subgradients of optimal value functions in dynamic programming the case of convex systems without optimal paths we study the first order behaviour of the optimal value function associated to a convex dynamic programming problem the optimization process takes place in a certain environment characterized by some perturbation parameters affecting the transition costs and or the evolution law of the dynamic system an important aspect of this work is that we do not assume the existence of optimal paths to the unperturbed problem

penalty functions in e programming and e minimax problems several authors have been intersted in optimality conditions in e programming and in e minimas problem see for example the references 11 16

an empirical bayes model for time varying paired comparisons ratings who is the greatest women s tennis player we present a methodology for fitting a time varying paired comparisons model using an empirical bayes approach the model simultaneously avoids two problems that typically arise with paired comparisons data first that extreme values of estimated strengths can occur for competitors appearing in and winning a small number of games producing absurd rankings and second that the time varying strengths balloon over time the empirical bayes approach automatically shrinks the strength estimates towards the mean thus avoiding both issues we present our model and demonstrate its use in the setting of tennis in search of an answer to the question who is the greatest women s player of all time our results suggest that steffi graf is a strong candidate but using confidence intervals on the rankings themselves others cannot be ruled out

extensions of frechet ϵ subdifferential calculus and applications abstract in this paper we establish some calculus rules for the limiting frechet subdifferentials of marginal functions and composite functions necessary conditions for approximate solutions of a constrained optimization problem are derived

mean value in invexity analysis abstract in this paper a generalization of the mean value theorem is considered in the case of functions defined on an invex set with respect to which is not necessarily connected

an interior proximal method in vector optimization abstract this paper studies the vector optimization problem of finding weakly efficient points for maps from r n to r m with respect to the partial order induced by a closed convex and pointed cone c r m with nonempty interior we develop for this problem an extension of the proximal point method for scalar valued convex optimization problem with a modified convergence sensing condition that allows us to construct an interior proximal method for solving vop on nonpolyhedral set

a brezis browder principle on partially ordered spaces and related ordering theorems abstract through a simple extension of brezis browder principle to partially ordered spaces a very general strong minimal point existence theorem on quasi ordered spaces is proved this theorem together with a generic quasi order and a new notion of strong approximate solution allow us to obtain two strong solution existence theorems and three general ekeland variational principles in optimization problems where the objective space is quasi ordered then they are applied to prove strong minimal point existence results generalizations of bishop phelps lemma in linear spaces and ekeland variational principles in set valued optimization problems through a set solution criterion

optimality conditions via scalarization for a new e efficiency concept in vector optimization problems abstract in this work necessary and sufficient conditions for approximate solutions of vector optimization problems are obtained via scalarization i e by considering approximate solutions of associated scalar optimization problems these conditions are proved through a new e efficiency concept and two very general assumptions on the scalarization that extend the usual order representing and monotonicity properties moreover neither solidness hypothesis on the order cone nor monotonicity property on the scalarization are assumed

generalized nonsmooth invexity over cones in vector optimization abstract in this paper k nonsmooth quasi invex and strictly or strongly k nonsmooth pseudo invex functions are defined by utilizing these new concepts the fritz john type and kuhn tucker type necessary optimality conditions and number of sufficient optimality conditions are established for a nonsmooth vector optimization problem wherein clarke s generalized gradient is used further a mond weir type dual is associated and weak and strong duality results are obtained

improvement sets and vector optimization abstract in this paper we focus on minimal points in linear spaces and minimal solutions of vector optimization problems where the preference relation is defined via an improvement set e to be precise we extend the notion of e optimal point due to chicco et al in 4 to a general non necessarily pareto quasi ordered linear space and we study its properties in particular we relate the notion of improvement set with other similar concepts of the literature and we characterize it by means of sublevel sets of scalar functions moreover we obtain necessary and sufficient conditions for e optimal solutions of vector optimization problems through scalarization processes by assuming convexity assumptions and also in the general nonconvex case by applying the obtained results to certain improvement sets we generalize well known results of the literature referred to efficient weak efficient and approximate efficient solutions of vector optimization problems

e optimality and duality for multiobjective fractional programming abstract using the scalar e parametric approach we establish the karush kuhn tucker which we call kkt necessary and sufficient conditions for an e pareto optimum of nondifferentiable multiobjective fractional objective functions subject to nondifferentiable convex inequality constraints linear equality constraints and abstract constraints these optimality criteria are utilized as a basis for constructing one duality model with appropriate duality theorems subsequently we employ scalar exact penalty function to transform the multiobjective fractional programming problem to an unconstrained problem under this case we derive the kkt necessary and sufficient conditions without a constraint qualification for e pareto optimality of multiobjective fractional programming

multistage ranking models abstract suppose that a sample of people independently examines a fixed set of k items and then ranks these items according to personal judgment the process of ranking the items is decomposed into k 1 stages in the forward model the most preferred item is selected at the first stage the best of the remaining items is selected at the second stage and so on until the least preferred item is selected by default various probability models are adopted at each stage and properties of the resulting models for randomly sampled rankings are investigated luce 1959 first proposed such a modeling scheme where each item i was thought to have an intrinsic value i and the probability of choosing a particular item i at any stage conditional on the set s of items not previously chosen was given by i i s i j s j where i is an indicator function plackett 1975 began with the same model but added interaction terms that would theoretically extend the usefulness of this approach when the basic m

pointwise well posedness in set optimization with cone proper sets abstract this paper deals with the well posedness property in the setting of set optimization problems by using a notion of well posed set optimization problem due to zhang et al 2009 18 and a scalarization process we characterize this property through the well posedness in the tykhonov sense of a family of scalar optimization problems and we show that certain quasiconvex set optimization problems are well posed our approach is based just on a weak boundedness assumption called cone properness that is unavoidable to obtain a meaningful set optimization problem

a class of b p r invex functions and mathematical programming abstract invexity of a function is generalized the new class of nonconvex functions called b p r invex functions with respect to and b being introduced includes many well known classes of generalized invex functions as its subclasses some properties of the introduced class of b p r invex functions with respect to and b are studied further mathematical programming problems involving b p r invex functions with respect to and b are considered the equivalence between saddle points and optima and different type duality theorems are established for this type of optimization problems

invertibility and spectral properties of dual toeplitz operators abstract in this paper we consider dual toeplitz operators with continuous symbols and bounded harmonic symbols on the orthogonal complement of the bergman space over the open unit disk we construct a dual toeplitz operator with continuous symbol such that its spectrum is disconnected on the other hand we show that the spectra of dual toeplitz operators with certain class of bounded harmonic symbols are connected which leads to some partial answers to the open question posed by stroethoff and zheng 18 in 2002 we obtain a complete characterization for the hyponormality of dual toeplitz operators with bounded harmonic symbols moreover we establish some sufficient conditions that are convenient to check for dual toeplitz operators with continuous symbols and bounded harmonic symbols to be invertible respectively

epsilon approximate solutions for multiobjective programming problems abstract in this paper approximate solutions for multiobjective programming problems up to 0 are investigated the concepts of approximate solutions were introduced by white and loridan the connections with these concepts are studied also the sufficient and necessary conditions to obtain the elements of the smallest approximate solution set are verified

a unified approach and optimality conditions for approximate solutions of vector optimization problems this paper deals with approximate v efficient solutions of vector optimization problems we introduce a new v efficiency concept which extends and unifies different approximate solution notions introduced in the literature we obtain necessary and sufficient conditions via nonlinear scalarization which allow us to study this new class of approximate solutions in a general framework since any convexity hypothesis is required several examples are proposed to show the concepts introduced and the results attained

global optimality conditions for discrete and nonconvex optimizationwith applications to lagrangian heuristics and column generation the well known and established global optimality conditions based on the lagrangian formulation of an optimization problem are consistent if and only if the duality gap is zero we develop a set of global optimality conditions that are structurally similar but are consistent for any size of the duality gap this system characterizes a primaldual optimal solution by means of primal and dual feasibility primal lagrangian e optimality and in the presence of inequality constraints a relaxed complementarity condition analogously called complementarity the total size e of those two perturbations equals the size of the duality gap at an optimal solution further the characterization is equivalent to a near saddle point condition which generalizes the classic saddle point characterization of a primaldual optimal solution in convex programming the system developed can be used to explain to a large degree when and why lagrangian heuristics for discrete optimization are successful in reaching near optimal solutions further experiments on a set covering problem illustrate how the new optimality conditions can be utilized as a foundation for the construction of new lagrangian heuristics finally we outline possible uses of the optimality conditions in column generation algorithms and in the construction of core problems

algebraic and spectral properties of dual toeplitz operators dual toeplitz operators on the orthogonal complement of the bergman space are defined to be multiplication operators followed by projection onto the orthogonal complement in this paper we study algebraic and spectral properties of dual toeplitz operators

nonlinear programming with e preinvex and local e preinvex functions abstract in this paper we extend the class of e convex sets e convex and e quasiconvex functions introduced by youness e a 1999 e convex sets e convex functions and e convex programming journal of optimization theory and applications 102 439 450 respectively by syau yu ru lee e stanley 2005 some properties of e convex functions applied mathematics letters 18 1074 1080 to e invex set e preinvex e prequasiinvex and corresponding local concepts some properties of these classes are studied as an application of our results we consider the nonlinear programming problem for which we establish that under mild conditions a local minimum is a global minimum

ϵ pareto optimality for nondifferentiable multiobjective programming via penalty function abstract necessary and sufficient conditions without a constraint qualification for pareto optimality of multiobjective programming are derived the necessary kuhn tucker condition suggests the establishment of a wolf type duality theorem for nondifferentiable convex multiobjective minimization problems the generalized saddle point for pareto optimality of the vector lagrangian is studied

pareto subdifferential calculus for convex vector mappings and applications to vector optimization this paper deals with the subdifferential of convex analysis defined in the pareto sense from a point of view of nonvacuity characterizations and calculus rules and their applications to the vector optimization the convex maps being vector valued in a finite or infinite dimensional ordered vector space subdifferentiability is characterized under conditions of attouch brezis type formulations by derivatives when they exist are provided concerning the calculus rules the first main result gives the gap between the pareto subdifferential and the ordinary one allowing thus the computation of the one from the other next as central results pareto subdifferentials of the sum and or composition of two convex vector mappings are developed the formulas are obtained under moreau rockafellar or attouch brezis type conditions revealing strangely the presence of the ordinary subdifferential these formulas actually allow the extension of the indicator function technique to the vector case so that pareto optimality efficiency conditions are easily derived and weakened with qualification conditions of the attouch brezis kind finally the gap between efficient and optimal sets is also deduced

an extension of ekeland s variational principle to locally complete spaces we prove an extension of ekeland s variational principle to locally complete spaces which uses subadditive strictly increasing continuous functions as perturbations

some nondifferentiable multiobjective programming problems in this paper it is shown that a relaxation defining the class of generalized d v type i functions leads to a new class of multi objective problems which preserves the sufficient optimality and duality results in the scalar non differentiable case and avoids the major difficulty of verifying that the inequality holds for the same kernel function the results obtained in this paper generalize and extend the previously known results in this area

on hadamard well posedness of families of pareto optimization problems abstract this paper deals with the well posedness of families of finite dimensional vector optimization problems ordered by components pareto problem for this problem two hadamard well posedness concepts are introduced these concepts involve the existence and uniqueness of efficient weak efficient solutions and also the continuous behavior of these solutions with respect to perturbations of the data the perturbations in the last property are formulated through a variational convergence notion of the objective functions and by considering approximate solutions of the perturbed problems necessary and sufficient conditions for the well posedness of pareto optimization problems are obtained in general and also under convexity and quasiconvexity assumptions to do this asymptotic mathematical tools are employed finally it is proved that the convex pareto optimization problems are essentially well posed in the sense of category theory

relative pareto minimizers for multiobjective problems existence and optimality conditions in this paper we introduce and study enhanced notions of relative pareto minimizers for constrained multiobjective problems that are defined via several kinds of relative interiors of ordering cones and occupy intermediate positions between the classical notions of pareto and weak pareto efficiency minimality using advanced tools of variational analysis and generalized differentiation we establish the existence of relative pareto minimizers for general multiobjective problems under a refined version of the subdifferential palais smale condition for set valued mappings with values in partially ordered spaces and then derive necessary optimality conditions for these minimizers as well as for conventional efficient and weak efficient counterparts that are new in both finite dimensional and infinite dimensional settings our proofs are based on variational and extremal principles of variational analysis in particular on new versions of the ekeland variational principle and the subdifferential variational principle for set valued and single valued mappings in infinite dimensional spaces

composite nonsmooth multiobjective programs with v ρ invexity abstract in this paper some problems consisting of nonsmooth composite multiobjective programs have been treated with v invexity type of conditions in particular we prove the generalized karush kuhn tucker sufficient optimality theorem and duality theorems for nonsmooth composite multiobjective programs also weak vector saddle point theorems are obtained for the composite programs under v invexity conditions the results obtained generalize the results of hun kuk et al to some extent

generating epsilon efficient solutions in multiobjective programming scalarization approaches to purposely generating e efficient solutions of multiobjective programs are investigated and a generic procedure for computing these solutions is proposed and illustrated with an example real life decision making situations in which the solutions are of significance are described

models for distributions on permutations abstract a parametric distribution on permutations of k objects is derived from gamma random variables the probability of a permutation is set equal to the probability that k independent gamma random variables with common shape parameter and different scale parameters are ranked according to that permutation this distribution is motivated by considering a competition in which k players scoring points according to independent poisson processes are ranked according to the time until r points are scored the distributions obtained in this way include the popular luce plackett and thurstone mosteller daniels ranking models these gamma based distributions can serve as alternatives to the null ranking model in which all permutations are equally likely here the gamma models are used to estimate the probability distribution of the order of finish in a horse race when only the probability of finishing first is given for each horse gamma models with shape parameters larger than 1 are found to be superior to

e optimality for multiobjective programming on a banach space abstract we establish a lagrange multiplier rule in terms of clarke generalized gradient that characterizes e efficiency of a nondifferentiable multiobjective programming problem on a real banach space this rule is then utilized to establish relationship between e efficient solution of multiobjective program and generalized saddle point of an appropriate vector valued lagrangian under invexity hypothesis

a unified concept of approximate and quasi efficient solutions and associated subdifferentials in multiobjective optimization in this paper we introduce some new notions of quasi efficiency and quasi proper efficiency for multiobjective optimization problems that reduce to the most important concepts of approximate and quasi efficient solutions given up to now we establish main properties and provide characterizations for these solutions by linear and nonlinear scalarizations with the help of quasi efficient solutions a generalized subdifferential of a vector mapping is introduced which generates a number of approximate subdifferentials frequently used in optimization in a unifying way the generalized subdifferential is related to the classical subdifferential of real functions by the method of scalarization an application of generalized subdifferential to express optimality conditions for quasi efficient solutions is also given

generalized e quasi solutions in multiobjective optimization problems existence results and optimality conditions abstract this paper is concerned with approximate solutions of multiobjective optimization problems whose order cone is not necessarily the nonnegative orthant we introduce the concept of generalized e quasi solution that extends other well known approximate efficiency notions of the literature and we study the limit behavior of these solutions as approximations to the efficient and weak efficient sets moreover we prove several existence results and a bound for the generalized e quasi solution set under convexity assumptions and by using asymptotic analysis tools finally we develop optimality conditions for a particular case of these kinds of e quasi solutions in nonsmooth convex and nonconvex problems

multiobjective programming under generalized type i invexity in this paper we extend a scalarized generalized type i invexity into a vector invexity v type i a number of sufficiency results are established using lagrange multiplier conditions and under various types of generalized v type i requirements weak strong and converse duality theorems are proved in the generalized v invexity type i setting

parametric approaches to fractional programs the fractional program p is defined by maxf x g x subject tox x a class of methods for solving p is based on the auxiliary problem q with a parameter maxf x g x subject tox x starting with two classical methods in this class the newton method and the binary search method a number of variations are introduced and compared among the proposed methods the modified binary search method is theoretically interesting because of its superlinear convergence and the capability to provide an explicit interval containing the optimum parameter value bar lambda computational behavior is tested by solving fractional knapsack problems and quadratic fractional programs the interpolated binary search method seems to be most efficient while other methods also behave surprisingly well

a theorem of brown halmos type for bergman space toeplitz operators abstract we study the analogues of the brown halmos theorem for toeplitz operators on the bergman space we show that for f and g harmonic t f t g t h only in the trivial case provided that h is of class c 2 with the invariant laplacian bounded here the trivial cases are f or g holomorphic from this we conclude that the zero product problem for harmonic symbols has only the trivial solution finally we provide examples that show that the brown halmos theorem fails for general symbols even for symbols continuous up to the boundary

subdifferentials of value functions and optimality conditions for dc and bilevel infinite and semi infinite programs the paper concerns the study of new classes of parametric optimization problems of the so called infinite programming that are generally defined on infinite dimensional spaces of decision variables and contain among other constraints infinitely many inequality constraints these problems reduce to semi infinite programs in the case of finite dimensional spaces of decision variables we focus on dc infinite programs with objectives given as the difference of convex functions subject to convex inequality constraints the main results establish efficient upper estimates of certain subdifferentials of intrinsically nonsmooth value functions in dc infinite programs based on advanced tools of variational analysis and generalized differentiation the value marginal functions and their subdifferential estimates play a crucial role in many aspects of parametric optimization including well posedness and sensitivity in this paper we apply the obtained subdifferential estimates to establishing verifiable conditions for the local lipschitz continuity of the value functions and deriving necessary optimality conditions in parametric dc infinite programs and their remarkable specifications finally we employ the value function approach and the established subdifferential estimates to the study of bilevel finite and infinite programs with convex data on both lower and upper level of hierarchical optimization the results obtained in the paper are new not only for the classes of infinite programs under consideration but also for their semi infinite counterparts

a chain rule for ɛ subdifferentials with applications to approximate solutions in convex pareto problems abstract in this work we obtain a chain rule for the approximate subdifferential considering a vector valued proper convex function and its post composition with a proper convex function of several variables nondecreasing in the sense of the pareto order we derive an interesting formula for the conjugate of a composition in the same framework and we prove the chain rule using this formula to get the results we require qualification conditions since in the composition the initial function is extended vector valued this chain rule extends analogous well known calculus rules obtained when the functions involved are finite and it gives a complementary simple expression for other chain rules proved without assuming any qualification condition as application we deduce the well known calculus rule for the addition and we extend the formula for the maximum of functions finally we use them and a scalarization process to obtain kuhn tucker type necessary and sufficient conditions for approximate solutions in convex pareto problems these conditions extend other obtained in scalar optimization problems

a method of linearizations for linearly constrained nonconvex nonsmooth minimization a readily implementable algorithm is given for minimizing a possibly nondifferentiable and nonconvex locally lipschitz continuous functionf subject to linear constraints at each iteration a polyhedral approximation tof is constructed from a few previously computed subgradients and an aggregate subgradient which accumulates the past subgradient information this aproximation and the linear constraints generate constraints in the search direction finding subproblem that is a quadratic programming problem then a stepsize is found by an approximate line search all the algorithm s accumulation points are stationary moreover the algorithm converges whenf happens to be convex

a generalized ekeland vector variational principle and its applications in optimization abstract in this paper we give a generalized ekeland vector variational principle by using the principle we extend and improve the related results in sharp efficiency in the framework of locally convex spaces we introduce two kinds of generalized sharp efficiencies and prove that they are equivalent in particular we show that a sharp efficient solution with respect to an interior point of the ordering cone is also one with respect to every interior point moreover we introduce the generalized takahashi s condition and the generalized hamel s condition for vector valued functions from the generalized ekeland principle we deduce that the two conditions are equivalent from this we discuss the relationship between the distance of f x from e f x and the distance of x from e f where e f x denotes the efficient point set of f x and e f denotes the efficient solution set

a successive quadratic programming method for a class of constrained nonsmooth optimization problems in this paper we present an algorithm for solving nonlinear programming problems where the objective function contains a possibly nonsmooth convex term the algorithm successively solves direction finding subproblems which are quadratic programming problems constructed by exploiting the special feature of the objective function an exact penalty function is used to determine a step size once a search direction thus obtained is judged to yield a sufficient reduction in the penalty function value the penalty parameter is adjusted to a suitable value automatically under appropriate assumptions the algorithm is shown to produce an approximate optimal solution to the problem with any desirable accuracy in a finite number of iterations

necessary and sufficient conditions in constrained optimization additional conditions are attached to the kuhn tucker conditions giving a set of conditions which are both necessary and sufficient for optimality in constrained optimization under appropriate constraint qualifications necessary and sufficient conditions are also given for optimality of the dual problem duality and converse duality are treated accordingly

ideal weakly efficient solutions for vector optimization problems we establish existence results for finite dimensional vector minimization problems allowing the solution set to be unbounded the solutions to be referred are ideal strong weakly efficient weakly pareto also several necessary and or sufficient conditions for the solution set to be non empty and compact are established moreover some characterizations of the non emptiness boundedness of the convex solution set in case the solutions are searched in a subset of the real line are also given however the solution set fails to be convex in general in addition special attention is addressed when the underlying cone is the non negative orthant and when the semi strict quasiconvexity of each component of the vector valued function is assumed our approach is based on the asymptotic description of the functions and sets some examples illustrating such an approach are also exhibited

stability results for ekeland s e variational principle and cone extremal solutions given x a banach space and f x a a proper lower semicontinuous function which is bounded from below the ekeland s e variational principle asserts the existence of a point xi in x which we call e extremal with respect to f which satisfies fu fxi e u xi for all u x u xi by using set convergence notions kuratowski painleve mosco bounded hausdorff and their epigraphical versions we study the semi continuity properties of the mapping which to f associates e ext f the set of such e extremal points the key for the geometrical understanding of such properties is to consider the equivalent phelps extremization principle which given a closed set d in x and a partial ordering with respect to a pointed cone associates the set of elements of d maximal with respect to this order direct or potential applications are given in various fields multicriteria optimization numerical algorithmic calculus of variations

optimality and duality in nonlinear programming involving semilocally b preinvex and related functions abstract a nonlinear programming problem is considered where the functions involved are semidifferentiable fritz john and karush kuhn tucker types necessary optimality conditions are obtained moreover a result concerning sufficiency of optimality conditions is given wolfe and mond weir types duality results are formulated in terms of semidifferentials the duality results are given using concepts of generalized semilocally b preinvex functions

quantifying heterogeneity in a meta analysis the extent of heterogeneity in a meta analysis partly determines the difficulty in drawing overall conclusions this extent may be measured by estimating a between study variance but interpretation is then specific to a particular treatment effect metric a test for the existence of heterogeneity exists but depends on the number of studies in the meta analysis we develop measures of the impact of heterogeneity on a meta analysis from mathematical criteria that are independent of the number of studies and the treatment effect metric we derive and propose three suitable statistics h is the square root of the chi2 heterogeneity statistic divided by its degrees of freedom r is the ratio of the standard error of the underlying mean from a random effects meta analysis to the standard error of a fixed effect meta analytic estimate and i2 is a transformation of h that describes the proportion of total variation in study estimates that is due to heterogeneity we discuss interpretation interval estimates and other properties of these measures and examine them in five example data sets showing different amounts of heterogeneity we conclude that h and i2 which can usually be calculated for published meta analyses are particularly useful summaries of the impact of heterogeneity one or both should be presented in published meta analyses in preference to the test for heterogeneity

on sufficiency of the kuhn tucker conditions if all the functions fo x f x f x are convex on c then these conditions are also sufficient various classes of functions have been defined for the purpose of weakening this limitation of convexity in mathematical programming mangasarian 11 has speculated that pseudo convexity of x and quasiconvexity off x are the weakest conditions that can be imposed so that the above conditions are sufficient for optimality it will be shown that there are other wide classes of functions for which the conditions are sufficient 545

nondifferentiable multiobjective programming under generalized di invexity abstract in this paper we are concerned with a nondifferentiable multiobjective programming problem with inequality constraints we introduce new concepts of d i invexity and generalized d i invexity in which each component of the objective and constraint functions is directionally differentiable in its own direction d i new fritz john type necessary and karush kuhn tucker type necessary and sufficient optimality conditions are obtained for a feasible point to be weakly efficient efficient or properly efficient moreover we prove weak strong converse and strict duality results for a mond weir type dual under various types of generalized d i invexity assumptions

algebraic properties of toeplitz operators on the dirichlet space we study some algebraic properties of toeplitz operators on the dirichlet space we first characterize semi commuting toeplitz operators with harmonic symbols next we study the product problem of when product of two toeplitz operators is another toeplitz operator as an application we show that the zero product of two toeplitz operators with harmonic symbol has only a trivial solution also the corresponding compact product problem is studied

ϵ optimality criteria for vector minimization problems via exact penalty functions abstract several authors have been interested in optimality criteria for scalar programming problems in this paper some optimality results for the scalar problems of ours math programming 56 1992 233 243 are extended to vector minimization problems via exact penalty functions similar to our paper the solvability of neither the vector problems nor the associated dual problems is assumed our criteria are derived by estimating the size of the penalty parameter in terms of approximate solutions for the dual problems

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

operators on the orthogonal complement of the dirichlet space ii in this paper we first prove that a dual hankel operator r on the orthogonal complement of the dirichlet space is compact for w 1 d and then that a semicommutator of two toeplitz operators on the dirichlet space or two dual toeplitz operators on the orthogonal complement of the dirichlet space in sobolev space is compact we also prove that a dual hankel operator r with w 1 d is of finite rank if and only if b is orthogonal to the dirichlet space for some finite blaschke product b and give a sufficient and necessary condition for the semicommutator of two dual toeplitz operators to be of finite rank

necessary suboptimality and optimality conditions via variational principles the paper aims to develop some basic principles and tools of nonconvex variational analysis with applications to necessary suboptimality and optimality conditions for constrained optimization problems in infinite dimensions we establish a certain subdifferential variational principle as a new characterization of asplund spaces this result is different from conventional support forms of variational principles and appears to be convenient for applications to nonsmooth optimization based on the subdifferential variational principle we obtain new necessary conditions for suboptimal solutions in general nonsmooth optimization problems with equality inequality and set constraints in asplund spaces in this way we establish the so called sequential normal compactness properties of constraint sets that play an essential role in infinite dimensional variational analysis and its applications as a by product of our approach we derive various forms of necessary optimality conditions for nonsmooth constrained problems in infinite dimensions which extend known results in that direction

e optimality criteria for convex programming problems via exact penalty functions in this paper we presente optimality criteria for convex programming problems associated with exact penalty functions several authors have given various criteria under the assumption that such convex problems and the associated dual problems can be solved we assume the solvability of neither the convex problem nor the dual problem to derive our criteria we estimate the size of the penalty parameter in terms of ane solution for the dual problem

nondifferentiable multiobjective programming under generalized d univexity abstract in this paper we are concerned with a nondifferentiable multiobjective programming problem with inequality constraints we introduce four new classes of generalized convex functions by combining the concepts of weak strictly pseudoinvex strong pseudoinvex weak quasi invex weak pseudoinvex and strong quasi invex functions in aghezzaf and hachimi numer funct anal optim 22 2001 775 d invex functions in antczak europ j oper res 137 2002 28 and univex functions in bector et al univex functions and univex nonlinear programming proc admin sci assoc canada 1992 p 115 by utilizing the new concepts we derive a karush kuhn tucker sufficient optimality condition and establish mond weir type and general mond weir type duality results for the nondifferentiable multiobjective programming problem

multiobjective programming under d invexity abstract in this paper a generalization of convexity is considered in the case of nonlinear multiobjective programming problem where the functions involved are nondifferentiable by considering the concept of pareto optimal solution and substituting d invexity for convexity the fritz john type and karush kuhn tucker type necessary optimality conditions and duality in the sense of mond weir and wolfe for nondifferentiable multiobjective programming are given

a dual differentiable exact penalty function a new penalty function is associated with an inequality constrained nonlinear programming problem via its dual this penalty function is globally differentiable if the functions defining the original problem are twice globally differentiable in addition the penalty parameter remains finite this approach reduces the original problem to a simple problem of maximizing a globally differentiable function on the product space of a euclidean space and the nonnegative orthant of another euclidean space many efficient algorithms exist for solving this problem for the case of quadratic programming the penalty function problem can be solved effectively by successive overrelaxation sor methods which can handle huge problems while preserving sparsity features

commuting dual toeplitz operators with pluriharmonic symbols abstract in this paper we completely characterize commuting dual toeplitz operators with bounded pluriharmonic symbols on the bergman space of the unit ball we show that for and pluriharmonic s s s on l a 2 b n only in the trivial case here the trivial case is or holomorphic

e optimal solutions in nondifferentiable convex programming and some related questions in this paper we present e optimality conditions of the kuhn tucker type for points which are within e of being optimal to the problem of minimizing a nondifferentiable convex objective function subject to nondifferentiable convex inequality constraints linear equality constraints and abstract constraints such e optimality conditions are of interest for theoretical consideration as well as from the computational point of view some illustrative applications are made thus we derive an expression for the e subdifferential of a general convex max function we also show how the e optimality conditions given in this paper can be mechanized into a bundle algorithm for solving nondifferentiable convex programming problems with linear inequality constraints

mm algorithms for generalized bradley terry models the bradley terry model for paired comparisons is a simple and muchstudied means to describe the probabilities of the possible outcomes when individuals are judged against one another in pairs among the many studies of the model in the past 75 years numerous authors have generalized it in several directions sometimes providing iterative algorithms for obtaining maximum likelihood estimates for the generalizations building on a theory of algorithms known by the initials mm for minorization maximization this paper presents a powerful technique for producing iterative maximum likelihood estimation algorithms for a wide class of generalizations of the bradley terry model while algorithms for problems of this type have tended to be custom built in the literature the techniques in this paper enable their mass production simple conditions are stated that guarantee that each algorithm described will produce a sequence that converges to the unique maximum likelihood estimator several of the algorithms and convergence results herein are new

an active set method for solving linearly constrained nonsmooth optimization problems an algorithm for solving linearly constrained optimization problems is proposed the search direction is computed by a bundle principle and the constraints are treated through an active set strategy difficulties that arise when the objective function is nonsmooth require a clever choice of a constraint to relax a certain nondegeneracy assumption is necessary to obtain convergence

on the existence and stability of approximate solutions of perturbed vector equilibrium problems abstract in this paper we consider several concepts of approximate minima of a set in normed vector spaces and we provide some results concerning the stability of these minima under perturbation of the underlying set with a sequence of sets converging in the sense of painleve kuratowski to the initial set then we introduce the concept of approximate solution for equilibrium problem governed by set valued maps and we study the stability of these solutions the particular case of linear continuous operators is considered as well

optimality and duality in nondifferentiable multiobjective optimization involvingd type i and related functions abstract in this paper new classes of functions namely d type i d quasitype i and d pseudo type i are defined for a multiobjective nondifferentiable programming problem kuhn tucker type necessary and sufficient optimality conditions are obtained for a feasible point to be a weak minimum for this problem two duals are formulated and various duality results are given by using the above defined classes of functions considering the concept of a weak minimum

a set valued ekeland s variational principle in vector optimization this paper deals with ekeland s variational principle for vector optimization problems by using a set valued metric a set valued perturbed map and a cone boundedness concept based on scalarization we introduce an original approach to extending the well known scalar ekeland s principle to vector valued maps as a consequence of this approach we obtain an ekeland s variational principle that does not depend on any approximate efficiency notion this result is related to other ekeland s principles proved in the literature and the finite dimensional case is developed via an varepsilon efficiency notion that we introduced in math methods oper res 64 2006 pp 165 185 siam j optim 17 2006 pp 688 710

sufficiency of exact penalty minimization by employing a recently obtained error bound for differentiable convex inequalities it is shown that under appropriate constraint qualifications a minimum solution of an exact penalty function for a single value of the penalty parameter which exceeds a certain threshold is also a solution of the convex program associated with the penalty function no a priori assumption is made regarding the solvability of the convex program if such a solvability assumption is made then we show that a threshold value of the penalty parameter can be used which is smaller than both the above mentioned value and that of zangwill these various threshold values of the penalty parameter also apply to the well known big m method of linear programming

pre invex functions in multiple objective optimization and called such functions convexlike if s is a convex set and if f is a convex function then clearly f is convexlike elster and nehse obtained a saddlepoint optimality condition for convexlike mathematical programs hayashi and komiya 9 also considered convexlike functions and developed a gordan type theorem of the alternative involving convexlike functions and in addition considered lagrangian duality for convexlike programs hanson 7 considered differentiable functions f s r for which there exists an n dimensional vector function q q u such that for all x u e s

permutation probabilities as models for horse races summary some properties of models for the outcomes of races are described these properties being consequences of a stochastic ordering of the permutations which define the outcomes of a race order statistics models which lead to stochastic ordering are also discussedparticular cases of these are the first order model of plackett 1975 and the normal model of upton and brook 1974 an approximation for the normal model is suggested

proper approximate solutions and e subdifferentials in vector optimization basic properties and limit behaviour abstract the aim of this work is to introduce a proper e subdifferential in the sense of benson for extended mappings between topological linear spaces it is defined by means of a new concept of a proper e efficient solution of a vector optimization problem the main results are hold by assuming a sort of nearly subconvexlikeness which is a very general concept of convexity some sufficient conditions for the existence of proper e subgradients are obtained our results generalize other similar results in the literature and improve them since the limit behaviour when the error e tends to zero of the involved concepts is more suitable

d invexity and optimality conditions abstract in this paper a generalization of convexity called d invexity is introduced substituting d invex for convex we get some optimality conditions for nondifferentiable multiobjective programming the application is demonstrated by an example