characterizations of the subdifferential of convex integral functions under qualification conditions abstract this work provides formulae for the e subdifferential of integral functions in the framework of complete finite measure spaces and locally convex spaces in this work we present here new formulae for this e subdifferential under the presence of continuity type qualification conditions relying on the data involved in the integrand

duality for e variational inequalities via the subdifferential calculus abstract based on the properties of the convex e subdifferential calculus we introduce to a general e variational inequality formulated with the help of a set valued operator and a perturbation function a dual one expressed by making use of the fenchel conjugate of the perturbation function under convexity hypotheses we show that the fulfillment of a regularity condition guarantees that the primal e variational inequality is solvable if and only if its dual one is solvable by particularizing the perturbation function we obtain several dual statements and we succeed to generalize and improve a duality scheme recently given by kum kim and lee an example justifying this generalization is also provided among the special instances of the general result we rediscover also the duality scheme concerning variational inequalities due to mosco

regularity conditions characterizing fenchel lagrange duality and farkas type results in dc infinite programming abstract in this paper we consider a dc infinite programming problem p with inequality constraints by using the properties of the epigraph of the conjugate functions we introduce some new notions of regularity conditions for p under these new regularity conditions we completely characterize the fenchel lagrange duality and the stable fenchel lagrange duality for p similarly we also completely characterize the farkas type results and the stable farkas type results for p as applications we obtain the corresponding results for conic programming problems

asymptotic closure condition and fenchel duality for dc optimization problems in locally convex spaces a b s t r a c t we consider the dc optimization problem p inf x f1 x f2 x g1 ax g2 ax where f1 f2 g1 and g2 are proper convex functions defined on locally convex hausdorff topological vector spaces x and y respectively and a is a linear continuous operator from x to y adopting the standard convexification technique a fenchel dual problem of p is given by using properties of the epigraph of conjugate functions some sufficient and necessary conditions for the fenchel duality and for the stable fenchel duality of p are provided

generalizations of slater s constraint qualification for infinite convex programs in this paper we study constraint qualifications and duality results for infinite convex programs p inf f x g x s x c whereg g1 g2 ands s1 s2 si are convex cones i 1 2 c is a convex subset of a vector spacex andf andgi are respectively convex andsi convex i 1 2 in particular we consider the special case whens2 is in afinite dimensional space g2 is affine ands2 is polyhedral we show that a recently introduced simple constraint qualification and the so called quasi relative interior constraint qualification both extend to p from the special case thatg g2 is affine ands s2 is polyhedral in a finite dimensional space the so called partially finite program this provides generalized slater type conditions for p which are much weaker than the standard slater condition we exhibit the relationship between these two constraint qualifications and show how to replace the affine assumption ong2 and the finite dimensionality assumption ons2 by a local compactness assumption we then introduce the notion of strong quasi relative interior to get parallel results for more general infinite dimensional programs without the local compactness assumption our basic tool reduces to guaranteeing the closure of the sum of two closed convex cones

strong conical hull intersection property bounded linear regularity jameson s property g and error bounds in convex optimization the strong conical hull intersection property and bounded linear regularity are properties of a collection of finitely many closed convex intersecting sets in euclidean space these fundamental notions occur in various branches of convex optimization constrained approximation convex feasibility problems linear inequalities for instance it is shown that the standard constraint qualification from convex analysis implies bounded linear regularity which in turn yields the strong conical hull intersection property jameson s duality for two cones which relates bounded linear regularity to property g is re derived and refined for polyhedral cones a statement dual to hoffman s error bound result is obtained a sharpening of a result on error bounds for convex inequalities by auslender and crouzeix is presented finally for two subspaces property g is quantified by the angle between the subspaces

a comparison of constraint qualifications in infinite dimensional convex programming in this paper the relationships between various constraint qualifications for infinite dimensional convex programs are investigated using robinson s refinement of the duality result of rockafellar it is demonstrated that the constraint qualification proposed by rockafellar provides a systematic mechanism for comparing many constraint qualifications as well as establishing new results in different topological environments

maximal spectral distance abstract the exact upper bound for the norm distance between two normal matrices is given as a function of their eigenvalues among the applications we obtain an inequality related to the cartesian decomposition of a matrix and the evaluation of the hadamard multiplier norms for matrices in a certain broad class

robust conjugate duality for convex optimization under uncertainty with application to data classification in this paper we present a robust conjugate duality theory for convex programming problems in the face of data uncertainty within the framework of robust optimization extending the powerful conjugate duality technique we first establish robust strong duality between an uncertain primal parameterized convex programming model problem and its uncertain conjugate dual by proving strong duality between the deterministic robust counterpart of the primal model and the optimistic counterpart of its dual problem under a regularity condition this regularity condition is not only sufficient for robust duality but also necessary for it whenever robust duality holds for every linear perturbation of the objective function of the primal model problem more importantly we show that robust strong duality always holds for partially finite convex programming problems under scenario data uncertainty and that the optimistic counterpart of the dual is a tractable finite dimensional problem as an application we also derive a robust conjugate duality theorem for support vector machines which are a class of important convex optimization models for classifying two labelled data sets the support vector machine has emerged as a powerful modelling tool for machine learning problems of data classification that arise in many areas of application in information and computer sciences

characterizations of convex approximate subdifferential calculus in banach spaces we establish subdifferential calculus rules for the sum of convex functions defined on normed spaces this is achieved by means of a condition relying on the continuity behaviour of the inf convolution of their corresponding conjugates with respect to any given topology intermediate between the norm and the weak topologies on the dual space such a condition turns out to also be necessary in banach spaces these results extend both the classical formulas by hiriart urruty and phelps and by thibault

free inertial processes driven by gaussian noise probability distributions anomalous diffusion and fractal behavior we study the motion of an unbound particle under the influence of a random force modeled as gaussian colored noise with an arbitrary correlation function we derive exact equations for the joint and marginal probability density functions and find the associated solutions we analyze in detail anomalous diffusion behaviors along with the fractal structure of the trajectories of the particle and explore possible connections between dynamical exponents of the variance and the fractal dimension of the trajectories

new sequential lagrange multiplier conditions characterizing optimality without constraint qualification for convex programs in this paper a new sequential lagrange multiplier condition characterizing optimality without a constraint qualification for an abstract nonsmooth convex program is presented in terms of the subdifferentials and the epsilon subdifferentials a sequential condition involving only the subdifferentials but at nearby points to the minimizer for constraints is also derived for a smooth convex program the sequential condition yields a limiting kuhn tucker condition at nearby points without a constraint qualification it is shown how the sequential conditions are related to the standard lagrange multiplier condition applications to semidefinite programs semi infinite programs and semiconvex programs are given several numerical examples are discussed to illustrate the significance of the sequential conditions

constraint qualifications for convex optimization without convexity of constraints new connections and applications to best approximation abstract we study constraint qualifications and necessary and sufficient optimality conditions for a convex optimization problem with inequality constraints where the constraint functions are continuously differentiable but they are not assumed to be convex we present constraint qualifications under which the karush kuhn tucker conditions are necessary and sufficient for optimality without the convexity of the constraint functions and establish new links among various known constraint qualifications that guarantee necessary karush kuhn tucker conditions we also present a new constraint qualification which is the weakest constraint qualification for the karush kuhn tucker conditions to be necessary for optimality of the convex optimization problem consequently we present lagrange multiplier characterizations for the best approximation from a convex set in the face of nonconvex inequality constraints extending corresponding known results in the literature we finally give a table summarizing various links among the constraint qualifications

best approximation from the intersection of a closed convex set and a polyhedron in hilbert space weak slater conditions and the strong conical hull intersection property let x be a real hilbert space c be a closed convex subset and h i x in x mid la x h i ra le b i quad i 1 2 m be a finite collection of half spaces under the assumption that k c cap cap 1 mh i is not empty the problem of characterizing the best approximation from k to any x in x is considered the strong conical hull intersection property strong chip which was introduced by us in 1997 is shown to be both necessary and sufficient for the following perturbation property to hold for each x in x an element x 0 in k satisfies x 0 p k x if and only if x 0 p c x sum 1 m lambda ih i for some scalars lambda i ge 0 with lambda i la x 0 h i ra b i 0 for each i here p d z denotes the unique best approximation from d to z in other words determining the best approximation from the set k to any point is equivalent to the generally easier problem of determining the best approximation from the set c to a perturbation of that point moreover even when the strong chip does not hold the perturbation property still holds except now c must be replaced by a certain convex extremal subset of c we also show that the strong chip is weaker than any of the weak slater conditions that one can naturally impose on the sets in question these results generalize the main results of our 1997 paper j approx theory 90 pp 385 444 and hence those of several other papers as well

enlargements and sums of monotone operators in this paper we study two important notions related to monotone operators one is the concept of enlargement of a given monotone operator which has turned out to be a useful tool in the analysis of approximate solutions to problems involving monotone operators the second one is the notion of sum of monotone operators for the latter we introduce and study a kind of extended sum of two monotone operators which in several important cases turns out to be a maximal monotone operator

constraint qualifications for extended farkas s lemmas and lagrangian dualities in convex infinite programming for an inequality system defined by a possibly infinite family of proper functions not necessarily lower semicontinuous we introduce some new notions of constraint qualifications in terms of the epigraphs of the conjugates of these functions under the new constraint qualifications we obtain characterizations of those reverse convex inequalities which are a consequence of the constrained system and we provide necessary and or sufficient conditions for a stable farkas lemma to hold similarly we provide characterizations for constrained minimization problems to have the strong or strong stable lagrangian dualities several known results in the conic programming problem are extended and improved

the strong conical hull intersection property for convex programming the strong conical hull intersection property chip is a geometric property of a collection of finitely many closed convex intersecting sets this basic property which was introduced by deutsch et al in 1997 is one of the central ingredients in the study of constrained interpolation and best approximation in this paper we establish that the strong chip of intersecting sets of constraints is the key characterizing property for optimality and strong duality of convex programming problems we first show that a sharpened strong chip is necessary and sufficient for a complete lagrange multiplier characterization of optimality for the convex programming model problem where c is a closed convex subset of a banach space x s is a closed convex cone which does not necessarily have non empty interior y is a banach space is a continuous convex function and g x y is a continuous s convex function we also show that the strong chip completely characterizes the strong duality for partially finite convex programs where y is finite dimensional and g x ax b and s is a polyhedral convex cone global sufficient conditions which are strictly weaker than the slater type conditions are given for the strong chip and for the sharpened strong chip

on extension of fenchel duality and its application by considering the epigraphs of conjugate functions we extend the fenchel duality applicable to a possibly infinite family of proper lower semicontinuous convex functions on a banach space applications are given in providing fuzzy kkt conditions for semi infinite programming

new dual constraint qualifications characterizing zero duality gaps of convex programs and semidefinite programs abstract in this paper we present new dual constraint qualifications which completely characterize the zero duality gap property for convex programming problems in general banach spaces as an application we derive constraint qualifications characterizing zero duality gaps of semidefinite programs our approach makes use of convex majorants of support functions together with conjugate analysis and approximate subdifferentials of convex functions

fenchel duality in infinite dimensional setting and its applications abstract we study fenchel duality problems in infinite dimensional spaces that involve the minimizing of a sum of two proper convex functions where one of which is polyhedral we use a constraint qualification with the notion of the strong quasi interior of a convex set and then deduce duality results and subgradient formula as applications we discuss the strong conical hull intersection property of convex sets finally by using a duality result due to rodriques and simons we establish several duality results for convex optimization over a finite intersection of closed convex sets

lagrange multiplier characterizations of robust best approximations under constraint data uncertainty abstract in this paper we explain how to characterize the best approximation to any x in a hilbert space x from the set c x x g i x 0 i 1 2 m in the face of data uncertainty in the convex constraints g i x 0 i 1 2 m where c is a closed convex subset of x following the robust optimization approach we establish lagrange multiplier characterizations of the robust constrained best approximation that is immunized against data uncertainty this is done by characterizing the best approximation to any x from the robust counterpart of the constraints where the constraints are satisfied for all possible uncertainties within the prescribed uncertainty sets unlike the traditional lagrange multiplier characterizations without data uncertainty for constrained best approximation problems in the face uncertainty we show that the strong conical hull intersection property strong chip alone is not sufficient to guarantee the lagrange multiplier characterizations we present conditions which guarantee that the strong chip is necessary and sufficient for the multiplier characterization we also establish that the strong chip is automatically satisfied for the cases of polyhedral constraints with polytope uncertainty and linear constraints with interval uncertainty as an application we show how robust solutions of shape preserving interpolation problems under ellipsoidal and box uncertainty cases can be obtained in terms of lagrange multipliers under strict robust feasibility conditions

non convex quadratic minimization problems with quadratic constraints global optimality conditions in this paper we first examine how global optimality of non convex constrained optimization problems is related to lagrange multiplier conditions we then establish lagrange multiplier conditions for global optimality of general quadratic minimization problems with quadratic constraints we also obtain necessary global optimality conditions which are different from the lagrange multiplier conditions for special classes of quadratic optimization problems these classes include weighted least squares with ellipsoidal constraints and quadratic minimization with binary constraints we discuss examples which demonstrate that our optimality conditions can effectively be used for identifying global minimizers of certain multi extremal non convex quadratic optimization problems

linear oscillators driven by gaussian colored noise crossovers and probability distributions abstract we study the statistical properties of linear oscillators driven by both internal and external gaussian colored noise we obtain the variances of displacement and velocity and analyze their asymptotic behavior and crossovers we also derive exact equations for the joint and marginal probability density functions of the system and find their solutions in order to test our general analysis we study the case of ornstein uhlenbeck driving noise

functional characterization of generalized langevin equations we present an exact functional formalism to deal with linear langevin equations with arbitrary memory kernels and driven by an arbitrary noise structure characterized through its characteristic functional no other hypothesis is assumed over the noise neither do we use the fluctuation dissipation theorem we find that the characteristic functional of the linear process can be expressed in terms of noise functional and the green function of the deterministic memory like dissipative dynamics this yields a procedure for calculating the full kolmogorov hierarchy of the non markov process as examples we have characterized through the 1 time probability a noise induced interplay between the dissipative dynamics and the structure of different noises conditions that lead to non gaussian statistics and distributions with long tails are analysed the introduction of arbitrary fluctuations in fractional langevin equations has also been pointed out

analysis of spectral variation and some inequalities a geometric method based on a decomposition of the space of complex matrices is employed to study the variation of the spectrum of a matrix when adapted to special cases this leads to some classical inequalities as well as some new ones as an example of the latter we show that if u v are unitary matrices and k is a skew hermitian matrix such that uv exp k then for every unitary invariant norm the distance between the eigenvalues of v and those of vis bounded by k this generalises two earlier results which used particular unitary invariant norms

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

volatility a hidden markov process in financial time series volatility characterizes the amplitude of price return fluctuations it is a central magnitude in finance closely related to the risk of holding a certain asset despite its popularity on trading floors volatility is unobservable and only the price is known diffusion theory has many common points with the research on volatility the key of the analogy being that volatility is a time dependent diffusion coefficient of the random walk for the price return we present a formal procedure to extract volatility from price data by assuming that it is described by a hidden markov process which together with the price forms a two dimensional diffusion process we derive a maximum likelihood estimate of the volatility path valid for a wide class of two dimensional diffusion processes the choice of the exponential ornstein uhlenbeck expou stochastic volatility model performs remarkably well in inferring the hidden state of volatility the formalism is applied to the dow jones index the main results are that i the distribution of estimated volatility is lognormal which is consistent with the expou model ii the estimated volatility is related to trading volume by a power law of the form sigma proportional variant v0 55 and iii future returns are proportional to the current volatility which suggests some degree of predictability for the size of future returns

stable zero duality gaps in convex programming complete dual characterisations with applications to semidefinite programs the zero duality gap that underpins the duality theory is one of the central ingredients in optimisation in convex programming it means that the optimal values of a given convex program and its associated dual program are equal it allows in particular the development of efficient numerical schemes however the zero duality gap property does not always hold even for finite dimensional problems and it frequently fails for problems with non polyhedral constraints such as the ones in semidefinite programming problems over the years various criteria have been developed ensuring zero duality gaps for convex programming problems in the present work we take a broader view of the zero duality gap property by allowing it to hold for each choice of linear perturbation of the objective function of the given problem globalising the property in this way permits us to obtain complete geometric dual characterisations of a stable zero duality gap in terms of epigraphs and conjugate functions for convex semidefinite programs we establish necessary and sufficient dual conditions for stable zero duality gaps as well as for a universal zero duality gap in the sense that the zero duality gap property holds for each choice of constraint right hand side and convex objective function zero duality gap results for second order cone programming problems are also given our approach makes use of elegant conjugate analysis and fenchel s duality

inequality systems and global optimization abstract solvability results for infinite inequality systems involving convex and difference of convex dc functions are given generalizations of farkas lemma are obtained these results are presented in terms of epigraphs of conjugate functions applications are given for characterizing global optimality and optimality of difference of convex dc optimization problems with convex inequality constraints solvability results for cone convex inequality systems are also given

subdifferential calculus using ϵ subdifferentials in applications of convex analysis it is important to be able to calculate the subdifferentials of various combinations of proper and lower semicontinuous convex functions such as the sum of two such functions or their inf convolution epi sum as well as the pre composition of a convex function with an affine map or the marginal function obtained from a convex function and a linear map the classical formulas for such calculations all require additional hypotheses same of which may be difficult to check or are not always satisfied in a given variational problem in this paper we present formulas for the subdifferentials of such combinations without assuming any additional hypotheses by utilizing subdifferentials this wider applicability comes at the price of somewhat more complicated formulas

quasi slater and farkas minkowski qualifications for semi infinite programming with applications the well known farkas minkowski fm type qualification plays an important role in linear semi infinite programming and has been extensively developed by many authors in establishing optimality conditions duality and stability for semi infinite programming in this paper we introduce the concept of the quasi slater condition for a semi infinite convex inequality system and present that the slater type conditions imply the fm qualification under some appropriate continuity assumption of the set valued mapping i mapsto f i x applying these relationships we establish dual characterizations both asymptotic and nonasymptotic for set containment problems and provide some sufficient conditions for ensuring the strong lagrangian duality and farkas lemma

constraint qualifications for optimality conditions and total lagrange dualities in convex infinite programming abstract for an inequality system defined by an infinite family of proper convex functions not necessarily lower semicontinuous we introduce some new notions of constraint qualifications under the new constraint qualifications we provide necessary and or sufficient conditions for the kkt rules to hold similarly we provide characterizations for constrained minimization problems to have total lagrangian dualities several known results in the conic programming problem are extended and improved

applications of convex analysis within mathematics in this paper we study convex analysis and its theoretical applications we first apply important tools of convex analysis to optimization and to analysis we then show various deep applications of convex analysis and especially infimal convolution in monotone operator theory among other things we recapture the minty surjectivity theorem in hilbert space and present a new proof of the sum theorem in reflexive spaces more technically we also discuss autoconjugate representers for maximally monotone operators finally we consider various other applications in mathematical analysis

nonlinearly constrained best approximation in hilbert spaces the strong chip and the basic constraint qualification we study best approximation problems with nonlinear constraints in hilbert spaces the strong conical hull intersection property chip and the basic constraint qualification bcq condition are discussed best approximations with differentiable constraints and convex constraints are characterized the analysis generalizes some linearly constrained results of recent works f deutsch w li and j ward j approx theory 90 1997 pp 385 444 f deutsch w li and j d ward siam j optim 10 1999 pp 252 268

numerical method for solving stochastic differential equations with dichotomous noise we propose a numerical method for solving stochastic differential equations with dichotomous markov noise the numerical scheme is formulated such that i the stochastic formula used follows the stratonovichtaylor form over the entire range of noise correlation times including the gaussian white noise limit and ii the method is readily applicable to dynamical systems driven by arbitrary types of noise provided there exists a way to describe the random increment of the stochastic process expressed in the stratonovich taylor form we further propose a simplified taylor scheme that significantly reduces the computation time while still satisfying the moment properties up to the required order the accuracies and efficiencies of the proposed algorithms are validated by applying the schemes to two prototypical model systems that possess analytical solutions

characterizing set containments involving infinite convex constraints and reverse convex constraints dual characterizations of the containment of a closed convex set defined by infinite convex constraints in an arbitrary polyhedral set in a reverse convex set defined by convex constraints and in another convex set defined by finite convex constraints are given a special case of these dual characterizations has played a key role in generating knowledge based support vector machine classifiers which are powerful tools in data classification and mining the conditions in these dual characterizations reduce to simple nonasymptotic conditions under slater s constraint qualification

sandwich theorem for quasiconvex functions and its applications abstract in convex programming sandwich theorem is very important because it is equivalent to fenchel duality theorem in this paper we investigate a sandwich theorem for quasiconvex functions also we consider some applications for quasiconvex programming

stable and total fenchel duality for convex optimization problems in locally convex spaces we consider the optimization problem p a inf x in x f x g ax where f and g are proper convex functions defined on locally convex hausdorff topological vector spaces x and y respectively and a is a linear operator from x to y by using the properties of the epigraph of the conjugated functions some sufficient and necessary conditions for the strong fenchel duality and the strong converse fenchel duality of p a are provided sufficient and necessary conditions for the stable fenchel duality and for the total fenchel duality are also derived

a new geometric condition for fenchel s duality in infinite dimensional spaces in 1951 fenchel discovered a special duality which relates the minimization of a sum of two convex functions with the maximization of the sum of concave functions using conjugates fenchel s duality is central to the study of constrained optimization it requires an existence of an interior point of a convex set which often has empty interior in optimization applications the well known relaxations of this requirement in the literature are again weaker forms of the interior point condition avoiding an interior point condition in duality has so far been a difficult problem however a non interior point type condition is essential for the application of fenchel s duality to optimization in this paper we solve this problem by presenting a simple geometric condition in terms of the sum of the epigraphs of conjugate functions we also establish a necessary and sufficient condition for the e subdifferential sum formula in terms of the sum of the epigraphs of conjugate functions our results offer further insight into fenchel s duality

duality gap of the conic convex constrained optimization problems in normed spaces in this paper motivated by a result due to champion math program 99 2004 we introduce a property for a conic quasi convex vector valued function in a general normed space we prove that this property characterizes the zero duality gap for a class of the conic convex constrained optimization problem in the sense that if this property is satisfied and the objective function f is continuous at some feasible point then the duality gap is zero and if this property is not satisfied then there exists a linear continuous function f such that the duality gap is positive we also present some sufficient conditions for the property

majorizations and inequalities in matrix theory abstract in matrix theory majorization plays a significant role for instance majorization relations among eigenvalues and singular values of matrices produce a lot of norm inequalities and even matrix inequalities this survey article is intended as a review of recent results in matrix theory related to majorization

strong chip for infinite system of closed convex sets in normed linear spaces for a general possibly infinite system of closed convex sets in a normed linear space we provide several sufficient conditions for ensuring the strong conical hull intersection property one set of sufficient conditions is given in terms of the finite subsystems while the other sets are in terms of the relaxed interior point conditions together with appropriate continuity of the associated set valued function on the topologized index set i in the special case when i is finite and x is finite dimensional one of these results reduces to a classical result of rockafellar

on permutations convex hulls and normal operators abstract a spectral characterization is obtained for those normal operators which belong to the convex hull of the unitary orbit of a given normal operator on a finite dimensional space this is used to prove the following if a and b are normal operators on an n dimensional complex hilbert space h with eigenvalues given by 1 n and 1 n respectively and if a b is also normal then a b max s n diag k k for any unitarily invariant norm on l h

duality and infinite dimensional optimization the purpose is to establish various duality theorems including a generalization of the fenchel duality theorem of attouch and brezis to convex processes under relaxed interiority conditions and show the usefulness of the attouch and brezis type conditions in optimization the significance of the interiority conditions is that they provide easily verifiable conditions for a large class of infinite dimensional convex programs whereas the standard slater constraint qualification fails the results are applied to obtain duality theorems for general systems of inequalities and minimax results

necessary and sufficient constraint qualifications for solvability of systems of infinite convex inequalities abstract in this paper we present constraint qualifications which completely characterize the farkas minkowski and the locally farkas minkowski convex possibly infinite inequality systems posed in topological vector spaces the number of constraints and the dimension of the linear space are arbitrary possibly infinite the constraint qualifications considered in this paper are expressed in terms of the solvability of certain parametric convex linear systems and the uniform strong duality or the uniform min max duality relative to the lagrange haar dual problems of suitable convex linear parametric optimization problems

constraint qualification the strong chip and best approximation with convex constraints in banach spaces several fundamental concepts such as the basic constraint qualification bcq the strong conical hull intersection property chip and the perturbations for convex systems of inequalities in banach spaces over mathbb r or mathbb c are extended and studied here the systems are not necessarily finite their relationships with each other in connection with the best approximations are investigated as applications we establish results on the unconstrained reformulation of best approximations with infinitely many constraints in hilbert spaces also we give several characterizations of best restricted range approximations in c q under quite general constraints

on majorization and schur products abstract suppose a d1 dm are n n matrices where a is self adjoint and let x m k 1 d k ad k it is shown that if d k d k d k d k i then the spectrum of x is majorized by the spectrum of a in general without assuming any condition on d1 dm a result is obtained in terms of weak majorization if each dk is a diagonal matrix then x is equal to the schur entrywise product of a with a positive semidefinite matrix thus the results are applicable to spectra of schur products of positive semidefinite matrices if a b are self adjoint with b positive semidefinite and if bii 1 for each i it follows that the spectrum of the schur product of a and b is majorized by that of a a stronger version of a conjecture due to marshall and olkin is also proved

on constraint qualification for an infinite system of convex inequalities in a banach space for a general infinite system of convex inequalities in a banach space we study the basic constraint qualification and its relationship with other fundamental concepts including various versions of conditions of slater type the mangasarian fromovitz constraint qualification as well as the pshenichnyi levin valadier property introduced by li nahak and singer applications are given in the restricted range approximation problem constrained optimization problems as well as in the approximation problem with constraints by conditionally positive semidefinite functions

robust linear semi infinite programming duality under uncertainty in this paper we propose a duality theory for semi infinite linear programming problems under uncertainty in the constraint functions the objective function or both within the framework of robust optimization we present robust duality by establishing strong duality between the robust counterpart of an uncertain semi infinite linear program and the optimistic counterpart of its uncertain lagrangian dual we show that robust duality holds whenever a robust moment cone is closed and convex we then establish that the closed convex robust moment cone condition in the case of constraint wise uncertainty is in fact necessary and sufficient for robust duality in other words the robust moment cone is closed and convex if and only if robust duality holds for every linear objective function of the program in the case of uncertain problems with affinely parameterized data uncertainty we establish that robust duality is easily satisfied under a slater type constraint qualification consequently we derive robust forms of the farkas lemma for systems of uncertain semi infinite linear inequalities

complete characterizations of stable farkas lemma and cone convex programming duality we establish necessary and sufficient conditions for a stable farkas lemma we then derive necessary and sufficient conditions for a stable duality of a cone convex optimization problem where strong duality holds for each linear perturbation of a given convex objective function as an application we obtain stable duality results for convex semi definite programs and convex second order cone programs

weaker conditions for subdifferential calculus of convex functions abstract in this paper we establish new rules for the calculus of the subdifferential mapping of the sum of two convex functions our results are established under conditions which are at an intermediate level of generality among those leading to the hiriart urruty and phelps formula hiriart urruty and phelps 1993 15 involving the approximate subdifferential and the stronger assumption used in the well known moreau rockafellar formula rockafellar 1970 23 moreau 1966 20 which only uses the exact subdifferential we give an application to derive asymptotic optimality conditions for convex optimization

strong duality for generalized monotropic programming in infinite dimensions abstract we establish duality results for the generalized monotropic programming problem in separated locally convex spaces we formulate the generalized monotropic programming gmp as the minimization of a possibly infinite sum of separable proper convex functions restricted to a closed and convex cone we obtain strong duality under a constraint qualification based on the closedness of the sum of the epigraphs of the conjugates of the convex functions when the objective function is the sum of finitely many proper closed convex functions we consider two types of constraint qualifications both of which extend those introduced in the literature the first constraint qualification ensures strong duality and is equivalent to the one introduced by bo and wanka the second constraint qualification is an extension of bertsekas constraint qualification and we use it to prove zero duality gap