cases of equality and strict inequality in the extended hardy littlewood inequalities extended hardy littlewood inequalities are where ui 1 i m are non negative functions and denote their schwarz symmetrization in this paper we determine appropriate conditions under which equality in occurs if and only if ui 1 i m are schwarz symmetric

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

tangential extremal principles for finite and infinite systems of sets ii applications to semi infinite and multiobjective optimization this paper contains selected applications of the new tangential extremal principles and related results developed in mordukhovich and phan math program 2011 to calculus rules for infinite intersections of sets and optimality conditions for problems of semi infinite programming and multiobjective optimization with countable constraints

on the partial realization problem abstract in this paper we take a unified approach to the partial realization problem in which we seek to incorporate ideas from numerical linear algebra most of which were originally developed in other contexts we approach the partial realization problem from several different angles and explore the connections to such topics as factoriza tion of hankel matrices block tridiagonalization generalizations of the lanczos process for biorthogonalization the euclidean algorithm and the principal part con tinued fractions of arne magnus the pade table and the berlekamp masseyalgo rithm in this way we are able to clarify some previous results by rissanen kalman and others and place them in a broader context this leads to several results and concepts which we think are new our analysis is restricted to the scalar case but some definitions and formulations have been rigged to facilitate an extension to the matrix case

uniform duality in semi infinite convex optimization we study infinite sets of convex functional constraints with possibly a set constraint under general background hypotheses which require closed functions and a closed set but otherwise do not require a slater point for example when the set constraint is not present only the consistency of the conditions is needed

new regularity conditions for strong and total fenchel lagrange duality in infinite dimensional spaces abstract we give new regularity conditions for convex optimization problems in separated locally convex spaces we completely characterize the stable strong and strong fenchel lagrange duality then we give similar statements for the case when a solution of the primal problem is assumed as known obtaining complete characterizations for the so called total and stable totalfenchel lagrange duality respectively for particular settings the conditions that we consider turn into some constraint qualifications already used by different authors like farkas minkowski cq locally farkas minkowski cq and basic cq and we rediscover and improve some recent results from the literature

a cell decomposition of the space of real hankel matrices of rank n and some applications abstract we construct a cellular decomposition of the space of all infinite real hankel matrices of rank n it induces cell decompositions on each manifold h m of hankel matrices of fixed rank m n the relationship between the cellular decomposition of h n and the decomposition of rat n into continued fraction cells is determined finally the cell decomposition is applied to derive some topological properties of the manifolds h n and rat n

connectivity properties of spaces of partial realizations abstract this paper addresses the problem of determining the number of connected components of the space of real scalar m n hankel matrices of rank n for n

gidas ni nirenberg results for finite difference equations estimates of approximate symmetry are positive solutions of finite difference boundary value problems hu f u in h u 0 on h as symmetric as the domain to answer this question we first show by examples that almost arbitrary non symmetric solutions can be constructed this is in striking difference to the continuous case where by the famous gidas ni nirenberg theorem b gidas wei ming ni l nirenberg symmetry and related problems via the maximum principle comm math phys 68 1979 209 243 positive solutions inherit the symmetry of the underlying domain then we prove approximate symmetry theorems for solutions on equidistantly meshed n dimensional cubes explicit estimates depending on the data are given which show that the solutions become more symmetric as the discretization gets finer the quality of the estimates depends on whether or not f 0 0 the one dimensional case stands out in two ways the proofs are elementary and the estimates for the defect of symmetry are o h compared to o 1 log h in the higher dimensional case

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 strong and total lagrange duality for convex optimization problems abstract we give some necessary and sufficient conditions which completely characterize the strong and total lagrange duality respectively for convex optimization problems in separated locally convex spaces we also prove similar statements for the problems obtained by perturbing the objective functions of the primal problems by arbitrary linear functionals in the particular case when we deal with convex optimization problems having infinitely many convex inequalities as constraints the conditions we work with turn into the so called farkas minkowski and locally farkas minkowski conditions for systems of convex inequalities recently used in the literature moreover we show that our new results extend some existing ones in the literature

minimal partial realization by descriptor systems abstract in this paper a generalization of kalman s partial realization theory is developed using partial realizations defined by descriptor systems the use of singular system realizations in contrast to regular linear systems enables us to circumvent certain technical difficulties inherent in the standard approach to partial realizations an existence and uniqueness result for minimal partial descriptor realizations is proven and a simple rank formula for the mcmillan degree is derived

the bruhat decomposition of finite hankel matrices abstract the bruhat decomposition of the space of n n hankel matrices is derived from the bruhat decomposition of the general linear group some new results concerning the hankel minors map h det h1 det hn are obtained det hi denotes the i th principal minor of h the adherence order of the bruhat decomposition is determined in the complex case

an approach to symmetrization via polarization we prove that the steiner symmetrization of a function can be approximated in l p r n by a sequence of very simple rearrangements which are called polarizations this result is exploited to develop elementary proofs of many inequalities including the isoperimetric inequality in euclidean space in this way we also obtain new symmetry results for solutions of some varia tional problems furthermore we compare the solutions of two boundary value problems one of them having a polarized geometry and we show some point wise inequalities between the solutions this leads to new proofs of well known functional inequalities which compare the solutions of two elliptic or parabolic problems one of them having a steiner symmetrized geometry the method also allows us to investigate the case of equality in the inequalities roughly speaking we prove that the equality sign is valid only if the original problem has the symmetrized geometry

numerical and analytical estimates of existence regions for semi linear elliptic equations with critical sobolev exponents in cuboid and cylindrical domains we use a variety of careful numerical and semi analytical methods to investigate two outstanding conjectures on the solutions of the parametrised semi linear elliptic equation u u u5 0 u 0 where u is defined to be zero on the boundary of a three dimensional domain this equation is important in analysis and in studies of combustion and polytropic gases it is known that there is a value 0 0 such that no solutions exist for 0 mcleod and schoen and bandle and flucher have given different estimates for 0 both of which have been conjectured to be exact we perform a semi analytic study of solutions on cylindrical domains and construct numerical approximations on cuboid domains using the finite element method combined with a careful post processing step to reduce the otherwise significant errors we compute these estimates for cylindrical and cuboid domains and show that on these domains the estimates do not agree we conclude that the conjecture of bandle and flucher is false our numerical computations on cuboid domains are consistent with mcleod s conjecture being true

on a theorem of hermite and hurwitz abstract the hermite hurwitz theorem computes the degree over r of a real rational function in terms of the signature of an associated quadratic form known today as the hankel matrix of this formula which hermite was led to by his work on the problem of representing integers as sums of squares gave rise to striking applications in the theory of equations and in the stability theory of ordinary differential equations in this paper this theorem and various generalizations to the matrix valued case are discussed and described in terms of signature formulae these include its relation to stability theory and the matrix hermite hurwitz theorem of bitmead anderson as applied to questions of circuit synthesis this also includes a global form of hormander s signature formula for the maslov index of a rational loop in a lagrangian grassmannian due to byrnes and duncan and applications to the topology of spaces of rational matrix valued functions following the work of brockett byrnes and duncan this includes in particular a topological proof of the matrix hermite hurwitz theorem

lagrange type duality in dc programming abstract we show a lagrange type duality theorem for a dc programming problem which is a generalization of previous results by j e martinez legaz m volle 5 and y fujiwara d kuroiwa 1 when all constraint functions are real valued to the purpose we decompose the dc programming problem into certain infinite convex programming problems

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

the finite element approximation of semilinear elliptic partial differential equations with critical exponents in the cube we consider the finite element solution of the parameterized semilinear elliptic equation delta u lambda u u 5 0 u 0 where u is defined in the unit cube and is 0 on the boundary of the cube this equation is important in analysis and it is known that there is a value lambda 0 0 such that no solutions exist for lambda lambda 0 and for the form of the spurious numerical solutions which are known to exist when lambda lambda 0 these estimates are then used to post process the numerical results to obtain a sharp estimate for lambda 0 which agrees with the conjectured value

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

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

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

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

rearrangement inequalities in the discrete setting and some applications abstract we prove extended hardy littlewood riesz and polya szeg inequalities in the discrete case we also establish cases of equality in these inqualities and give some applications of our result

an alternative formulation for a new closed cone constraint qualification abstract we give an alternative formulation for the so called closed cone constraint qualification cccq related to a convex optimization problem in banach spaces recently introduced in the literature this new formulation allows to prove in a simple way that cccq is weaker than some generalized interior point constraint qualifications given in the past by means of some insights from the theory of conjugate duality we also show that strong duality still holds under some weaker hypotheses than the ones considered so far in the literature

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

necessary and sufficient conditions for some constraint qualifications in quasiconvex programming abstract in this paper we investigate relations between constraint qualifications in quasiconvex programming at first we show a necessary and sufficient condition for the closed cone constraint qualification for quasiconvex programming q cccq and investigate some sufficient conditions for the q cccq also we consider a relation between the q cccq and the basic constraint qualification for quasiconvex programming q bcq and we compare the q bcq with some constraint qualifications

lagrange duality in canonical dc programming abstract we study lagrange duality theorems for canonical dc programming problems we show two types lagrange duality results by using a decomposition method to infinite convex programming problems and by using a previous result by lemaire 1998 6 also we observe these constraint qualifications for the duality theorems