free energy and correlations of the number theoretical spin chain we analyze the number theoretical spin chain with partition function z 1 using the polymer model technique the finite grand canonical chains give bounds for the limit free energy and internal energy the correlation functions for inverse temperature 1 are products of two point functions a combinatorial result for general interval graphs is derived

benchmarking optimization software with performance profiles we propose performance profiles distribution functions for a performance metric as a tool for benchmarking and comparing optimization software we show that performance profiles combine the best features of other tools for performance evaluation

fine tuning nesterov s steepest descent algorithm for differentiable convex programming we modify the first order algorithm for convex programming described by nesterov in his book in introductory lectures on convex optimization a basic course kluwer boston 2004 in his algorithms nesterov makes explicit use of a lipschitz constant l for the function gradient which is either assumed known nesterov in introductory lectures on convex optimization a basic course kluwer boston 2004 or is estimated by an adaptive procedure nesterov 2007 we eliminate the use of l at the cost of an extra imprecise line search and obtain an algorithm which keeps the optimal complexity properties and also inherit the global convergence properties of the steepest descent method for general continuously differentiable optimization besides this we develop an adaptive procedure for estimating a strong convexity constant for the function numerical tests for a limited set of toy problems show an improvement in performance when compared with the original nesterov s algorithms

phases of the number theoretic spin chain we present numerical and analytical evidence for a first order phase transition of the ferromagnetic spin chain with partition functionz 1 at the inverse temperature cr 2

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

bundle level type methods uniformly optimal for smooth and nonsmooth convex optimization the main goal of this paper is to develop uniformly optimal first order methods for convex programming cp by uniform optimality we mean that the first order methods themselves do not require the input of any problem parameters but can still achieve the best possible iteration complexity bounds by incorporating a multi step acceleration scheme into the well known bundle level method we develop an accelerated bundle level method and show that it can achieve the optimal complexity for solving a general class of black box cp problems without requiring the input of any smoothness information such as whether the problem is smooth nonsmooth or weakly smooth as well as the specific values of lipschitz constant and smoothness level we then develop a more practical restricted memory version of this method namely the accelerated prox level apl method we investigate the generalization of the apl method for solving certain composite cp problems and an important class of saddle point problems recently studied by nesterov math program 103 127 152 2005 we present promising numerical results for these new bundle level methods applied to solve certain classes of semidefinite programming and stochastic programming problems

on a ferromagnetic spin chain the quotient s 1 s of riemann zeta functions is shown to be the partition function of a ferromagnetic spin chain for inverse temperatures

the identification problem in systems nonlinear in the variables this paper examines the identifiability of the coefficients of a single equation in a simultaneous equation model which is nonlinear only in the variables the concept of identifiability in this model is motivated and developed using the closely related concept of observational equivalence this framework is then utilized to develop necessary and sufficient conditions for identifiability when the disturbances are required to be independent of the exogenous variables the approach recommended by fisher is shown to yield sufficient but not necessary conditions for identifiability for several relatively common special cases the necessary and sufficient conditions are found to simplify to the familiar rank condition for identifiability in the linear model the simultaneous equation model that is nonlinear only in the variables has enjoyed widespread application in economics such models are linear in the parameters and typically seem to be linear in the variables as well when viewing a single equation in many models the nonlinearity in the variables arises due to endogenous variables entering in different forms in different equations logged and unlogged form for example in macroeconometric models nonlinearity in the variables arises when the model includes endogenous real nominal and price variables which are nonlinearly related 2 whatever the reason for the nonlinearity in the variables it is important to determine the conditions under which the equations of such models can be identified

efficient estimation in models with independence restrictions abstract unconditional and conditional independence restrictions are used in many econometric models to identify their parameters however there are few results about efficient estimation procedures for finite dimensional parameters under these independence restrictions this paper computes the efficiency bound for finite dimensional parameters under independence restrictions and proposes an estimator that is consistent asymptotically normal and which achieves the efficiency bound the estimator is based on a growing number of zero covariance conditions that are asymptotically equivalent to the independence restriction the results are illustrated with examples including an instrumental variables regression model and partially linear regression models a small monte carlo study is performed to investigate the estimator s small sample properties and to quantify the efficiency gains that can be made by using the proposed efficient estimator

discontinuity of the magnetization in one dimensional 1 x y 2 ising and potts models results from percolation theory are used to study phase transitions in one dimensional ising and q state potts models with couplings of the asymptotic form j sub x y approx const absolute value x y sup 2 for translation invariant systems with well defined lim sub x infinity x sup 2 j sub x j sup possibly o or infinity we establish 1 there is no long range order at inverse temperatures beta with beta j sup less than or equal to1 2 if beta j sup q then by sufficiently increasing j sub 1 the spontaneous magnetization m is made positive 3 in models with 0 j sup infinity the magnetization is discontinuous at the transition point as originally predicted by thouless and obeys m beta sub c less than or equal to 1 beta sub c j sup sup 1 2 4 for ising q 2 models with j sup infinity it is noted that the correlation function decays as beta approx c beta absolute value x y sup 2 whenever beta beta sub c points 1 3 are deduced from previous percolation results by utilizing the fortuin kasteleyn representation which also yields other results of independent interest relating potts models with different values of q

a strictly contractive peaceman rachford splitting method with logarithmic quadratic proximal regularization for convex programming recently a strictly contractive peaceman rachford splitting method prsm was proposed for a separable convex minimization model whose variables are subject to some linear constraints and two additional generic constraints in general the strictly contractive prsm requires to solve two constrained minimization subproblems at each iteration in this paper we consider the case where the additional constraints on variables are positive orthants and apply the well developed logarithmic quadratic proximal lqp regularization to regularize the subproblems of the strictly contractive prsm a new algorithm in combination with the strictly contractive prsm and the lqp regularization is thus proposed the new algorithm only needs to solve two unconstrained subproblems at each iteration an inexact version allowing the unconstrained subproblems to be solved approximately subject to certain inexactness criterion is also studied we prove the global convergence and establish a worst case convergence rate measured

estimation of semiparametric models when the criterion function is not smooth we provide easy to verify sufficient conditions for the consistency and asymptotic normality of a class of semiparametric optimization estimators where the criterion function does not obey standard smoothness conditions and simultaneously depends on some nonparametric estimators that can themselves depend on the parameters to be estimated our results extend existing theories such as those of pakes and pollard 1989 andrews 1994a and newey 1994 we also show that bootstrap provides asymptotically correct confidence regions for the finite dimensional parameters we apply our results to two examples a hit rate and a partially linear median regression with some endogenous regressors

weighted minimum mean square distance from independence estimation in this paper we introduce a family of semi parametric estimators suggested by manski s minimum mean square distance from independence estimator we establish the strong consistency asymptotic normality and consistency of bootstrap estimates of the sampling distribution and the asymptotic variance of these estimators this abstract was borrowed from another version of this item

on the nonparametric identification of nonlinear simultaneous equations models comment on brown 1983 and roehrig 1988 this note revisits the identification theorems of brown 1983 and roehrig 1988 we describe an error in the proofs of the main identification theorems in these papers and provide an important counterexample to the theorems on the identification of the reduced form specifically the reduced form of a nonseparable simultaneous equations model is not identified even under the assumptions of these papers we provide conditions under which the reduced form is identified and is recoverable using the distribution of the endogenous variables conditional on the exogenous variables however these conditions place substantial limitations on the structural model we conclude the note with a conjecture that it may be possible to use classical exclusion restrictions to recover some of the key implications of the theorems in more general settings

a cyclic block coordinate descent method with generalized gradient projections the aim of this paper is to present the convergence analysis of a very general class of gradient projection methods for smooth constrained possibly nonconvex optimization the key features of these methods are the armijo linesearch along a suitable descent direction and the non euclidean metric employed to compute the gradient projection we develop a very general framework from the point of view of block coordinate descent methods which are useful when the constraints are separable in our numerical experiments we consider a large scale image restoration problem to illustrate the impact of the metric choice on the practical performances of the corresponding algorithm

a farey fraction spin chain we introduce a new number theoretic spin chain and explore its thermodynamics and connections with number theory the energy of each spin configuration is defined in a translation invariant manner in terms of the farey fractions and is also expressed using pauli matrices we prove that the free energy exists and a phase transition occurs for positive inverse temperature 2 the free energy is the same as that of related non translation invariant number theoretic spin chain using a number theoretic argument the low temperature 3 state is shown to be completely magnetized for long chains the number of states of energy e log n summed over chain length is expressed in terms of a restricted divisor problem we conjecture that its asymptotic form is n log n consistent with the phase transition at 2 and suggesting a possible connection with the riemann function the spin interaction coefficients include all even many body terms and are translation invariant computer results indicate that all the interaction coefficients except the constant term are ferromagnetic

discontinuity of the percolation density in one dimensional 1 x y 2 percolation models we consider one dimensional percolation models for which the occupation probability of a bond kx y has a slow power decay as a function of the bond s length for independent models and with suitable reformulations also for more general classes of models it is shown that i no percolation is possible if for short bondskx y p 1 this dichotomy resembles one for the magnetization in 1 x y 2 ising models which was first proposed by thouless and further supported by the renormalization group flow equations of anderson yuval and hamann the proofs of the above percolation phenomena involve rigorous renormalization type arguments of a different sort

conditions for identification in nonparametric and parametric models on presente une definition formelle de l identification dans un contexte non parametrique sur la base des travaux de koopmans et reiersol 1950

primal dual first order methods with mathcal o 1 epsilon iteration complexity for cone programming in this paper we consider the general cone programming problem and propose primal dual convex smooth and or nonsmooth minimization reformulations for it we then discuss first order methods suitable for solving these reformulations namely nesterov s optimal method nesterov in doklady an sssr 269 543 547 1983 math program 103 127 152 2005 nesterov s smooth approximation scheme nesterov in math program 103 127 152 2005 and nemirovski s prox method nemirovski in siam j opt 15 229 251 2005 and propose a variant of nesterov s optimal method which has outperformed the latter one in our computational experiments we also derive iteration complexity bounds for these first order methods applied to the proposed primal dual reformulations of the cone programming problem the performance of these methods is then compared using a set of randomly generated linear programming and semidefinite programming instances we also compare the approach based on the variant of nesterov s optimal method with the low rank method proposed by burer and monteiro math program ser b 95 329 357 2003 math program 103 427 444 2005 for solving a set of randomly generated sdp instances

bootstrapping general empirical measures it is proved that the bootstrapped central limit theorem for empirical processes indexed by a class of functions mathscr f and based on a probability measure p holds a s if and only if mathscr f in mathrm clt p and int f 2 dp infty where f sup f in mathscr f f and it holds in probability if and only if mathscr f in mathrm clt p thus for a large class of statistics no local uniformity of the clt about p is needed for the bootstrap to work consistency of the bootstrap the bootstrapped law of large numbers is also characterized these results are proved under certain weak measurability assumptions on mathscr f

system identification via sparse multiple kernel based regularization using sequential convex optimization techniques model estimation and structure detection with short data records are two issues that receive increasing interests in system identification in this paper a multiple kernel based regularization method is proposed to handle those issues multiple kernels are conic combinations of fixed kernels suitable for impulse response estimation and equip the kernel based regularization method with three features first multiple kernels can better capture complicated dynamics than single kernels second the estimation of their weights by maximizing the marginal likelihood favors sparse optimal weights which enables this method to tackle various structure detection problems e g the sparse dynamic network identification and the segmentation of linear systems third the marginal likelihood maximization problem is a difference of convex programming problem it is thus possible to find a locally optimal solution efficiently by using a majorization minimization algorithm and an interior point method where the cost of a single interior point iteration grows linearly in the number of fixed kernels monte carlo simulations show that the locally optimal solutions lead to good performance for randomly generated starting points

minimum distance from independence estimation of nonseparable instrumental variables models abstract i develop a semiparametric minimum distance from independence estimator for a nonseparable instrumental variables model an independence condition identifies the model for many types of discrete and continuous instruments the estimator is taken as the parameter value that most closely satisfies this independence condition implementing the estimator requires a quantile regression of the endogenous variables on the instrument so the procedure is two step with a finite or infinite dimensional nuisance parameter in the first step i prove consistency and establish asymptotic normality for a parametric but flexibly nonlinear outcome equation the consistency of the nonparametric bootstrap is also shown i illustrate the use of the estimator by estimating the returns to schooling using data from the 1979 national longitudinal survey

accelerated schemes for a class of variational inequalities we propose a novel stochastic method namely the stochastic accelerated mirror prox samp method for solving a class of monotone stochastic variational inequalities svi the main idea of the proposed algorithm is to incorporate a multi step acceleration scheme into the stochastic mirror prox method the developed samp method computes weak solutions with the optimal iteration complexity for svis in particular if the operator in svi consists of the stochastic gradient of a smooth function the iteration complexity of the samp method can be accelerated in terms of their dependence on the lipschitz constant of the smooth function for svis with bounded feasible sets the bound of the iteration complexity of the samp method depends on the diameter of the feasible set for unbounded svis we adopt the modified gap function introduced by monteiro and svaiter for solving monotone inclusion and show that the iteration complexity of the samp method depends on the distance from the initial point to the set of strong solutions it is worth noting that our study also significantly improves a few existing complexity results for solving deterministic variational inequality problems we demonstrate the advantages of the samp method over some existing algorithms through our preliminary numerical experiments

optimal primal dual methods for a class of saddle point problems we present a novel accelerated primal dual apd method for solving a class of deterministic and stochastic saddle point problems spps the basic idea of this algorithm is to incorporate a multistep acceleration scheme into the primal dual method without smoothing the objective function for deterministic spp the apd method achieves the same optimal rate of convergence as nesterov s smoothing technique our stochastic apd method exhibits an optimal rate of convergence for stochastic spp not only in terms of its dependence on the number of the iteration but also on a variety of problem parameters to the best of our knowledge this is the first time that such an optimal algorithm has been developed for stochastic spp in the literature furthermore for both deterministic and stochastic spp the developed apd algorithms can deal with the situation when the feasible region is unbounded as long as a saddle point exists in the unbounded case we incorporate the modified termination criterion introduced b

osga a fast subgradient algorithm with optimal complexity this paper presents an algorithm for approximately minimizing a convex function in simple not necessarily bounded convex finite dimensional domains assuming only that function values and subgradients are available no global information about the objective function is needed apart from a strong convexity parameter which can be put to zero if only convexity is known the worst case number of iterations needed to achieve a given accuracy is independent of the dimension and apart from a constant factor best possible under a variety of smoothness assumptions on the objective function

the low activity phase of some dirichlet series we show that a rigorous statistical mechanics description of some dirichlet series is possible using the abstract polymer model language of statistical mechanics and the polymer expansion theory we characterize the low activity phase by the suitable exponential decay of the truncated correlation functions

inexact proximal point methods for variational inequality problems we present a new family of proximal point methods for solving monotone variational inequalities our algorithm has a relative error tolerance criterion in solving the proximal subproblems our convergence analysis covers a wide family of regularization functions including double regularizations recently introduced by silva eckstein and humes jr siam j optim 12 2001 pp 238 261 and the bregman distance induced by h x sum i 1 n x i log x i we do not use in our analysis the assumption of paramonotonicity which is standard in proving convergence of bregman based proximal methods

identification of nonparametric simultaneous equations models with a residual index structure we present new identification results for a class of nonseparable nonparametric simultaneous equations models introduced by matzkin 2008 these models combine traditional exclusion restrictions with a requirement that each structural error enter through a residual index our identification results are constructive and encompass a range of special cases with varying demands on the exogenous variation provided by instruments and the shape of the joint density of the structural errors the most important results demonstrate identification when instruments have only limited variation even when instruments vary only over a small open ball relatively mild conditions on the joint density suffice we also show that the primary sufficient conditions for identification are verifiable and that the maintained hypotheses of the model are falsifiable

identification and estimation of time varying nonseparable panel data models without stayers abstract this paper explores the identification and estimation of nonseparable panel data models we show that the structural function is nonparametrically identified when it is strictly increasing in a scalar unobservable variable the conditional distributions of unobservable variables do not change over time and the joint support of explanatory variables satisfies some weak assumptions to identify the target parameters existing studies assume that the structural function does not change over time and that there are stayers namely individuals with the same regressor values in two time periods our approach by contrast allows the structural function to depend on the time period in an arbitrary manner and does not require the existence of stayers in estimation part of the paper we propose parametric and nonparametric estimators that implement our identification results monte carlo studies indicate that our parametric estimator performs well in finite samples finally we extend our identification results to models with discrete outcomes and show that the structural function is partially identified

applications of statistical mechanics in number theory abstract the links between statistical physics and number theory are discussed first the attempts to prove the riemann hypothesis by means of the suitable spin model and the lee yang theorem about zeros of the partition function are shortly reviewed next the analogies between random walks and prime numbers are mentioned in the last section the partition function of the system whose energies are defined by the distances between consecutive primes is calculated the arguments are given that such a prime numbers gas behaves like a set of noninteracting harmonic oscillators

simulation and the asymptotics of optimization estimators on demontre un theoreme de limite centrale general pour des estimateurs definis par minimisation de la longueur d une fonction de critere aleatoire a valeurs vectorielles aucune hypothese de regularite suffisante n est imposee sur la fonction de critere pour que les resultats puissent etre appliques a une classe d estimlateurs de simulation assez large des analyses completes de deux estimateurs de simulation l un introduit par pakes l autre par mcfadden illustrent l application des theoremes generaux

on a ferromagnetic spin chain ii thermodynamic limit the existence of the thermodynamic limit of the free energy for the ferromagnetic spin chain connected with the riemann zeta function is proven

interior gradient and proximal methods for convex and conic optimization interior gradient subgradient and proximal methods for convex constrained minimization have been much studied in particular for optimization problems over the nonnegative octant these methods are using non euclidean projections and proximal distance functions to exploit the geometry of the constraints in this paper we identify a simple mechanism that allows us to derive global convergence results of the produced iterates as well as improved global rates of convergence estimates for a wide class of such methods and with more general convex constraints our results are illustrated with many applications and examples including some new explicit and simple algorithms for conic optimization problems in particular we derive a class of interior gradient algorithms which exhibits an o k 2 global convergence rate estimate

a unified continuous optimization framework for center based clustering methods center based partitioning clustering algorithms rely on minimizing an appropriately formulated objective function and different formulations suggest different possible algorithms in this paper we start with the standard nonconvex and nonsmooth formulation of the partitioning clustering problem we demonstrate that within this elementary formulation convex analysis tools and optimization theory provide a unifying language and framework to design analyze and extend hard and soft center based clustering algorithms through a generic algorithm which retains the computational simplicity of the popular k means scheme we show that several well known and more recent center based clustering algorithms which have been derived either heuristically or and have emerged from intuitive analogies in physics statistical techniques and information theoretic perspectives can be recovered as special cases of the proposed analysis and we streamline their relationships

an optimal algorithm for constrained differentiable convex optimization we describe two algorithms for solving differentiable convex optimization problems constrained to simple sets in mathbb r n i e sets on which it is easy to project an arbitrary point the algorithms are optimal in the sense that they achieve an absolute precision of varepsilon in relation to the optimal value in o 1 sqrt varepsilon iterations using only first order information this complexity depends on an unknown lipschitz constant l for the function derivatives and on a known strong convexity constant mu geq 0 the algorithms are extensions of well known methods devised by nesterov introductory lectures on convex optimization kluwer academic boston 2004 without the need for estimates of l and including in the second algorithm line searches and an adaptive procedure for estimating a strong convexity constant the algorithms are such that all computed points are feasible and the complexity analysis follows a simple geometric approach numerical tests for

an inexact proximal method for quasiconvex minimization abstract in this paper we propose an inexact proximal point method to solve constrained minimization problems with locally lipschitz quasiconvex objective functions assuming that the function is also bounded from below lower semicontinuous and using proximal distances we show that the sequence generated for the method converges to a stationary point of the problem