual gap function of a variational inequality abstract in terms of the mapping involved in a variational inequality we characterize the g teaux differentiability of the dual gap function g and present several sufficient conditions for its directional derivative expression including one weaker than that of danskin j m danskin the theory of max min with applications siam journal on applied mathematics 14 1966 641 664 when the solution set of a variational inequality problem is contained in that of its dual problem the g teaux differentiability of g on the latter turns out to be equivalent to the conditions appearing in the authors recent results about the weakly sharp solutions of the variational inequality problem

characterizations of weakly sharp solutions for a variational inequality with a pseudomonotone mapping abstract for a variational inequality problem with a pseudomonotone mapping f we characterize the weak sharpness of its solution set c without using gap functions when the mapping f is also constant on c the characterizations of weak sharpness of c become more succinct as an example a pseudomonotone mapping on c is shown to be constant on c consequently the weak sharpness of c can further be described by a g teaux differentiable function which itself characterizes the pseudomonotonicity of f on c it turns out that several existing relevant results with differentiable gap functions can be obtained from ours