monotonicity of lowner operators and its applications to symmetric cone complementarity problems we prove necessary and sufficient conditions for locally lipschitz lowner operators to be monotone strictly monotone and strongly monotone utilizing our characterization of the strict monotonicity of lowner operators we generalize mangasarian class of nonlinear complementarity problem ncp functions to the setting of symmetric cone complementarity problem this affirmatively answers a question of tseng math program 83 159 185 1998

an extension of sums of squares relaxations to polynomial optimization problems over symmetric cones this paper is based on a recent work by kojima which extended sums of squares relaxations of polynomial optimization problems to polynomial semidefinite programs let varepsilon and varepsilon be a finite dimensional real vector space and a symmetric cone embedded in varepsilon examples of varepsilon and varepsilon include a pair of the n dimensional euclidean space and its nonnegative orthant a pair of the n dimensional euclidean space and n dimensional second order cones and a pair of the space of m m real symmetric or complex hermitian matrices and the cone of their positive semidefinite matrices sums of squares relaxations are further extended to a polynomial optimization problem over varepsilon i e a minimization of a real valued polynomial a x in the n dimensional real variable vector x over a compact feasible region bf x b bf x in varepsilon where b x denotes an varepsilon valued polynomial in x it is shown under a certain moderate assumption on the varepsilon valued polynomial b x that optimal values of a sequence of sums of squares relaxations of the problem which are converted into a sequence of semidefinite programs when they are numerically solved converge to the optimal value of the problem

convexity and differentiability properties of spectral functions and spectral mappings on euclidean jordan algebras abstract we study in this paper several properties of the eigenvalues function of a euclidean jordan algebra extending several known results in the framework of symmetric matrices in particular we give a concise form for the directional differential of a single eigenvalue we especially focus on spectral functions f on euclidean jordan algebras which are the composition of a symmetric real valued function f with the eigenvalues function we explore several properties of f that are transferred to f in particular convexity strong convexity and differentiability spectral mappings are also considered a special case of which is the gradient mapping of a spectral function answering a problem proposed by h sendov we give a formula for the jacobian of these functions

some p properties for linear transformations on euclidean jordan algebras a real square matrix is said to be a p matrix if all its principal minors are positive it is well known that this property is equivalent to the nonsign reversal property based on the componentwise product of vectors the order p property based on the minimum and maximum of vectors uniqueness property in the standard linear complementarity problem lipschitzian homeomorphism property of the normal map corresponding to the nonnegative orthant in this article we extend these notions to a linear transformation defined on a euclidean jordan algebra we study some interconnections between these extended concepts and specialize them to the space s n of all n n real symmetric matrices with the semidefinite cone s n and to the space r n with the lorentz cone

estimation and testing for lattice conditional independence models on euclidean jordan algebras in this paper we generalize the major results of andersson and perlman on lci models to the setting of symmetric cones and give an explicit closed form formula for the estimate of the covariance matrix in the generalized lci models that we define to this end we replace the cone h i r sitting inside the jordan algebra of symmetric real i x i matrices by the symmetric cone of an euclidean jordan algebra v we introduce the andersson perlman cone k which generalizes f k h i r we prove several characterizations and properties of k which allows us to recover though with different proofs the main results of andersson and perlman regarding p k the new lattice conditional independence models are defined assuming that the euclidean jordan algebra v has a symmetric representation using standard results from the theory of jordan algebras we can reduce the general model to the case where v is the jordan algebra of hermitian matrices over the real complex or quaternionic numbers and is the corresponding cone of positive definite matrices our main statistical result is a closed form formula for the estimate of the covariance matrix in the generalized lci model we also give the likelihood ratio test for testing a given model versus another one nested within the first

the resolvent average on symmetric cones abstract recently bauschke et al introduced a new notion of proximal average in the context of convex analysis and studied this subject systemically in bauschke et al 2004 2007 2008 2009 3 7 from various viewpoints in addition this new concept was applied to positive semidefinite matrices under the name of resolvent average and basic properties of the resolvent average are successfully established by themselves from a totally different view and techniques of convex analysis rather than the classical matrix analysis 8 bauschke et al 2010 inspired by their works and the well known fact that the convex cone of positive definite matrices is a typical example of a symmetric cone self dual homogeneous cone we study the resolvent average on symmetric cones and derive corresponding results in a different manner based on a purely jordan algebraic technique

some majorization inequalities induced by schur products in euclidean jordan algebras abstract in a euclidean jordan algebra v of rank n an element x is said to be majorized by an element y symbolically x y if the corresponding eigenvalue vector x is majorized by y in r n in this article we describe pointwise majorization inequalities of the form t x s x where t and s are linear transformations induced by schur products specializing we recover analogs of majorization inequalities of schur hadamard and oppenheimer stated in the setting of euclidean jordan algebras as well as majorization inequalities connecting quadratic and lyapunov transformations on v we also show how schur products induced by certain scalar means such as arithmetic geometric harmonic and logarithmic means naturally lead to majorization inequalities

lowner s operator and spectral functions in euclidean jordan algebras we study analyticity differentiability and semismoothness of lowner s operator and spectral functions under the framework of euclidean jordan algebras in particular we show that many optimization related classical results in the symmetric matrix space can be generalized within this framework for example the metric projection operator over any symmetric cone defined in a euclidean jordan algebra is shown to be strongly semismooth the research also raises several open questions whose answers would be of strong interest for optimization research

derivatives of spectral functions a spectral function of a hermitian matrix x is a function which depends only on the eigenvalues of x 1x 2x nx and hence may be written f 1x 2x nx for some symmetric function f such functions appear in a wide variety of matrix optimization problems we give a simple proof that this spectral function is differentiable at x if and only if the function f is differentiable at the vector x and we give a concise formula for the derivative we then apply this formula to deduce an analogous expression for the clarke generalized gradient of the spectral function a similar result holds for real symmetric matrices

exact solution of smoluchowski s equation for reorientational motion in maier saupe potential abstract the analytic treatment of the non inertial rotational diffusion equation i e of smoluchowski s one se in a symmetric genuinely double well maier saupe uniaxial potential of mean torque is considered such potential may find applications to reorientations of the fragments of structure in polymers and proteins we obtain the exact solution of se via the confluent heun s function the solution is uniformly valid for any barrier height we apply the obtained solution to the calculation of the mean first passage time and the longitudinal correlation time and obtain their precise dependence on the barrier height in the intermediate to high barrier low temperature region the results of our approach are in full agreement with those of the approach developed by coffey kalmykov dejardin and their coauthors in the low barrier high temperature region our results noticeably distinguish from the predictions of the literature formula and give appreciably greater values for the transition rates from the potential well the reason is that the above mentioned formula is obtained in the stationary limit we conclude that for very small barrier heights the transient dynamics plays a crucial role and has to be taken into account explicitly when this requirement is satisfied as e g at the calculation of the longitudinal correlation time we obtain absolute identity of our results with the literature formula in the whole range of barrier heights the drawbacks of our approach are its applicability only to the symmetric potential and its inability to yield an analytical expression for the smallest non vanishing eigenvalue

extension of primal dual interior point algorithms to symmetric cones in this paper we show that the so called commutative class of primal dual interior point algorithms which were designed by monteiro and zhang for semidefinite programming extends word for word to optimization problems over all symmetric cones the machinery of euclidean jordan algebras is used to carry out this extension unlike some non commutative algorithms such as the xs sx method this class of extensions does not use concepts outside of the euclidean jordan algebras in particular no assumption is made about representability of the underlying jordan algebra as a special case we prove polynomial iteration complexities for variants of the short semi long and long step path following algorithms using the nesterov todd xs or sx directions

commutation principle for variational problems on euclidean jordan algebras this paper establishes a commutation result for variational problems involving spectral sets and spectral functions the discussion takes places in the context of a general euclidean jordan algebra

commutation principles in euclidean jordan algebras and normal decomposition systems the commutation principle of ramirez seeger and sossa siam j optim 23 2013 pp 687 694 proved in the setting of euclidean jordan algebras says that when the sum of a frechet differentiable function theta and a spectral function f is minimized maximized over a spectral set omega any local minimizer respectively maximizer a operator commutes with the frechet derivative theta a in this paper we extend this result to sets and functions which are just invariant under algebra automorphisms we also consider a similar principle in the setting of normal decomposition systems

convex analysis on the hermitian matrices there is growing interest in optimization problems with real symmetric matrices as variables generally the matrix functions involved are spectral they depend only on the eigenvalues of the matrix it is known that convex spectral functions can be characterized exactly as symmetric convex functions of the eigenvalues a new approach to this characterization is given via a simple fenchel conjugacy formula we then apply this formula to derive expressions for subdifferentials and to study duality relationships for convex optimization problems with positive semidefinite matrices as variables analogous results hold for hermitian matrices

non euclidean restricted memory level method for large scale convex optimization we propose a new subgradient type method for minimizing extremely large scale nonsmooth convex functions over simple domains the characteristic features of the method are a the possibility to adjust the scheme to the geometry of the feasible set thus allowing to get nearly dimension independent and nearly optimal in the large scale case rate of convergence results for minimization of a convex lipschitz continuous function over a euclidean ball a standard simplex and a spectahedron the set of positive semidefinite symmetric matrices of given size with unit trace b flexible handling of accumulated information allowing for tradeoff between the level of utilizing this information and iteration s complexity we present extensions of the scheme for the cases of minimizing non lipschitzian convex objectives finding saddle points of convex concave functions and solving variational inequalities with monotone operators finally we report on encouraging numerical results of experiments with test problems of dimensions up to 66 000

a new variational result for quasi newton formulae the recent measure function of byrd and nocedal siam j numer anal 26 1989 pp 727 739 is considered and simple proofs of some of its properties are given it is then shown that the bfgs and dfp formulae satisfy a least change property with respect to this new measure

a jordan algebraic approach to potential reduction algorithms we consider the linear monotone complementarity problem for domains obtained as the intersection of an affine subspace and the cartesian product of symmetric cones a primal dual potential reduction algorithm is described and its complexity estimates are established with the help of the jordan algebraic technique

convex functions with unbounded level sets and applications to duality theory a class of convex functions with unbounded level sets but good behavior at infinity analyse non lineaire gauthier villars paris 1989 pp 101 122 is investigated characterizations and properties are given the results are then applied to studying sequential approximation schemes for optimization problems and to duality theory when the involved functions have unbounded level sets in particular the convergence properties of stationary sequences for the dual of a convex program are studied and methods for associating with it a primal sequence converging to a solution of the primal problem are demonstrated

convex analysis of spectrally defined matrix functions the purpose of this work is to carry out a systematic study of a special class of convex functions defined over the space s n of symmetric matrices of order n ti n the functions under consideration phi s n ra r cup infty are spectrally defined in the sense that the value phi a depends only on the spectrum lam 1 a ldots lam n a of the matrix a in s n fenchel legendre conjugation first and second order subdifferentiability asymptotic behavior and other concepts of convex analysis are the main ingredients of our exposition

a one parametric class of merit functions for the symmetric cone complementarity problem abstract in this paper we extend the one parametric class of merit functions proposed by kanzow and kleinmichel c kanzow h kleinmichel a new class of semismooth newton type methods for nonlinear complementarity problems comput optim appl 11 1998 227 251 for the nonnegative orthant complementarity problem to the general symmetric cone complementarity problem sccp we show that the class of merit functions is continuously differentiable everywhere and has a globally lipschitz continuous gradient mapping from this we particularly obtain the smoothness of the fischer burmeister merit function associated with symmetric cones and the lipschitz continuity of its gradient in addition we also consider a regularized formulation for the class of merit functions which is actually an extension of one of the ncp function classes studied by c kanzow y yamashita m fukushima new ncp functions and their properties j optim theory appl 97 1997 115 135 to the sccp by exploiting the cartesian p properties for a nonlinear transformation we show that the class of regularized merit functions provides a global error bound for the solution of the sccp and moreover has bounded level sets under a rather weak condition which can be satisfied by the monotone sccp with a strictly feasible point or the sccp with the joint cartesian r 02 property all of these results generalize some recent important works in j s chen p tseng an unconstrained smooth minimization reformulation of the second order cone complementarity problem math program 104 2005 293 327 c k sim j sun d ralph a note on the lipschitz continuity of the gradient of the squared norm of the matrix valued fischer burmeister function math program 107 2006 547 553 p tseng merit function for semidefinite complementarity problems math program 83 1998 159 185 under a unified framework

spectral sets and functions on euclidean jordan algebras abstract spectral sets functions in euclidean jordan algebras are generalizations of permutation invariant sets respectively functions in r n in this article we study properties of such sets and functions and show how they are related to algebra automorphisms and majorization we show that spectral sets functions are indeed invariant under automorphisms but the converse may not hold unless the algebra is r n or simple we study schur convex spectral functions and provide some applications we also discuss the transfer principle and a related metaformula

smoothing functions for second order cone complementarity problems smoothing functions have been much studied in the solution of optimization and complementarity problems with nonnegativity constraints in this paper we extend smoothing functions to problems in which the nonnegative orthant is replaced by the direct product of second order cones these smoothing functions include the chen mangasarian class and the smoothed fischer burmeister function we study the lipschitzian and differential properties of these functions and in particular we derive computable formulas for these functions and their jacobians these properties and formulas can then be used to develop and analyze noninterior continuation methods for solving the corresponding optimization and complementarity problems in particular we establish the existence and uniqueness of the newton direction when the underlying mapping is monotone

first and second order epi differentiability in eigenvalue optimization abstract the subject of this work is the first and second order sensitivity analysis of some spectral functions which are essential in eigenvalue optimization by the way of epi differentiability we show that the sum of the m largest eigenvalues of a real symmetric matrix is twice epi differentiable and we derive an explicit expression of its second order epi derivative we also prove that the mth largest eigenvalue function is twice epi differentiable if and only if it ranks first in a group of equal eigenvalues finally we derive chain rules and then we obtain optimality conditions for an important class of eigenvalue optimization problems

sensitivity analysis of all eigenvalues of a symmetric matrix given a x a ij x a n by n symmetric matrix depending smoothly on a parameter x we study the first order sensitivity of all the eigenvalues lambda m x of a x 1 le m le n under some smoothness assumption like the a ij be c 1 we prove that the directional derivatives d mapsto lambda prime m x d lim t to o lambda m x td lambda m x t do exist and give an explicit expression of them in terms of the data of the parametrized matrix the key idea to circumvent the difficulties inherent to the study of each lambda m taken separately is to consider the functions f m x 1 le m le n defined as the sums of the m largest eigenvalues of a x based on ky fan s variational formulation of f m and some chain rule from nonsmooth analysis we derive an explicit formula for the generalized gradient of f m and a computationally useful formula for the directional derivative of f m using these formulas and the relation lambda m f m f m 1 we then derive the directional derivative of lambda m some properties of this directional derivative as well as an illustrative example are presented

a holder type inequality and an interpolation theorem in euclidean jordan algebras abstract in a euclidean jordan algebra v of rank n which carries the trace inner product to each element x we associate the eigenvalue vector x whose components are the eigenvalues of x written in the decreasing order for any p 1 we define the spectral p norm of x to be the p norm of x in r n in this paper we show that x y 1 x p y q where x y denotes the jordan product of two elements x and y in v and q is the conjugate of p for a linear transformation on v we state and prove an interpolation theorem relative to these spectral norms in addition we compute estimate the norms of lyapunov transformations quadratic representations and positive transformations on v

some inertia theorems in euclidean jordan algebras abstract this paper deals with some inertia theorems in euclidean jordan algebras first based on the continuity of eigenvalues we give an alternate proof of kaneyuki s generalization of sylvester s law of inertia in simple euclidean jordan algebras as a consequence we show that the cone spectrum of any quadratic representation with respect to a symmetric cone is finite second we present ostrowski schneider type inertia results in euclidean jordan algebras in particular we relate the inertias of objects a and x in a euclidean jordan algebra when l a x 0 or s a x 0 where l a and s a denote lyapunov and stein transformations respectively

approximation of norms on banach spaces abstract relatively recently it was proved that if is an arbitrary set then any equivalent norm on c 0 can be approximated uniformly on bounded sets by polyhedral norms and c smooth norms with arbitrary precision we extend this result to more classes of spaces having uncountable symmetric bases such as preduals of the discrete lorentz spaces d w 1 and certain symmetric nakano spaces and orlicz spaces we also show that given an arbitrary ordinal number there exists a scattered compact space k having cantor bendixson height at least such that every equivalent norm on c k can be approximated as above

weak majorization doubly substochastic maps and some related inequalities in euclidean jordan algebras abstract in this paper we extend the notion of weak majorization and doubly substochastic maps and the hardy littlewood polya theorem on majorization to euclidean jordan algebras we also provide a characterization of doubly substochastic maps on euclidean jordan algebras in terms of jordan algebra automorphisms and quadratic representations in conjunction with that various weak majorization inequalities of aujla silva are generalized to euclidean jordan algebras

measures for symmetric rank one updates measures of deviation of a symmetric positive definite matrix from the identity are derived they give rise to symmetric rank one sr1 type updates the measures are motivated by considering the volume of the symmetric difference of the two ellipsoids which arise from the current and updated quadratic models in quasi newton methods the measure defined by the problem maximize the determinant subject to a bound of 1 on the largest eigenvalue yields the sr1 update the measure a 1 a det a 1 n yields the optimally conditioned sized symmetric rank one updates the volume considerations also suggest a correction for the initial stepsize for these sized updates it is then shown that the optimal updates as well as the oren luenberger self scaling updates are all optimal updates for the measure the l2 condition number moreover all four sized updates result in the same largest and smallest scaled eigenvalue and corresponding eigenvector in fact the inversesized bfgs is the mean of the

the fixed point property under renorming in some classes of banach spaces abstract assume that y is a banach space such that r y 2 where r is garcia falset s coefficient and x is a banach space which can be continuously embedded in y we prove that x can be renormed to satisfy the weak fixed point property w fpp on the other hand assume that k is a scattered compact topological space such that k 0 and c k is the space of all real continuous functions defined on k with the supremum norm we will show that c k can be renormed to satisfy r c k 2 thus both results together imply that any banach space which can be continuously embedded in c k k as above can be renormed to satisfy the w fpp these results extend a previous one about the w fpp under renorming for banach spaces which can be continuously embedded in c 0 furthermore we consider a metric in the space p of all norms in c k which are equivalent to the supremum norm and we show that for almost all norms in p in the sense of porosity c k satisfies the w fpp

distortion and stability of the fixed point property for non expansive mappings abstract let x be a banach space we say that x satisfies the fixed point property weak fixed point property if every non expansive mapping defined from a convex closed bounded convex weakly compact subset of x into itself has a fixed point we say that x satisfies the stable fixed point property stable weak fixed point property if the same is true for every equivalent norm which is close enough to the original one denote by p x the set formed by all equivalent norms with the topology of the uniform convergence on the unit ball of x we prove that the subset of p x formed by the norms failing the fixed point property is dense in p x when x is a non distortable space which fails the fixed point property in particular no renorming of l 1 can satisfy the stable fixed point property furthermore we show some examples of distortable spaces failing the weak fixed point property which can be renormed to satisfy the stable weak fixed point property as a consequence we prove that every separable banach space can be renormed to satisfy the stable weak fixed point property

spectral cones in euclidean jordan algebras abstract a spectral cone in a euclidean jordan algebra v of rank n is of the form k 1 q where q is a permutation invariant convex cone in r n and v r n is the eigenvalue map which takes x to x the vector of eigenvalues of x with entries written in the decreasing order in this paper we describe some properties of spectral cones we show for example that spectral cones are invariant under automorphisms of v that the dual of a spectral cone is a spectral cone when v is simple or carries the canonical inner product and characterize the pointedness solidness of a spectral cone we also show that for any spectral cone k in v dim k 0 1 m 1 m where dim k denotes the dimension of k and m is the dimension of v

calculus rules on a new set valued second order derivative for convex functions etant donnee une fonction convexe f qui a ete construite a partir d autres fonctions f i dont les differentielles secondes sont aisees a determiner on cherche a exprimer les differentielles secondes de f en fonction des 2 f i

convex graph invariants the structural properties of graphs are usually characterized in terms of invariants which are functions of graphs that do not depend on the labeling of the nodes in this paper we study convex graph invariants which are graph invariants that are convex functions of the adjacency matrix of a graph some examples include functions of a graph such as the maximum degree the maxcut value and its semidefinite relaxation and spectral invariants such as the sum of the k largest eigenvalues such functions can be used to construct convex sets that impose various structural constraints on graphs and thus provide a unified framework for solving a number of interesting graph problems via convex optimization we give a representation of all convex graph invariants in terms of certain elementary invariants and we describe methods to compute or approximate convex graph invariants tractably we discuss the interesting subclass of spectral invariants and also compare convex and nonconvex invariants finally we

a riesz thorin type interpolation theorem in euclidean jordan algebras abstract in a euclidean jordan algebra v of rank n which carries the trace inner product to each element a we associate the eigenvalue vector a in r n whose components are the eigenvalues of a written in the decreasing order for any p 1 we define the spectral p norm of a to be the p norm of a in r n in a recent paper based on the k method of real interpolation theory and a majorization technique we described an interpolation theorem for a linear transformation on v relative to the same spectral p norm in this paper using complex function theory methods we describe a riesz thorin type interpolation theorem relative to two spectral p norms we illustrate the result by estimating the norms of certain special linear transformations such as lyapunov transformations quadratic representations and positive transformations

on the connectedness of spectral sets and irreducibility of spectral cones in euclidean jordan algebras abstract let v be a euclidean jordan algebra of rank n a set e in v is said to be a spectral set if there exists a permutation invariant set q in r n such that e 1 q where v r n is the eigenvalue map that takes x v to x the vector of eigenvalues of x written in the decreasing order if the above q is also a convex cone we say that e is a spectral cone this paper deals with connectedness and arcwise connectedness properties of spectral sets by relying on the result that in a simple euclidean jordan algebra every eigenvalue orbit x y y x is arcwise connected we show that if a permutation invariant set q is connected arcwise connected then 1 q is connected respectively arcwise connected a related result is that in a simple euclidean jordan algebra every pointed spectral cone is irreducible

comparison of methods for the calculation of superparamagnetic relaxation times a general expression for the correlation time of the decay of the magnetization of an assembly of singledomain noninteracting ferromagnetic particles is given in terms of the inverse of the fokker planck operator the results of moro and nordio g moro and p l nordio mol phys 56 255 1985 given in the context of dielectric relaxation are recovered when the fokker planck operator is axially symmetric their result is a particular example of szabo s calculation of the correlation times of the autocorrelation functions of the legendre polynomials by means of a generalization of the theory of first passage times a szabo j chem phys 72 4620 1980 likewise the results of garanin ischenko and panina nd a garanin v v ischenko and l v panina teor mat fiz 82 242 1990 theor math phys 82 169 1990 o for the integral relaxation time i e the area under the curve of the normalized decay of the magnetization are regained in the axially symmetric case where it is possible to integrate the fokker planck equation directly it is shown by manipulation of kummer s functions that the exact integral expression for the correlation time for simple uniaxial anisotropy derived by coffey et al w t coffey d s f crothers yu p kalmykov e s massawe and j t waldron phys rev e 49 1869 1994 by representing the fokker planck equation as a differential recurrence relation is identical to the integral relaxation time originally derived by garanin et al by direct integration of the fokker planck equation s1063 651x 96 08811 3

renormings failing to have asymptotically isometric copies of l1 or c0 abstract it is proved that every equivalent norm in l 1 is the starting point of a ray of norms which fail to have an asymptotically isometric copy of l 1 in the case where x is separable it is proved that the set of all equivalent norms in x which fail to have an asymptotically isometric copy of l 1 is residual similar results are obtained for the case of c 0 in particular we deduce that every separable banach space has an equivalent norm for which it fails to contain an asymptotically isometric copy of l 1 and c 0

linear systems in jordan algebras and primal dual interior point algorithms we discuss a possibility of the extension of a primal dual interior point algorithm suggested recently by alizadeh et al 1994 we consider optimization problems defined on the intersection of a symmetric cone and an affine subspace the question of solvability of a linear system arising in the implementation of the primal dual algorithm is analyzed a nondegeneracy theory for the considered class of problems is developed the jordan algebra technique suggested by faybusovich 1995 plays major role in the present paper

probability distribution function for reorientations in maier saupe potential abstract exact analytic solution for the probability distribution function of the non inertial rotational diffusion equation i e of the smoluchowski one in a symmetric maier saupe uniaxial potential of mean torque is obtained via the confluent heun s function both the ordinary maier saupe potential and the double well one with variable barrier width are considered thus the present article substantially extends the scope of the potentials amenable to the treatment by reducing smoluchowski equation to the confluent heun s one the solution is uniformly valid for any barrier height we use it for the calculation of the mean first passage time also the higher eigenvalues for the relaxation decay modes in the case of ordinary maier saupe potential are calculated the results obtained are in full agreement with those of the approach developed by coffey kalmykov dejardin and their coauthors in the whole range of barrier heights